Conservapedia - User contributions [en]

Talk:Center

2008-08-29T17:47:44Z

DiEb: A Question about Averaging Points

Responding to [[User:SamHB|SamHB]]'s criticism of this article:

:I'll agree this isn't the best written article. However, it has to have a definition that works for all plane figures, from circles to polygons and everything in between. Now there are easy definitions of center for some of these figures:
:* a circle's center is the unique point that is equidistant to every point on the circle.
:* a quadrilateral's center is the intersection of the two diagonals.
:* an ellipse's center is the intersection of the major and minor axes.
:But even these three definitions alone are not easy to resolve into one that works in all cases. They also make strong use of the symmetry of the figures to get a simple definition. Throw in pentagons, hearts, stars, and strange figures such as cardioids, and the definition becomes more complicated. Basically, for this article I had to approach the formal definition of [[centroid]] while avoiding the use of integrals. That is to say, I had to explain the intuition of the integration involved in finding the centroid, which is not easy without actually writing an integral. Realizing that my "all-purpose" definition was somewhat hard to digest, I tried to clarify the definition with the examples of a circle and square.

:It's ironic that I had to invoke the spirit of integration for the center article, since the [[centroid]] article should have been the more technical one. But the already-existing centroid article avoided integrals by referring to the term "center", so we've got a case of mathematical definitions by synonyms. I could have continued this trend by writing "the '''center''' of a geometric shape is the middle of the shape", but that's just avoiding the task of having to give a proper definition.

:I welcome improvements to the article and ''constructive'' criticism. -[[User:Foxtrot|Foxtrot]] 04:22, 28 August 2008 (EDT)

:My question for Foxtrot: How can points be ''on average'' equidistant? What points do you take the average of? What do you want to say? IMO, this formulation is even less clear than "the '''center''' is in the middle"... --[[User:DiEb|DiEb]] 13:47, 29 August 2008 (EDT)

Proof by induction

2008-08-29T17:28:53Z

DiEb:

A '''proof by induction''' is a technique of mathematical [[proof]]. It is one of the axioms of [[Zermelo-Fraenkel]] set theory, allowing a mathematician to have proofs that count in order through all the [[Natural_number|natural numbers]]. Proofs by induction come in three flavors: weak (or classic) induction, strong induction, and transfinite induction. As the names suggest, weak induction proofs require fewer assumptions than strong induction proofs. Sometimes these assumptions are too weak, in which case strong induction is necessary. Transfinite induction involves infinitary mathematics, including the [[Axiom of Choice]]. Many mathematicians avoid transfinite induction when possible.

=== Weak Induction ===

Hypothesis A is true for 0. Then assume [[hypothesis]] A is true for the number n. Prove that it must be true for n+1.

=== Strong Induction ===

Hypothesis A is true for 0. Then assume hypothesis A is true for all the numbers less than n. Prove that it must be true for n.

=== Technique of Induction ===
Both Weak and Strong Induction are based on the [[Peano's Axioms|Peano Axioms]], especially the ''Induction Axiom'':<blockquote>If a set S of numbers contains zero and the successor of every number in S, then S contains every number. </blockquote>
If you want to show that an hypothesis A is true for all [[Natural Numbers]], you look at the set S of all numbers for which the hypothesis is true: you show, that <ol><li>zero is an element of the set S 
<math>0 \in S</math><li>with any number n, S contains the successor of n <math> n \in S \Rightarrow (n+1) \in S </math></ol>
By the ''Induction Axiom'', S then includes all Natural Numbers, i.e., the hypothesis A hold for all Natural Numbers.
A standard example for this kind of reasoning is the following: 
Hypothesis A: ''For any Natural Number n, the sum of the numbers less or equal to n is <math>\frac{n(n+1)}{2}</math>''
<ol><li>Show: A holds for zero, i.e., zero is in S:
That's easy: The sum of all natural numbers less or equal to zero '''is''' zero, and this equals <math>\frac{0(0+1)}{2}</math>
<li>Show: If A holds for n, then A holds for n+1:
So, we assume that A is true for n: 
(*) <math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}</math> 
We have to show, that then A is true for n+1: 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{(n+1)((n+1)+1}{2}</math> 
How to do this? For example, the following way, starting with (*): 
<math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}\quad |</math>adding (n+1) on both sides 
<math>\Leftrightarrow</math> 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{n(n+1)}{2} + (n+1)\quad|</math>simplify the right side 
<math>\Leftrightarrow</math> 
<math>1 + 2 + 3 + ... + n + (n+1)=\frac{n(n+1)}{2} + \frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}</math>
Now, we're finished: the hypothesis A holds for all the Natural Numbers.

=== Transfinite Induction ===

Assume hypothesis A is true for a finite number n or an infinite [[cardinal]] k. Prove it must be true for n+1 (respectively, k+1).

[[Category:Mathematics]]

Talk:Prime number

2008-08-29T17:03:53Z

DiEb: Why Mathematicians Prefer the Alternative Definition

== Primality testing - Problems with the number 4 ==

When testing whether a number ''N'' is divisible by ''s'', using values of ''s'' from 2 to ''N'' - 1, when ''N'' is 4, the method correctly determines that 4 is not prime, because it is divisible by 2. This example is in the C programming language.
<blockquote><pre>
for (s = 2; s < N - 1; ++s) {
if (N % s == 0) {
printf("N is not a prime number.");
}
}
</pre></blockquote>

However, when using values of ''m'' from 2 to the square root of ''n'', when ''n'' is 4, an error can occur. Since 2 is the square root of 4, the test range is 2 to 2, and, if the range is exclusive, as in a for-loop, the whole test is skipped, and 4 is incorrectly determined to be a prime number.
<blockquote><pre>
for (s = 2; s < sqrt(N); ++s) {
if (N % s == 0) {
printf("N is not a prime number.");
}
}
</pre></blockquote>

[[User:Isaac4given|Isaac4given]] 11:28, 10 January 2008 (EST)

I'm surprised that 9 worked. Try.
<blockquote><pre>
for (s = 2; s <= sqrt(N); ++s) { // less than or equal to catch perfect squares
if (N % s == 0) {
printf("N is not a prime number.");
}
}
</pre></blockquote>

[[User:Ga ohoyt|Ga ohoyt]] 13:32, 10 January 2008 (EST)

== Flaw in proof ==

"It is easy to prove that there are an infinite number of primes using Euclid's second theorem. If there were a finite number of primes, you could multiply them all together and add 1. The resulting number would show the existence of a new prime, since it would not be divisible by any smaller prime (it would always have a remainder of 1)." This argument doesn't work. 510511 = 2 * 3 * 5 * 7 * 11 * 13 * 17 + 1, but it's not prime. [[User:Sepura|Sepura]] 13:21, 20 February 2008 (EST)

:You misunderstand the argument. Since you multiplied only the first 7 primes to get 510510, your hypothesis (the thing to be contradicted in this proof by contradiction) is that those 7 numbers constitute the complete finite set of primes. And in fact, 510511 is not divisible by any of them. Therefore they are not the complete set of primes. And the same argument can be made for any hypothesized finite set. Therefore, no finite set can be the complete set of primes. [[User:Ga ohoyt|Ga ohoyt]] 20:15, 20 February 2008 (EST)

==Why Mathematicians Prefer the Alternative Definition==
Because mathematician make a distinction between irreducible elements (p irred. <=> (a|p => a unit or a=p)) and prime elements (p prime <=> (for all a,b: p|ab => p|a or p|b). While these two properties coincide on <math>\mathbf{Z}</math>, they differ e.g., on <math>\mathbf{Z}[\sqrt{-5}]</math>: here 3 is irreducible, but not a prime, as <math>21 = 3 \cdot 7 = (1 + 2 \sqrt{-5})(1 - 2 \sqrt{-5})</math> --[[User:DiEb|DiEb]] 13:03, 29 August 2008 (EDT)

Talk:Fourier transformation

2008-08-22T15:14:15Z

DiEb: Top. Groups & Fourier Transforms

==MishMash==
The Fourier transformation (often called the "Fourier transform" or "Fourier integral") is an invertible integral transformation that decomposes a square integrable, piecewise continuous functions on a topological group into a linear combination of basis elements.<ol>
<li>Shouldn't we start with classes of functions on <math>\mathbf{C}^n</math>? Test functions <math>\mathcal{D}</math>? And then perhaps tempered distributions <math>\mathcal{S}</math>?
<li>What's the "piecewise continuous" part about?
<li>Yes, it can be done on a locally compact abelian group, but just to throw in half of the necessary conditions isn't helpful, IMO </ol>
I'd like to rewrite the article as stated above... Any objections? --[[User:DiEb|DiEb]] 11:14, 22 August 2008 (EDT)

Fourier series

2008-08-22T14:53:59Z

DiEb: minor correction

'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions.

The Fourier [[series (mathematics)|series]] of a function ''f(t)'' is of the form:
:<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math>
where, ''n'' is an integer and
:<math> \omega_n = n\frac{2\pi}{T}</math>
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>

Fourier series can be generalized to [[Fourier transformation]]s for other classes of functions, for instance [[square integrable function]]s.

[[category: mathematics]]

User talk:DeanS

2008-08-22T14:32:50Z

DiEb: /* Thank you! */

Just tell me how could we protect better the Mormon articles? --[[User:Joaquín Martínez]], [[User talk:Joaquín Martínez|talk]] 12:31, 1 July 2008 (EDT)

:What do you mean by "protect better" Joaquín? This is a wiki and we try to avoid protecting the articles. --[[User:DeanS|DeanStalk]] 13:23, 1 July 2008 (EDT)

: OK, I will do my best to patroll them. At present I am out of home. Just keep doing your wonderfull work. --[[User:Joaquín Martínez]], [[User talk:Joaquín Martínez|talk]] 13:31, 1 July 2008 (EDT)

:: I am out of town. I will be back untill August 6th. Thanks for your help. --[[User:Joaquín Martínez]], [[User talk:Joaquín Martínez|talk]] 14:16, 30 July 2008 (EDT)

::: Good to hear from you Joaquín. I hope you have a great vacation! --[[User:DeanS|DeanStalk]] 17:26, 30 July 2008 (EDT)

Hi DeanS, what a great article 'Breaking Speaker Pelosi's News Blackout'. I love the strategy, proud of my GOP 'ers, it is so patriotic!--[[User:Jpatt|jp]] 00:50, 5 August 2008 (EDT)

:Thanks Jpatt! Now if only President Bush would make Congress come back in session to deal with this issue. They don't deserve a five-week vacation. --[[User:DeanS|DeanStalk]] 02:11, 5 August 2008 (EDT)

== Thanks, Bro! ==

I thought your guys are gonna ignore me again, but Nah, I got wrong ideas. Thank you, May the Lord bless you! [http://www.conservapedia.com/index.php?title=Template:Mainpageright&oldid=497248] I wish I was brave enough to do that! But only I'm could is to support them to do so, "sigh".--[[User:TagoPagdaluhong|TagoPagdaluhong]] 11:36, 8 August 2008 (EDT)

:Your welcome Tago. I think the pic is a nice addition as well. --[[User:DeanS|DeanStalk]] 11:39, 8 August 2008 (EDT)

::I must say, that picture is much better view then the source had.--[[User:TagoPagdaluhong|TagoPagdaluhong]] 11:50, 8 August 2008 (EDT)

:::I agree. I followed the pic to its source and was rewarded with this version. --[[User:DeanS|DeanStalk]] 11:58, 8 August 2008 (EDT)

==Welcome==

Thanks for the welcome! This website is actually a god-send, a perfect oppurtunity to actually make use of my summer holidays. Hey by the way, I was wondering what CP's policy on science articles are, as this will probably be the aspect I'll be contributing to quite a lot. Thanks again- [[User:Jl2000|Jl2000]]

:Just start editing and you'll find out soon enough ;-) --[[User:DeanS|DeanStalk]] 15:21, 8 August 2008 (EDT)

:I researched this person's edits and found much of the material to be false. There was also information on that other site which made it very clear that he was just another of their parodists. [[User:Jinxmchue|Jinxmchue]] 15:58, 8 August 2008 (EDT)

::Interesting. They're supposed to be on a boycott of CP right now. ;-) --[[User:DeanS|DeanStalk]] 16:29, 8 August 2008 (EDT)

==Christian solidarity==

Hi, Dean. I like your [[:Image:Beijing arrests.jpg]] but I couldn't edit the caption. Can you crop the image and put the caption in the image link? --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 09:30, 9 August 2008 (EDT)

:Hi Ed. How about this cropped image? The caption is in the image info. [[:Image:Beijing arrests2.jpg]] --[[User:DeanS|DeanStalk]] 13:24, 9 August 2008 (EDT)

== Thanks and a request or two. ==

Thanks for pointing me at that list of admins. Two things - [[Conservapedia:User rights]] is full of broken links but is locked - can you or someone else fix these? Also can you unlock [[Rape]] for me as well - the current wording includes the phrase "seducing a child" - as seduction as a concept can suggest an element of consent I'd like to rephrase the article. Thanks in advance. [[User:DefenderofTrue|DefenderofTrue]] 21:39, 14 August 2008 (EDT)

==Pre-screening of News stories==

Just curious - why the "No article prescreening" policy, and why delete what I thought were constructive comments, instead of responding to them on the page? --[[User:DinsdaleP|DinsdaleP]] 20:44, 20 August 2008 (EDT)

:The reason there will be no article pre-screening by you (or other liberals) is because myself or another admin will be prescreening the articles.

:I deleted your comments because those project pages are only for project members, and I'm not going to have nonmembers influencing the project. --[[User:DeanS|DeanStalk]] 20:58, 20 August 2008 (EDT)

::Okay, all that makes sense, but if you read my post, I wasn't suggesting that I pre-screen, because I know non-conservatives like me aren't welcome on the sub-project. I had the sense that there was no pre-screening by anyone, and your edit-comment seemed to bear that out. Thanks for clarifying.

::Also, can I ask a blunt but serious question? Do you ever find that the labels of "liberal" and "conservative" are taken wayyy too seriously here, resulting in an all-or-none, black-and-white division between the two perspectives? I never post here to make trouble, only to offer a viewpoint that's usually more nuanced than the black-hat/white-hat exchanges that often take place. I hate labels myself, because in my case I'm libertarian on most social issues, conservative on fiscal ones, and still trying to figure out the spiritual - "Unitarian/Universalist" is probably closer than "Agnostic", but neither one is a strong enough fit to apply to myself.

::So if you want to create sub-projects for the true-believers only, go for it. I've just observed that while this is Conservapedia and not Neutralapedia, the "feel" of the place has become more polarized and combative, with less tolerance for dissent or even respectful disagreement, and a corresponding drop in the number and quality of contributors.

::As I said though, good luck with the news project, and I'll post my comments on the Main:Talk page. --[[User:DinsdaleP|DinsdaleP]] 21:17, 20 August 2008 (EDT)

:::Dinsdale, You said, "I wasn't suggesting that I pre-screen". What is this? [http://www.conservapedia.com/index.php?title=Talk:Wikiproject:News&diff=prev&oldid=502069 "... I'd be glad to participate in article screening..."]

:::No, I don't find the labels are taken ''too'' seriously here. I have a problem with liberals and their way of thinking. Liberal viewpoints are wrong.

:::I find very little of the "respectful disagreement" from liberals; mostly I see great examples of [[Liberal Style]], vandalism, parody, troublemaking, obscenity, etc.

:::Your comments here are respectful; therefore I have responded in kind. --[[User:DeanS|DeanStalk]] 21:39, 20 August 2008 (EDT)

::::Thanks for the courteous response. As for my statement on volunteering, the ''complete'' statement was "I'd be glad to participate in article screening but I don't qualify to be on this sub-project." Hey, I'd love night-editing rights, too, but I'm about as qualified to get that, too ;-) --[[User:DinsdaleP|DinsdaleP]] 22:05, 20 August 2008 (EDT)

:::::LOL --[[User:DeanS|DeanStalk]] 22:21, 20 August 2008 (EDT)
==Didn't mean to step on toes==
I'm just a huge fan of Cherone's and his ability to not alienate people when discussing Christianity and why abortion is wrong. So, I'd rather stick his "line in the sand" in there, if you don't mind. He has stated that anyone who really wants to be taken seriously in this debate at least needs to recognize that a woman has a right to live, if / when a baby directly threatens that. It's a position, being a woman wanting children, I happen to share. :-) If you really think it's liberal bias, let me know or a third party know, and we can "let the majority vote" (as it were).--[[User:MHayes|MHayes]] 22:43, 20 August 2008 (EDT)

: Btw, I DO mind. There is NO majority vote when an administrator removes liberal bias. --[[User:DeanS|DeanStalk]] 22:59, 20 August 2008 (EDT)

::I guess I don't understand what "liberal bias" is in this case, then. Are you asserting that Cherone is a liberal? I really have no idea his political preference, just what he has said about abortion. --[[User:MHayes|MHayes]] 23:01, 20 August 2008 (EDT)

:::No, I'm not saying that Cherone is a liberal. See my comments at [[Talk:Extreme]] --[[User:DeanS|DeanStalk]] 23:16, 20 August 2008 (EDT)

==Wikinews==
DeanS, Wikinews, my spelling error on the Ayers story. Unrepentent should be unrepentant.-- [[Image:50 star flag.png|14px]] [[User:Jpatt|jp]] 10:31, 21 August 2008 (EDT)

Could you add int link for Stephanie Tubbs Jones? Tanks in advance!--[[Image:50 star flag.png|14px]] [[User:Jpatt|jp]] 12:43, 21 August 2008 (EDT)

:Thanks Jp. This mistake has been [http://www.conservapedia.com/index.php?title=Template:Mainpageright&curid=66589&diff=502572&oldid=502384 corrected].

:The wikilink was added [http://www.conservapedia.com/index.php?title=Template:Mainpageright&curid=66589&diff=502578&oldid=502572 here]. --[[User:DeanS|DeanStalk]] 12:53, 21 August 2008 (EDT)

==Thank you!==
For stepping in. Whoever that person was, he was doing bad things to my blood pressure :) --[[User:JessicaT|KotomiT]][[User talk:JessicaT|''Hajimemashite!'']] 10:07, 22 August 2008 (EDT)

:You're welcome Jessica. He was an imposter and was dealt with accordingly. --[[User:DeanS|DeanStalk]] 10:10, 22 August 2008 (EDT)
I got really annoyed... --[[User:DiEb|DiEb]] 10:32, 22 August 2008 (EDT)

Proof by induction

2008-08-22T13:21:15Z

DiEb:

A '''proof by induction''' is a technique of mathematical [[proof]]. It is one of the axioms of [[Zermelo-Fraenkel]] set theory, allowing a mathematician to have proofs that count in order through all the [[Natural_number|natural numbers]]. Proofs by induction come in three flavors: weak (or classic) induction, strong induction, and transfinite induction. As the names suggest, weak induction proofs require fewer assumptions than strong induction proofs. Sometimes these assumptions are too weak, in which case strong induction is necessary. Transfinite induction involves infinitary mathematics, including the [[Axiom of Choice]]. Many mathematicians avoid transfinite induction when possible.

=== Weak Induction ===

Hypothesis A is true for 0. Then assume [[hypothesis]] A is true for the number n. Prove that it must be true for n+1.

=== Strong Induction ===

Hypothesis A is true for 0. Then assume hypothesis A is true for all the numbers less than n. Prove that it must be true for n.

=== Technique of Induction ===
Both Weak and Strong Induction are based on the [[Peano's Axioms|Peano Axioms]], especially the ''Induction Axiom'':<blockquote>If a set S of numbers contains zero and the successor of every number in S, then S contains every number. This is known as the induction axiom. </blockquote>
If you want to show that an hypothesis A is true for all [[Natural Numbers]], you look at the set S of all numbers for which the hypothesis is true: you show, that <ol><li>zero is an element of the set S 
<math>0 \in S</math><li>with any number n, S contains the successor of n <math> n \in S \Rightarrow (n+1) \in S </math></ol>
By the ''Induction Axiom'', S then includes all Natural Numbers, i.e., the hypothesis A hold for all Natural Numbers.
A standard example for this kind of reasoning is the following: 
Hypothesis A: ''For any Natural Number n, the sum of the numbers less or equal to n is <math>\frac{n(n+1)}{2}</math>''
<ol><li>Show: A holds for zero, i.e., zero is in S:
That's easy: The sum of all natural numbers less or equal to zero '''is''' zero, and this equals <math>\frac{0(0+1)}{2}</math>
<li>Show: If A holds for n, then A holds for n+1:
So, we assume that A is true for n: 
(*) <math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}</math> 
We have to show, that then A is true for n+1: 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{(n+1)((n+1)+1}{2}</math> 
How to do this? For example, the following way, starting with (*): 
<math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}\quad |</math>adding (n+1) on both sides 
<math>\Leftrightarrow</math> 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{n(n+1)}{2} + (n+1)\quad|</math>simplify the right side 
<math>\Leftrightarrow</math> 
<math>1 + 2 + 3 + ... + n + (n+1)=\frac{n(n+1)}{2} + \frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}</math>
Now, we're finished: the hypothesis A hold for all the Natural Numbers.

=== Transfinite Induction ===

Assume hypothesis A is true for a finite number n or an infinite [[cardinal]] k. Prove it must be true for n+1 (respectively, k+1).

[[Category:Mathematics]]

Proof by induction

2008-08-22T13:19:48Z

DiEb: Technique of Induction

A '''proof by induction''' is a technique of mathematical [[proof]]. It is one of the axioms of [[Zermelo-Fraenkel]] set theory, allowing a mathematician to have proofs that count in order through all the [[Natural_number|natural numbers]]. Proofs by induction come in three flavors: weak (or classic) induction, strong induction, and transfinite induction. As the names suggest, weak induction proofs require fewer assumptions than strong induction proofs. Sometimes these assumptions are too weak, in which case strong induction is necessary. Transfinite induction involves infinitary mathematics, including the [[Axiom of Choice]]. Many mathematicians avoid transfinite induction when possible.

=== Weak Induction ===

Hypothesis A is true for 0. Then assume [[hypothesis]] A is true for the number n. Prove that it must be true for n+1.

=== Strong Induction ===

Hypothesis A is true for 0. Then assume hypothesis A is true for all the numbers less than n. Prove that it must be true for n.

=== Technique of Induction ===
Both Weak and Strong Induction are based on the [[Peano's Axioms|Peano Axioms]], especially the ''Induction Axiom'':<blockquote>If a set S of numbers contains zero and the successor of every number in S, then S contains every number. This is known as the induction axiom. </blockquote>
If you want to show that an hypothesis A is true for all [[Natural Numbers]], you look at the set S of all numbers for which the hypothesis is true: you show, that <ol><li>zero is an element of the set S 
<math>0 \in S</math><li>with any number n, S contains the successor of n <math> n \in S \Rightarrow (n+1) \in S </math></ol>
By the ''Induction Axiom'', S then includes all Natural Numbers, i.e., the hypothesis A hold for all Natural Numbers.
A standard example for this kind of reasoning is the following: 
Hypothesis A: ''For any Natural Number n, the sum of the numbers less or equal to n is <math>\frac{n(n+1)}{2}</math>''
<ol><li>Show: A holds for zero, i.e., zero is in S:
That's easy: The sum of all natural numbers less or equal to zero '''is''' zero, and this equals <math>\frac{0(0+1)}{2}</math>
<li>Show: If A holds for n, then A holds for n+1:
So, we assume that A is true for n: 
(*) <math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}</math> 
We have to show, that then A is true for n+1: 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{(n+1)((n+1)+1}{2}</math> 
How to do this? For example, the following way, starting with (*): 
<math>1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}\quad |</math>adding (n+1) on both sides 
<math>\Leftrightarrow</math> 
<math>1 + 2 + 3 + ... + n + (n+1)= \frac{n(n+1)}{2} + (n+1)\quad|</math>simplify the right side 
<math>\Leftrightarrow</math> 
<math>\frac{n(n+1)}{2} + \frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}</math>
Now, we're finished: the hypothesis A hold for all the Natural Numbers.

=== Transfinite Induction ===

Assume hypothesis A is true for a finite number n or an infinite [[cardinal]] k. Prove it must be true for n+1 (respectively, k+1).

[[Category:Mathematics]]

Natural number

2008-08-22T12:41:02Z

DiEb: Axiomatization

In [[mathematics]], a '''natural number''' is a a number from the set {0,1,2,...} or (1,2,3...}, depending on the context and the reference.<ref>0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). + Older books sometimes exclude [[zero]], as there is a long history of people thinking that zero is unnatural or not really a number. Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." [http://mathworld.wolfram.com/NaturalNumber.html (Wolfram)] </ref> Natural numbers were used initially for counting ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of [[cardinality]] and [[ordinality]] which slowly become more complicated as one treats [[infinite]] numbers as well. The set of natural numbers is [[countable]]- via [[bijection]], this property can be used to prove the countability of the [[integer]]s and [[rational number]]s.

==Axiomatization==
In the late 19th century, [[Giuseppe Peano]] (August 27, 1858 – April 20, 1932) elaborated ''the'' axiomatic system for the Natural Numbers, later named [[Peano's Axioms|Peano Axioms]] in his honor.
==Reference==
<references/>

[[Category:Mathematics]]

Limit (mathematics)

2008-08-21T19:00:48Z

DiEb:

{{Template:Math-h}}

The concept of the '''limit''' is the cornerstone of [[calculus]], [[analysis]], and [[topology]]. At the simplest intuitive level, the limit of a function at a point is the value that the function "approaches" as its argument "approaches" that point. But that is an unsatisfactory definition, and a much more careful definition is required.

==Limit of a Sequence==
Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...),</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a\,</math>, i.e.,
<math>\lim_{n\to \infty} = a,\,</math>
if for any <math>\varepsilon > 0\,</math>, there exists a number <math>N\,</math>, such that
<math>
|a_n - a| < \varepsilon\,</math> for every <math>n > N\,</math>. Intuitively, this means that if you take an [[interval]] as small as you like, [[center]]ed at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval.

==Limit of a Function==
Here is an example. Suppose a function is
:<math>f(x) = \frac{x^2-x-6}{x^2-2x-3}\,</math>
What is the limit of f(x) as x approaches 3? This could be written
:<math>\lim_{x\to 3}f(x)\,</math>
We can't just evaluate f(3), because the numerator and denominator are both zero. But there's a trick here—we can divide (x-3) into both numerator and denominator, getting
:<math>f(x) = \frac{x+2}{x+1}\,</math>
so the limit is 5/4.
(That does't actually prove that the limit is 5/4. Once we have defined the limit correctly, we will need a few theorems about limits and [[continuous function]]s to establish this result. It is nevertheless true.)

Now try an example that isn't trivial. Let
:<math>f(x) = x^x\,</math>
This function is well-defined for x>0, using the general definition that involves the [[exponential]] and [[natural logarithm]] functions. We can calculate f(x) for various values of x:
f(5) = 3125
f(2) = 4
f(0.5) = 0.7071
f(0.3) = 0.6968
f(0.2) = 0.7248
It got smaller, but now it's getting bigger. What's happening?
f(0.1) = 0.7943
f(0.01) = 0.95499
f(0.0001) = 0.999079
f(0.000001) = 0.99998618
It looks as though it's approaching 1. Is it? And what does that mean?
f(1 trillionth) = 0.999999999972369

Remember that f(0) doesn't exist. So we really have to be careful.

We are going to say that the limit of f(x), as x approaches 0, is 1. What that means is this:

We can get f(x) ''arbitrarily close to 1'' if we choose an x sufficiently close to zero. If we want f(x) within one quadrillionth of 1, <math>x = 10^{-17}\,</math> will do. We never have to set x to zero exactly, and we never have to get f(x) = 1 exactly.

In the general case, for arbitrary functions, we might want to say something like
:<math>\lim_{x\to X}f(x) = Y\,</math>

===For Every Epsilon ...===

Stated precisely, given any tolerance ε (by tradition, the letter ε is always used) x being sufficiently close to X will get f(x) within ε of Y. That condition is formally written:
:<math>|f(x) - Y| < \varepsilon\,</math>
In this example (<math>f(x) = x^x\,</math>, X=0, Y=1), when ε is <math>10^{-15}\,</math> (one quadrillionth), <math>x < 10^{-17}\,</math> satisfies the condition.

===... There Exists a Delta===

The way we formalize the notion of x being very close to X is:

:"There is a number δ (by tradition it's always δ) such that, whenever x is within δ of X, f(x) is within ε of Y".

That is written:
:Whenever <math>0 < |x - X| < \delta, |f(x) - Y| < \varepsilon\,</math>

In our example, if ε is <math>10^{-15}\,</math>, δ = <math>10^{-17}\,</math> works. That is, any <math>x < 10^{-17}\,</math> will guarantee <math>|f(x)-1| < 10^{-15}\,</math>.

===The Full Definition===
So here is the full definition:

:<math>\lim_{x\to X} f(x) = Y\,</math> means

:For every ε > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, |f(x)-Y| < \varepsilon\,</math>

So the definition is sort of like a bet or a contract—"For any epsilon you can give me, I can come up with a delta."

A few things to note:
*We require ε > 0. Specifying a required tolerance of zero is not allowed. We only have to be able to get f(x) within an arbitrarily close but nonzero tolerance of Y. We don't ever have to get it exactly equal to Y.
*We have 0 < |x-X| < δ, not just |x-X| < δ. That is, we never have to calculate f(X) exactly. f(X) doesn't need to be defined. In the example we are considering, <math>0^0\,</math> isn't defined.

This definition, and variations of it, are the central point of calculus, analysis, and topology. The phrase "For every ε there is a δ" is ingrained into the consciousness of every mathematics student. This notion of a "bet" could be considered to set the branches of mathematics that follow (calculus, topology, ...) apart from the earlier branches (arithmetic, algebra, geometry, ...) Students who have mastered the notion of "For every ε there is a δ" are ready for higher mathematics.

In our example of <math>f(x) = x^x\,</math>, we haven't actually satisfied the definition of the limit, because f(x) isn't defined for negative x. There are more restrictive notions of "limit from the left" and "limit from the right". We have the limit from the right of <math>x^x = 1\,</math>, which means
:For every ε > 0, there is a δ > 0 such that, whenever 0 < x-X < δ, |f(x)-Y| < ε

We still haven't proved that the limit is actually 1. We just gave some accurate calculations strongly suggesting that it is. In fact it is, and the proof requires a few theorems about limits, [[continuous function]]s, and the [[exponential]] and [[natural logarithm]] functions.

==The Limit of f(x) is Infinity==

With this precise (or, as mathematicians say, ''rigorous'') definition of a limit, we can examine variations that involve "infinity". Remember, '''Infinity is not a number.''' It is only through the magic of the epsilon-delta formulation that we can make sense of it.

We might say something like "The limit of f(x), as x approaches X, is infinity." What that means is that we replace the
:<math>|f(x) - Y| < \varepsilon\,</math>
with
:<math>f(x) > M\,</math>

That is,
:For every M > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) > M\,</math>

Consider the function
:<math>f(x) = \frac{1}{(x-3)^2}\,</math>
If you graph this, there is an "infinitely high" peak at x=3. We have:
:<math>\lim_{x\to 3}f(x) = \infty\,</math>
If we set <math>M = 10^{20}\,</math>, <math>\delta = 10^{-10}\,</math> will win the bet.

For a limit of minus infinity, we set M to some huge negative number, so
:<math>\lim_{x\to X}f(x) = - \infty\,</math>
means
:For every M < 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) < M.</math>

[[category:mathematics]]

Limit (mathematics)

2008-08-21T19:00:14Z

DiEb:

{{Template:Math-h}}

The concept of the '''limit''' is the cornerstone of [[calculus]], [[analysis]], and [[topology]]. At the simplest intuitive level, the limit of a function at a point is the value that the function "approaches" as its argument "approaches" that point. But that is an unsatisfactory definition, and a much more careful definition is required.

==Limit of a Sequence==
Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...,</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a\,</math>, i.e.,
<math>\lim_{n\to \infty} = a,\,</math>
if for any <math>\varepsilon > 0\,</math>, there exists a number <math>N\,</math>, such that
<math>
|a_n - a| < \varepsilon\,</math> for every <math>n > N\,</math>. Intuitively, this means that if you take an [[interval]] as small as you like, [[center]]ed at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval.

==Limit of a Function==
Here is an example. Suppose a function is
:<math>f(x) = \frac{x^2-x-6}{x^2-2x-3}\,</math>
What is the limit of f(x) as x approaches 3? This could be written
:<math>\lim_{x\to 3}f(x)\,</math>
We can't just evaluate f(3), because the numerator and denominator are both zero. But there's a trick here—we can divide (x-3) into both numerator and denominator, getting
:<math>f(x) = \frac{x+2}{x+1}\,</math>
so the limit is 5/4.
(That does't actually prove that the limit is 5/4. Once we have defined the limit correctly, we will need a few theorems about limits and [[continuous function]]s to establish this result. It is nevertheless true.)

Now try an example that isn't trivial. Let
:<math>f(x) = x^x\,</math>
This function is well-defined for x>0, using the general definition that involves the [[exponential]] and [[natural logarithm]] functions. We can calculate f(x) for various values of x:
f(5) = 3125
f(2) = 4
f(0.5) = 0.7071
f(0.3) = 0.6968
f(0.2) = 0.7248
It got smaller, but now it's getting bigger. What's happening?
f(0.1) = 0.7943
f(0.01) = 0.95499
f(0.0001) = 0.999079
f(0.000001) = 0.99998618
It looks as though it's approaching 1. Is it? And what does that mean?
f(1 trillionth) = 0.999999999972369

Remember that f(0) doesn't exist. So we really have to be careful.

We are going to say that the limit of f(x), as x approaches 0, is 1. What that means is this:

We can get f(x) ''arbitrarily close to 1'' if we choose an x sufficiently close to zero. If we want f(x) within one quadrillionth of 1, <math>x = 10^{-17}\,</math> will do. We never have to set x to zero exactly, and we never have to get f(x) = 1 exactly.

In the general case, for arbitrary functions, we might want to say something like
:<math>\lim_{x\to X}f(x) = Y\,</math>

===For Every Epsilon ...===

Stated precisely, given any tolerance ε (by tradition, the letter ε is always used) x being sufficiently close to X will get f(x) within ε of Y. That condition is formally written:
:<math>|f(x) - Y| < \varepsilon\,</math>
In this example (<math>f(x) = x^x\,</math>, X=0, Y=1), when ε is <math>10^{-15}\,</math> (one quadrillionth), <math>x < 10^{-17}\,</math> satisfies the condition.

===... There Exists a Delta===

The way we formalize the notion of x being very close to X is:

:"There is a number δ (by tradition it's always δ) such that, whenever x is within δ of X, f(x) is within ε of Y".

That is written:
:Whenever <math>0 < |x - X| < \delta, |f(x) - Y| < \varepsilon\,</math>

In our example, if ε is <math>10^{-15}\,</math>, δ = <math>10^{-17}\,</math> works. That is, any <math>x < 10^{-17}\,</math> will guarantee <math>|f(x)-1| < 10^{-15}\,</math>.

===The Full Definition===
So here is the full definition:

:<math>\lim_{x\to X} f(x) = Y\,</math> means

:For every ε > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, |f(x)-Y| < \varepsilon\,</math>

So the definition is sort of like a bet or a contract—"For any epsilon you can give me, I can come up with a delta."

A few things to note:
*We require ε > 0. Specifying a required tolerance of zero is not allowed. We only have to be able to get f(x) within an arbitrarily close but nonzero tolerance of Y. We don't ever have to get it exactly equal to Y.
*We have 0 < |x-X| < δ, not just |x-X| < δ. That is, we never have to calculate f(X) exactly. f(X) doesn't need to be defined. In the example we are considering, <math>0^0\,</math> isn't defined.

This definition, and variations of it, are the central point of calculus, analysis, and topology. The phrase "For every ε there is a δ" is ingrained into the consciousness of every mathematics student. This notion of a "bet" could be considered to set the branches of mathematics that follow (calculus, topology, ...) apart from the earlier branches (arithmetic, algebra, geometry, ...) Students who have mastered the notion of "For every ε there is a δ" are ready for higher mathematics.

In our example of <math>f(x) = x^x\,</math>, we haven't actually satisfied the definition of the limit, because f(x) isn't defined for negative x. There are more restrictive notions of "limit from the left" and "limit from the right". We have the limit from the right of <math>x^x = 1\,</math>, which means
:For every ε > 0, there is a δ > 0 such that, whenever 0 < x-X < δ, |f(x)-Y| < ε

We still haven't proved that the limit is actually 1. We just gave some accurate calculations strongly suggesting that it is. In fact it is, and the proof requires a few theorems about limits, [[continuous function]]s, and the [[exponential]] and [[natural logarithm]] functions.

==The Limit of f(x) is Infinity==

With this precise (or, as mathematicians say, ''rigorous'') definition of a limit, we can examine variations that involve "infinity". Remember, '''Infinity is not a number.''' It is only through the magic of the epsilon-delta formulation that we can make sense of it.

We might say something like "The limit of f(x), as x approaches X, is infinity." What that means is that we replace the
:<math>|f(x) - Y| < \varepsilon\,</math>
with
:<math>f(x) > M\,</math>

That is,
:For every M > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) > M\,</math>

Consider the function
:<math>f(x) = \frac{1}{(x-3)^2}\,</math>
If you graph this, there is an "infinitely high" peak at x=3. We have:
:<math>\lim_{x\to 3}f(x) = \infty\,</math>
If we set <math>M = 10^{20}\,</math>, <math>\delta = 10^{-10}\,</math> will win the bet.

For a limit of minus infinity, we set M to some huge negative number, so
:<math>\lim_{x\to X}f(x) = - \infty\,</math>
means
:For every M < 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta,\quad f(x) < M.</math>

[[category:mathematics]]

Albert Einstein

2008-08-21T18:20:31Z

DiEb: /* Religious views */ Quote re personal God

[[Image:600px-Albert Einstein Head.jpg|right|thumb|250px]]
'''Albert Einstein''' (1879-1955) was the most famous [[physicist]] of the 20th century. He was born in Ulm, in Württemberg, [[Germany]] in a [[Judaism|Jewish]] family. During his stay at the [[Switzerland|Swiss]] [[Patent and Trademark Office|Patent Office]] Einstein produced much of his remarkable work. In 1905 he obtained his [[Doctor|doctor's degree]]. He won the [[Nobel Prize]] for his work on the Photoelectric Effect in 1921. <ref>http://nobelprize.org/nobel_prizes/physics/laureates/1921/</ref> Einstein [[Immigration|immigrated]] to the [[United States of America|United States]] in the late 1930s to take the position of Professor of Theoretical Physics at [[Princeton University|Princeton]]; upon the urging of a colleague in physics, recommended that [[President of the United States of America|President]] [[Franklin Roosevelt|Franklin D. Roosevelt]] develop an [[atomic bomb]] project. <ref>[http://www.dannen.com/ae-fdr.html]</ref> He later regretted this and became a passionate [[Pacifism|pacifist]]. In 1952, he was offered the Presidency of [[Israel]], but he turned it down. Einstein collaborated with Dr. Chaim Weizmann in establishing the [[Hebrew]] [[University]] of [[Jerusalem]]. Albert Einstein died on April 18, 1955 at Princeton, [[New Jersey]] at the age of 76.

==Anecdotes==
*IN 1898, young Albert Einstein applied for admission to the Munich Technical Institute and was turned down. The young man, the Institute declared, "showed no promise" as a student. By 1905, he had formulated his special [[theory of relativity]]. <ref>http://oaks.nvg.org/sa5ra17.html</ref>
*Einstein came to Princeton University in 1935 and was asked what he would require for his study. he replied: "A desk, some pads and a pencil, and a large wastebasket to hold all of my mistakes." <ref>http://oaks.nvg.org/sa5ra17.html</ref>
*THE SCULPTOR Jacob Epstein tells this story:
*:"When I was doing Professor Albert Einstein's bust he had many a jibe at the Nazi professors, one hundred of whom had condemned his theory of relativity in a book.
*:"Were I wrong," he said, "one professor would have been enough." [ibid]

== Works ==
Einstein contributed to the theory of a [[Special Relativity]] and the theory of [[General Relativity]]. He also contributed to the study of Photoelectric Effect, Brownian Motion and Bose-Einstein Statistics. He was critical of quantum mechanics and has proposed the EPR-Paradox in attempt to point out spooky aspects of the theory. The last fifteen years of his life were spent in unsuccessfully searching for a [[Unified Field Theory]].
Einstein's best-known scientific work is his 1905 papers on [[Special Relativity]]. The theory of relativity had already been published by [[Hendrik Lorentz]] and [[Henri Poincaré]]. Einstein's contribution was to give a description of the Lorentz-Poincare theory without reference to the luminiferous aether, and to clarify some of the physical consequences of the theory.

Einstein then wrote a series of papers applying relativity to gravity using the mathematics of Riemannian Geometry, resulting in [[General Relativity]]. Much of the actual mathematics of general relativity was worked out by Marcel Grossman, David Hilbert, and others. Einstein's main contribution was to give an alternate derivation of the field equations, and to apply them to explain an [[anomaly]] in the advance of the perihelion in Mercury's orbit.
Arthur Eddington popularized General Relativity by publishing an explanation of it and by attempting to prove the theory experimentally in a dramatic test. Eddington embarked on a highly publicized trip into the Atlantic Ocean off Africa in order to observe the bending of [[star]]light passing nearby the [[Sun]] during the total [[solar eclipse]] of May 29, 1919. Eddington returned with a declaration that he had proven Einstein's theory of [[General Relativity]], making Einstein extremely famous<ref>On November 8, 1919, for example, the ''London Times'' had an article headlined: "The Revolution In Science/Einstein Versus Newton." Two days later, ''The New York Times''' headlines read: "Lights All Askew In The Heavens/Men Of Science More Or Less Agog Over Results Of Eclipse Observations/Einstein Theory Triumphs." [http://www.pbs.org/wgbh/nova/einstein/relativity/]</ref>, when in fact he had selectively used data to attain this result. No Nobel Prize was given for relativity.

In April 2007, NASA reported early results from their satellite Gravity Probe B, confirming 'to a precision of better than 1 per cent' Einstein's assertion that large objects such as the Earth distort the fabric of space and time.<ref> http://observer.guardian.co.uk/uk_news/story/0,,2057529,00.html</ref> The Gravity Probe B project, a joint mission between NASA and Stanford University, used the most precise gyroscopes ever manufactured to detect minute distortions in the fabric of the universe.

Einstein compared the effects of large objects on space and time to the way that a large ball placed on a sheet of rubber stretches the material and causes it to sag, and drew the analogy with the way stars and planets warp space-time. A smaller ball rolling along the sagging fabric will be drawn towards the larger ball, in the same way that the Earth is drawn towards the sun, but will not fall into it as long as it continues to move at sufficient speed. This was a change in thinking from the Newtonian cosmos, in which gravity was viewed purely as an attractive force between bodies.

Robert Massey of the Royal Astronomical Society said: 'From the most esoteric aspects of time dilation through to the beautiful and simple equation, E=mc², the vast bulk of Einstein's ideas about the universe are standing up to the test of time,'

===E=mc2===
Einstein’s famous formula E=mc2 says that mass and energy can turn into each other. It was first published by H. Poincare. Einstein published an alternate derivation that was corrected and improved by M. Planck. E stands for [[energy]], m is [[mass]], and c is the [[speed of light]]. So, if you increase the energy of a substance, for example by heating it, it gains a minute amount of mass. It was later realized that if you split an atom apart, as in a nuclear explosion, it loses a tiny amount of mass and you release a huge amount of [[energy]].

The connection E=mc2 to nuclear energy is not as direct as pop culture sometimes represents. Einstein did not discover the huge energy available from nuclear reactions. Early workers in radioactivity, such as Ernest Rutherford, understood clearly that the natural radioactive decay of radium released quantities of energy that could not be explained by chemistry, and were much larger than those of chemical reactions. Neither did Einstein originate the key idea that made nuclear energy a possible weapon or a practical energy source. That was Leo Szilard, who acknowledged having been inspired by a 1914 H. G. Wells novel. Szilard discovered or invented the chain reaction (and patented it). What Einstein did was to provide a theoretical framework that accounted for the release of energy, and a quantitative theory that was used by the scientists at Los Alamos who designed and built the bomb.

Energy binds together the parts of an [[atom]], including the [[nucleus]]. The nucleus of an atom is tiny, but very little mass could release a huge amount of energy. When one nucleus splits apart, it splits other nuclei around it. This is called a [[chain reaction]].

===Cosmological Constant===

Einstein revised his work on [[General Relativity]] to include a 'fudge-factor' which he termed the [[Cosmological Constant]]. He was trying to account for the apparent steady-state nature of the universe.

[[Edwin Hubble]], in whose name [[NASA]]'s famous [[Hubble Space Telescope]] is named, later convinced him otherwise. At one point he had Einstein actually making direct observations alongside him, and Hubble's findings showed conclusively that the [[universe]] was indeed expanding, as predicted by certain solutions of GR's field equations in the absence of a cosmological constant. Einstein recanted his original [[philosophy]], and accepted as true the notion of a changing universe having a definite [[begin]]ning.

He later said of his Cosmological Constant, "It was the biggest blunder of my life."

An interesting end-note on this term: Principle investigators from [[Princeton University]] first published findings in 1998 apparently revealing an accelerating expansion for the universe<ref>http://map.gsfc.nasa.gov/m_mm.html</ref>. Because this initially seemed like it might be a sort of 'anti-gravity' effect (similar in concept to Einstein's original notion), it briefly became quite popular for scientists to dub the phenomenon the 'Cosmological Constant' as well. For more information, see: [[Dark Energy]].

== Writings ==

*Special Theory of Relativity (1905)
*Relativity (English translations, 1920 and 1950)
*General Theory of Relativity (1916)
*Investigations on Theory of Brownian Movement (1926)
*The Evolution of Physics (1938)
*About Zionism (1930)
*Why War? (1933),
*My Philosophy (1934)
*(The World As I See It) (1934)
*Out of My Later Years (1950)

''...the daily effort comes from no deliberate intention or program, but straight from the heart.''

== Honors ==
Einstein received honorary doctorate degrees in science, medicine and philosophy from many American and European universities.

He won the Nobel Prize in 1921 for his work on the Photoelectric Effect.
*"It strikes me as unfair, and even in bad taste, to select a few individuals for boundless admiration, attributing superhuman powers of mind and character to them. This has been my fate, and the contrast between the popular assessment of my powers and achievements and the reality is simply grotesque." <ref>http://www.infinite-energy.com/iemagazine/issue38/einstein.html</ref>

Einstein told a friend in 1930, "To punish me for my contempt of authority, Fate has made me an authority myself." <ref>http://www.infinite-energy.com/iemagazine/issue38/einstein.html</ref>

He gained the Copley Medal of the Royal Society of London in 1925 and the Franklin Medal of the Franklin Institute in 1935.

Einstein was offered the presidency of Israel in 1952 <ref>[http://www.einstein-website.de/z_information/variousthings.html#president]</ref> but respectfully declined because he believed his life-long scientific attitude of objectivity made him an inappropriate candidate for such a high political position, he was also strongly opposed to nationalism.

In 1999, he was named "Person of the Century" by "Time" magazine.

==Einstein and the A-bomb==
In 1939 research done by several leading scientists including [[Enrico Fermi]] regarding chain reactions of nuclear fission prompted [[Leo Szilard]] to persuade Einstein to write a letter to President <ref>[[Roosevelt]] [http://hypertextbook.com/eworld/einstein.shtml]</ref>, to warn the President about the possibility of nuclear weapons and that Nazi [[Germany]] had already taken an interest in this technology.

Einstein himself had never been involved in the [[Manhattan project]]. He was considered a security risk, as he was a member, sponsor, or affiliated with thirty-four communist fronts between 1937 and 1954.<ref>http://foia.fbi.gov/foiaindex/einstein.htm</ref> Later he condemned the use of nuclear weapons.

==Controversy Surrounding His Death==
Dr. Thomas Stoltz Harvey, then a pathologist, performed the autopsy on Einstein in the [[Princeton Medical Center]] several hours after his death. Upon completion, Dr. Harvey removed the brain and eyes <ref>NPR, [http://www.npr.org/templates/story/story.php?storyId=4602913]</ref>. There is controversy about whether or not Harvey had consent from the Einstein family to remove the brain and keep it, though letters from Otto Nathan, executer of the Einstein estate, suggest that Harvey did indeed have permission to keep the brain, as long as research performed on it was only described in scientific journals.

The brain was photographed and sliced into 240 pieces in order to be preserved and further studied. Harvey hoped some uniqueness would be discovered that distinguished Einstein's brain as "genius".

Although he sent pieces away for further study, the bulk of the brain remained with Harvey for over 40 years. His once-ambitious plans for doing research on Einstein's brain were never actualized, though there were at least two controversial studies performed which suggested a physical reason for Einstein's intellegence. In 1996 Harvey entrusted Elliot Krauss, cheif [[pathologist]] at the Princeton Medical Center, with the brain. The brain now sits in a secret location, possibly only a few feet away from where it was first removed from Einstein's head. Krauss loans out pieces of the brain for scientific research, but he is very discriminating in deciding whom shall receive a piece. <ref>Abraham, Carolyn. ''Possessing Genius''. Icon Books Ltd (2005)</reF>

==Religious views==
[[Image:Einstein inshul1930.jpg|right|thumb|Einstein wearing a kippah in a Berlin Synagogue, 1930.|300px]]
There is some controversy surrounding the religion of Albert Einstein.
He explained his religious views to Time magazine in this manner: “I'm not an atheist. I don't think I can call myself a pantheist. The problem involved is too vast for our limited minds. We are in the position of a little child entering a huge library filled with books in many languages. The child knows someone must have written those books. It does not know how. It does not understand the languages in which they are written. The child dimly suspects a mysterious order in the arrangement of the books but doesn't know what it is. That, it seems to me, is the attitude of even the most intelligent human being toward God. We see the universe marvelously arranged and obeying certain laws but only dimly understand these laws." <ref>http://www.time.com/time/printout/0,8816,1607298,00.html</ref>

Einstein also said
<blockquote>
"In view of such harmony in the cosmos which I, with my limited human mind, am able to recognise, there are yet people who say there is no God. But what makes me really angry is that they quote me for support of such views."<ref>http://www.amazon.com/Einstein-Life-Universe-Walter-Isaacson/dp/0743264738</ref>,
</blockquote>
<blockquote>
"The fanatical atheists, are like slaves who are still feeling the weight of their chains which they have thrown off after hard struggle. They are creatures who--in their grudge against traditional religion as the 'opium of the masses'-- cannot hear the music of the spheres."<ref>http://www.amazon.com/Einstein-Life-Universe-Walter-Isaacson/dp/0743264738</ref>
</blockquote>
and
<blockquote>
"science without religion is lame. Religion without science is blind" <ref>http://www.amazon.com/Einstein-Life-Universe-Walter-Isaacson/dp/0743264738</ref>
</blockquote>
<blockquote>
"The word god is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honourable, but still primitive legends which are nevertheless pretty childish. No interpretation no matter how subtle can (for me) change this."<ref>http://www.guardian.co.uk/science/2008/may/12/peopleinscience.religion</ref>
</blockquote>

Taken as a whole, it appears Einstein rejected the literal stories of God's personal involvement with the Hebrews, but he did believe in the existence of a higher power.

This, he expressed in a letter on March 24th, 1954
<blockquote>"It was, of course, a lie what you read about my religious convictions, a lie which is being systematically repeated. I do not believe in a personal God and I have never denied this but have expressed it clearly. If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it."<ref>as quoted in "Albert Einstein - The Human Side", Selected and Edited by Helen Dukas and Banesh Hoffman, Princeton University Press, 1979.</ref></blockquote>

Famed [[atheist]] [[Richard Dawkins]] suggested that Einstein used the word "god" not in the sense of a higher power but rather a shorthand term for the inherent beauty of the universe.

==Views On Judaism and Zionism==
[[Image:Einst 22.jpg|right|thumb|Einstein with [[David Ben-Gurion]].|300px]]
In 1920, in a letter to the Central Association of German Citizens of the Jewish Faith, Einstein wrote
<blockquote>
"I am neither a German citizen, nor is there in me anything that can be described as ‘Jewish faith.’ But I am happy to belong to the Jewish people, even if I do not consider them in any way God's elect."
</blockquote>
Einstein do not view fellow Jews as co-religionists, but as tribal companions.
Einstein was made an honorary citizen of [[Tel Aviv]] during his visit in 1923.
In 1952, Einstein was offered the position of president of Israel by the 1st prime minster of [[Israel]] David Ben-Gurion, but declined the offer. Before his death in 1955, Einstein wrote to the Zionist leader Kurt Blumenfeld:
<blockquote>
"I thank you, even at this late hour, for having made me conscious of the Jewish soul."
</blockquote>
Einstein willed that all his scholarly work be archieved in Israel's Hebrew University after his death.

Whatever his views on Zionism were, Einstein has always expressed his anti-war views and staunchly opposed nationalism:

"He who joyfully marches to rank and file, has already earned my contempt. He has been given a large brain by mistake, since for him the spinal cord would surely suffice. This disgrace to civilization should be done away with at once. Heroism on command, how violently I hate all this, how despicable and ignoble war is; I would rather be torn to shreds than be a part of so a base of action. It is my conviction that killing under the cloak of war is nothing but an act of murder."

==Einstein on Germany and the rise of Hitler==
In 1944, in a message to a bulletin run by Polish Jews in New York, Einstein wrote
<blockquote>
The Germans as an entire people are responsible for the mass murders and must be punished as a people if there is justice in the world and if the consciousness of collective responsibility in the nations is not to perish from the earth entirely. Behind the Nazi party stands the German people, who elected Hitler after he had in his book [Mein Kampf] and in his speeches made his shameful [genocidal] intentions clear beyond the possibility of misunderstanding.
</blockquote>

== False attributions ==
Many ideas and quotes are falsely attributed to Einstein. He did not invent very much of we now call [[special relativity]]. The Principle of Relativity, that the law of physics should be the same in all inertial frames, had already been published before Einstein.<ref>[http://arxiv.org/abs/physics/0408077]</ref> He did not discover the [[Lorentz transformation]], or the Lorentz invariance of [[Maxwell's equations]] for electromagnetism.<ref>[http://eprint.uq.edu.au/archive/00002307/]</ref> He was not the first to propose that the [[speed of ligh]]t is constant for all observers, or that "[[the aether]]" is superfluous and not observable.<ref>Lost in Einstein's Shadow, American Scientist, 2006. [http://www.americanscientist.org/template/AssetDetail/assetid/49611/page/1]</ref> He was not the first to recognize and explain how special relativity causes an ambiguity in defining simultaneity.<ref>[http://www.everythingimportant.org/relativity/Poincare.htm]</ref> He did not combine space and time into a four-dimensional [[spacetime]] in his special relativity papers until others had been doing it for a couple of years.

Einstein was not the first to observe the equation E=mc2<ref>H. Poincare published it in 1900. [http://www.serve.com/herrmann/ein.htm]</ref> as a consequence of special relativity, or to foresee its application to [[nuclear binding]] energies or [[antimatter]] annihilation. He did not foresee a nuclear [[chain reaction]] and had to be persuaded about the possibility of an [[atomic bomb]]. However, while Poincare's version of E = mc2 was mathematically the same as Einstein's, [[Poincare]] couldn't explain this without referring to a fictitious fluid or an ether in order for the motion of the [[center of mass]] of energy to remain uniform.
Einstein's theory that a body losing energy through heat or [[radiation]] was losing mass is the one that's currently accepted.<ref>Einstein, A., 1905, [http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig?], Annalen der Physik, volume 18, pages 639–643. See also [http://www.fourmilab.ch/etexts/einstein/specrel/www English translation].</ref>

Einstein did not originate the idea of using [[metric tensor]]s to reconcile gravity with special relativity.<ref>[http://www.univ-nancy2.fr/DepPhilo/walter/papers/b4.pdf] It was not his idea to describe gravity in [[covariant equations]], and he published papers supposedly showing that it was impossible.<ref>[http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf]</ref> He did not discover the Lagrangian formulation of [[general relativity]], and was not the first to publish the field equations. He did not foresee the [[expansion of the universe]] (although this expansion is consistent with Einstein's equations), the possibility of black holes, or [[dark energy]].

Einstein never developed a [[unified field theory]] of any significance. He published papers on [[quantum mechanics]], but always insisted on a deterministic interpretation.

Many quotes are falsely attributed to Einstein. Here are a few. "If the bee disappeared off the surface of the globe then man would only have four years of life left." "We use only 10% of our brains." "Evil is simply the absence of God." "Originality is the art of concealing your sources." 

==References==
{{reflist|2}}

*"The Nobel Prize in Physics 1921"[http://nobelprize.org/nobel_prizes/physics/laureates/1921/]
*"On The Electrodynamics Of Moving Bodies"[http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf]
*"Does The Inertia Of A Body Depend Upon Its Energy-Content?"[http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
*Michael Paterniti, Driving Mr. Albert: A Trip Across America With Einstein's Brain
*"Collected Quotes from Albert Einstein" [http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html]
*"Confusion regarding Einstein's Religion"[http://onegoodmove.org/1gm/1gmarchive/2004/04/confusion.html]

{{DEFAULTSORT:Einstein, Albert}}

[[Category:Nobel laureates in Physics]]
[[Category:Physicists]]
[[Category:German People]]

Sine

2008-08-18T21:53:45Z

DiEb: obvious error

'''Sine''' is a [[trigonometry|trigonometric]] function.

<math>\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}\,</math>

==Sine Graph==
The basic sine graph is as follow:
{| style="background:none"
|
{| class="wikitable" style="text-align:center"
|-
!x values
!y values
|-
|width="50%"|0
|width="50%"|0
|-
|π/2
|1
|-
|π
|0
|-
|3π/2
| -1
|-
|2π
|0
|}
|[[Image:Sine2.gif|350px]]
|}
That would be a full period of the graph - as 2π [[radians]] is equivalent to one circle. The graph continues in that pattern.
[[Category:Mathematics]]

Imperial Diet

2008-08-18T21:22:45Z

DiEb: Holy Roman Empire of German Nation

The '''Imperial Diet''' was the assembly of the [[Holy Roman Empire]] from circa 1100 to 1806. After 1489 the Diet was divided into three chambers or colleges.

Seven ''Electors'' (lay and ecclesiastical [[prince]]s) formed the electoral (first) college and were charged with electing a new [[emperor]]. The second college, also of princes, was composed of the other noble rulers, both lay and ecclesiastical. The third college represented delegates from the Imperial towns.

With the collapse of central authority during the [[Thirty Years' War]], the Diet lost its legislative powers and became a largely ceremonial body.

[[Category:German History]]

Talk:Limit (mathematics)

2008-08-18T21:00:51Z

DiEb: /* Funtion vs. Sequence */

who writes this stuff??--[[User:Lemonpeel|Lemonpeel]] 21:41, 3 July 2008 (EDT)

The [[math]] articles were written by random editors of [[conservapedia]], I actually liked the earlier version of the article as it was easier to understand that the current version -- [[Image:50 star flag.png|12px]] [[User:Deborah|Deborah]] [[Special:Contributions/Deborah|(contributions)]] [[User_talk:Deborah|(talk)]] 22:05, 3 July 2008 (EDT)

:Deborah while the previous article was easier to read, it was easier to read because it was wrong. [[User:DanielB|DanielB]] 00:04, 4 July 2008 (EDT)

== Function vs. Sequence ==

''Function should be first. It is more fundamental, even if it might be considered harder.''
I'd like to object. The idea of the limit of a sequence is the more basic one - and it's taught first, I suppose. Generally, you introduce the limit of a function in a point via the limit of sequences: What does it mean (in your example) that <math>x \rightarrow 3</math>? It means, you look at all sequences <math>(x_1, x_2,...)</math> with <math>\lim x_n = x</math>. And if for all these sequences, the corresponding sequence <math>(f(x_1),f(x_2), ...)</math> converges to the same value <math>z</math>, you'd say that the function has limit in x.
But without this underlying concept, the expression <math> x \rightarrow 3</math> is unmotivated and baseless. --[[User:DiEb|DiEb]] 18:11, 17 August 2008 (EDT)

Talk:Limit (mathematics)

2008-08-17T22:11:30Z

DiEb: /* Funtion vs. Sequence */ new section

who writes this stuff??--[[User:Lemonpeel|Lemonpeel]] 21:41, 3 July 2008 (EDT)

The [[math]] articles were written by random editors of [[conservapedia]], I actually liked the earlier version of the article as it was easier to understand that the current version -- [[Image:50 star flag.png|12px]] [[User:Deborah|Deborah]] [[Special:Contributions/Deborah|(contributions)]] [[User_talk:Deborah|(talk)]] 22:05, 3 July 2008 (EDT)

:Deborah while the previous article was easier to read, it was easier to read because it was wrong. [[User:DanielB|DanielB]] 00:04, 4 July 2008 (EDT)

== Funtion vs. Sequence ==

''Function should be first. It is more fundamental, even if it might be considered harder.''
I'd like to object. The idea of the limit of a sequence is the more basic one - and it's taught first, I suppose. Generally, you introduce the limit of a function in a point via the limit of sequences: What does it mean (in your example) that <math>x \rightarrow 3</math>? It means, you look at all sequences <math>(x_1, x_2,...)</math> with <math>\lim x_n = x</math>. And if for all these sequences, the corresponding sequence <math>(f(x_1),f(x_2), ...)</math> converges to the same value <math>z</math>, you'd say that the function has limit in x.
But without this underlying concept, the expression <math> x \rightarrow 3</math> is unmotivated and baseless. --[[User:DiEb|DiEb]] 18:11, 17 August 2008 (EDT)

Talk:Real number

2008-08-17T21:52:19Z

DiEb:

==Moved comment from Anjruu==
The introduction is actually a blatant misrepresentation of the truth. Real numbers are no more or less "abstract" than imaginary numbers. The author, one must conclude, was trying to allude to the fact that there cannot be an imaginary number of objects, but there cannot be a negative number of objects either. Or perhaps s/he was trying to say that the imaginary numbers cannot be represented on the number line, which is irrelevant, since they can be represented on the complex plane, an equally valid, and more complete, numerical device. Finally, perhaps the author simply meant that the square root of a negative number does not exist, which is wrong. It does exist, in the guise of an imaginary number. While this may seem cyclical, simply because a number is "unusual" does not mean that it is abstract and non-existent.

Illustrations of this fact, and the incorrectness of the introduction, can be seen when solving differential equations which contain a bifurcation parameter, where the method of eigenvalues must be used. There, an complex, "imaginary" solution to one of several equations often contains eigenvalues and thus a corresponding eigenfunction. Therefore, the real-life existent solution to a real-life existent problem depends on an imaginary number. This exemplifies why complex numbers are just as valid as real numbers, and are not merely an "abstract concept," as the introduction erroneously points out. Or to be more precise, ALL of mathematics is an abstract concept, but there exists an isomorphism to the "real world"; an isomorphism that holds just as strongly in the domain of complex numbers as in the subset of complex numbers, the reals.

----

==The "real" world==
Re: this sentence from the intro:
<blockquote>
''They are called "real" because they actually exist as quantities in the real world, i.e. as measures, weights, temperatures. In contrast, the imaginary numbers are just an abstract concept, but do not exist concretely in the real world.''
</blockquote>
The bulk of that statement simply isn't correct, which is why I removed it earlier. Historically, it was thought that imaginary numbers didn't "really exist," but for the last 200 years or so it's been understood that they do; I believe Cauchy and Gauss were heavily involved in putting that question to bed. Furthermore, there are indeed physical applications of complex numbers. Anything with a magnitude and phase (think alternating current, any harmonic oscillator, communications signals, etc.) is conveniently represented by complex numbers, and they are absolutely essential in quantum mechanics.
:: I second this motion: this sentence should be erased. ''Real numbers'' are as real - or imaginary - as complex numbers. --[[User:DiEb|DiEb]] 15:41, 15 August 2008 (EDT)
'''EDIT:''' I've altered the sentence in question to reflect my comment.--[[User_talk:Recorder|Recorder]] 15:30, 15 August 2008 (EDT)

==Again: Real vs. Imaginary==
The statement ''The term "real number" is in contrast to the imaginary numbers.'' (RSchlafly) is IMO a false dichotomy and shouldn't be put in the article.
I like the sentence: ''In classical physics, measurements of real world physical objects that can vary smoothly and continuously, like speed or temperature, are treated as real numbers. '' as it doesn't presuppose the existence of real numbers, but - analogously - you can state: ''In physics, measurements of real world physical objects that can vary smoothly and continuously - and exist from an angular compound and an absolute value - like current or voltage, are treated as complex numbers.'' --[[User:DiEb|DiEb]] 16:30, 16 August 2008 (EDT)

: I don't understand your point. The terms ''real number'' and ''imaginary number'' are in common use. Are you denying that? Are you saying that they are not in contrast? What do you want? [[User:RSchlafly|RSchlafly]] 18:35, 16 August 2008 (EDT)

:: Of course, I don't deny the terms. And I acknowledge that the ''terms'' - which can be understood in their historical context - are contrasting. But I don't like the idea to think of the ''real numbers'' as more '''real''' than the ''imaginary numbers'' - especially not because of an etymological reasoning: To quote Kronecker: ''"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"''--[[User:DiEb|DiEb]] 16:47, 17 August 2008 (EDT)
:::Huh? Real numbers are those people concretely use. You can express any amount of dollars with real numbers. What would 3 + 2i dollars mean? Or the length of a segment or a curve: it is "real" and expressed in real numbers. Imaginary numbers are just ''imaginary'' constructions to help our calculations. They don't refer to "real" concepts people use. Thus, the historical meaning of real vs. imaginary is still valid for normal people - and homeschooled students who use Conservapedia to learn. I absolutely agree with RSchlafly. [[User:SilvioB|SilvioB]] 16:53, 17 August 2008 (EDT)
::::Actually, you can express any amount of currency with rational numbers. That doesn't make the real numbers any less "real," does it? Using your logic, I could say "Pi is a real number. What does pi dollars mean? Nothing. Therefore, the real numbers don't exist." Again, imaginary numbers are ''not'' "just imaginary." They exist just as much as any other number, and have several practical applications.--[[User_talk:Recorder|Recorder]] 17:21, 17 August 2008 (EDT)
::::For currency, you use a subset of the rational numbers, i.e., finite decimal fractions (even less, generally: <math>\mathbb{Z} / 100</math>. To speak of <math>\pi</math> dollars (or even $<math>\sqrt{2}</math>) is as surreal as to speak of ''3 + 2i'' dollars. ''Real numbers'' which aren't rational, are just a figment of our imagination, invented to ''help our calculations'', too--[[User:DiEb|DiEb]] 17:25, 17 August 2008 (EDT)
:::::I realized that and that's why I gave a second example, that of the length of a segment. Do you people really consider real numbers and imaginary numbers equal from the point of view of normal people? Let's not forget who is the target of this project. [[User:SilvioB|SilvioB]] 17:30, 17 August 2008 (EDT)
::::::Presumably the purpose of articles like this is to provide accurate information, not to confirm misconceptions caused by terminology. Plenty of people use complex numbers regularly, and just because many people don't isn't a reason to have an inaccuracy in the article.--[[User_talk:Recorder|Recorder]] 17:42, 17 August 2008 (EDT)
::::::In the physical world, it is not possible to find a line segment of length <math>\pi cm</math> (or even of length 1 cm), you may find approximations. For ''normal people'', this may be of no interest. So, for them, we can skip the hole ''real is more real than imaginary'' stuff. We shouldn't mislead more interested pupils by this entry: in mathematics, you go from the simple to the more complex, but not from the ''real'' to the ''surreal''. --[[User:DiEb|DiEb]] 17:52, 17 August 2008 (EDT)

Talk:Real number

2008-08-17T21:25:44Z

DiEb:

Talk:Real number

2008-08-17T21:25:18Z

DiEb:

==Moved comment from Anjruu==
The introduction is actually a blatant misrepresentation of the truth. Real numbers are no more or less "abstract" than imaginary numbers. The author, one must conclude, was trying to allude to the fact that there cannot be an imaginary number of objects, but there cannot be a negative number of objects either. Or perhaps s/he was trying to say that the imaginary numbers cannot be represented on the number line, which is irrelevant, since they can be represented on the complex plane, an equally valid, and more complete, numerical device. Finally, perhaps the author simply meant that the square root of a negative number does not exist, which is wrong. It does exist, in the guise of an imaginary number. While this may seem cyclical, simply because a number is "unusual" does not mean that it is abstract and non-existent.

Illustrations of this fact, and the incorrectness of the introduction, can be seen when solving differential equations which contain a bifurcation parameter, where the method of eigenvalues must be used. There, an complex, "imaginary" solution to one of several equations often contains eigenvalues and thus a corresponding eigenfunction. Therefore, the real-life existent solution to a real-life existent problem depends on an imaginary number. This exemplifies why complex numbers are just as valid as real numbers, and are not merely an "abstract concept," as the introduction erroneously points out. Or to be more precise, ALL of mathematics is an abstract concept, but there exists an isomorphism to the "real world"; an isomorphism that holds just as strongly in the domain of complex numbers as in the subset of complex numbers, the reals.

----

==The "real" world==
Re: this sentence from the intro:
<blockquote>
''They are called "real" because they actually exist as quantities in the real world, i.e. as measures, weights, temperatures. In contrast, the imaginary numbers are just an abstract concept, but do not exist concretely in the real world.''
</blockquote>
The bulk of that statement simply isn't correct, which is why I removed it earlier. Historically, it was thought that imaginary numbers didn't "really exist," but for the last 200 years or so it's been understood that they do; I believe Cauchy and Gauss were heavily involved in putting that question to bed. Furthermore, there are indeed physical applications of complex numbers. Anything with a magnitude and phase (think alternating current, any harmonic oscillator, communications signals, etc.) is conveniently represented by complex numbers, and they are absolutely essential in quantum mechanics.
:: I second this motion: this sentence should be erased. ''Real numbers'' are as real - or imaginary - as complex numbers. --[[User:DiEb|DiEb]] 15:41, 15 August 2008 (EDT)
'''EDIT:''' I've altered the sentence in question to reflect my comment.--[[User_talk:Recorder|Recorder]] 15:30, 15 August 2008 (EDT)

==Again: Real vs. Imaginary==
The statement ''The term "real number" is in contrast to the imaginary numbers.'' (RSchlafly) is IMO a false dichotomy and shouldn't be put in the article.
I like the sentence: ''In classical physics, measurements of real world physical objects that can vary smoothly and continuously, like speed or temperature, are treated as real numbers. '' as it doesn't presuppose the existence of real numbers, but - analogously - you can state: ''In physics, measurements of real world physical objects that can vary smoothly and continuously - and exist from an angular compound and an absolute value - like current or voltage, are treated as complex numbers.'' --[[User:DiEb|DiEb]] 16:30, 16 August 2008 (EDT)

: I don't understand your point. The terms ''real number'' and ''imaginary number'' are in common use. Are you denying that? Are you saying that they are not in contrast? What do you want? [[User:RSchlafly|RSchlafly]] 18:35, 16 August 2008 (EDT)

:: Of course, I don't deny the terms. And I acknowledge that the ''terms'' - which can be understood in their historical context - are contrasting. But I don't like the idea to think of the ''real numbers'' as more '''real''' than the ''imaginary numbers'' - especially not because of an etymological reasoning: To quote Kronecker: ''"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"''--[[User:DiEb|DiEb]] 16:47, 17 August 2008 (EDT)
:::Huh? Real numbers are those people concretely use. You can express any amount of dollars with real numbers. What would 3 + 2i dollars mean? Or the length of a segment or a curve: it is "real" and expressed in real numbers. Imaginary numbers are just ''imaginary'' constructions to help our calculations. They don't refer to "real" concepts people use. Thus, the historical meaning of real vs. imaginary is still valid for normal people - and homeschooled students who use Conservapedia to learn. I absolutely agree with RSchlafly. [[User:SilvioB|SilvioB]] 16:53, 17 August 2008 (EDT)
::::Actually, you can express any amount of currency with rational numbers. That doesn't make the real numbers any less "real," does it? Using your logic, I could say "Pi is a real number. What does pi dollars mean? Nothing. Therefore, the real numbers don't exist." Again, imaginary numbers are ''not'' "just imaginary." They exist just as much as any other number, and have several practical applications.--[[User_talk:Recorder|Recorder]] 17:21, 17 August 2008 (EDT)
::::Actually, you can express any amount of currency with rational numbers. That doesn't make the real numbers any less "real," does it? Using your logic, I could say "Pi is a real number. What does pi dollars mean? Nothing. Therefore, the real numbers don't exist." Again, imaginary numbers are ''not'' "just imaginary." They exist just as much as any other number, and have several practical applications.--[[User_talk:Recorder|Recorder]] 17:21, 17 August 2008 (EDT)

Talk:Kreisau Circle

2008-08-17T21:10:45Z

DiEb: New page: ==Count vs. Graf== If you prefer the English titles, the name should be ''Helmuth James count von Moltke'', not ''count Helmuth James von Moltke'': Since 1918, Graf (count) is a part of t...

==Count vs. Graf==

If you prefer the English titles, the name should be ''Helmuth James count von Moltke'', not ''count Helmuth James von Moltke'': Since 1918, Graf (count) is a part of the name in Germany, and not longer a title. So, while he was born as ''Graf James v. Moltke'', the name became ''James Graf v. Moltke'' after the Great War. --[[User:DiEb|DiEb]] 17:10, 17 August 2008 (EDT)

Kreisau Circle

2008-08-17T21:06:11Z

DiEb: count vs. graf

The '''Kreisau Circle''' (German: ''Kreisauer Kreis'') was a [[Germany|German]] anti-[[Nazi]] resistance group, founded in 1940 by Helmuth James Graf von Moltke (11 March 1907 – 23 January 1945) on his estate at Kreisau. The Circle drafted a [[Christian]]-[[conservative]] [[politics|political]] program for post-[[war]] Germany.

Von Moltke was arrested in January 1944 after which most of his colleagues attached themselves to [[Claus von Stauffenberg|Colonel Claus Schenk Graf von Stauffenberg ]]'s group which tried, unsuccessfully, to kill [[Adolf Hitler|Hitler]] in July 1944.

[[Category:History]]
[[Category:Germany]]
[[Category:Conservative]]
[[Category:Christians]]
[[Category:War]]

Talk:Derivative (calculus)

2008-08-17T21:01:29Z

DiEb: /* Integrability */ new section

What a surprise--once again, a page in desperate need of revision. I'll do my best to delete absurd and meaningless statements, but someone else needs to come here and do a complete rewrite.--[[User:Mathoreilly|Mathoreilly]] 13:42, 2 July 2008 (EDT)

So the page is at least a little less wrong now, but someone still needs to do something about the max/min section which is, to put it bluntly, very poorly written.--[[User:Mathoreilly|Mathoreilly]] 13:53, 2 July 2008 (EDT)

:Someone other than you, apparently. Sorry, but I think a red link would be better than what you left us with.

:I have taught SAT math and I earned AP credit for calculus. This was simply unacceptable quality. Do not edit any more math articles. --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 14:41, 2 July 2008 (EDT)

I have taken this out as suggested by Mathoeilly as it needs work, possiblly a different article,

Thus the derivative is a measurement of how a function changes when the values of its inputs vary. Derivatives are helpful in determining the [[maximum|maxima]] and [[minimum|minima]] of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called ''critical'' points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the ''second derivative test''. The second derivative is used to determine the [[concavity]], or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of ''inflection,'' meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.

==The current article==
Okay I will make a list of questions about the article
#In what contacts do you mean "measure"? Measure has a very specific mathematical meaning how do you mean it.
#You link to tangent but if you look at the tangent article it discusses the tangent function and even then not in a way you can under stand the tangent function.
#"differentiation is the reverse process of integration" In all my years of doing maths I have never heard anyone use the word process unless they are discussing and actual physical process. I think the term you are looking for is operation.
#You then go on to talk about maxima and minima. This is all good but as we at now stage have found out anything about how to differentiate it is usless to the reader.
This article would be greatly help with some actual maths and some pictures (as no one who knows anything about maths have upload rights it is not going to happen.
I would make these changes myself but every time I have made one minor change to this article it is deleted or reverted ny Ed Poor, so I am not going to bother. This article is in no way going to help with your stated aims of creating a home schooling resource, but that is okay as I am not going to help either. [[User:DanielB|DanielB]] 19:40, 4 August 2008 (EDT)

== Integrability ==

The notion ''"Differentiability implies continuity as well as integrability"'' is at least misleading. You want to say - I suppose - that, if a function is differentiable (and therefore continuous) in a point (and therefore in a neighborhood of this point), then there exists a primitive for this function on this neighborhood.
Integrability of a function <math>f(x)</math> on <math>\mathbb{R}</math> OTOH is commonly understood to be the existence of <math>\int_{\mathbb{R}}f(x) dx</math> - and the simple example of <math>f(x)= \sin(x)</math> shows that the Integral of this differentiable function does not exist.--[[User:DiEb|DiEb]] 17:01, 17 August 2008 (EDT)

Talk:Real number

2008-08-17T20:47:53Z

DiEb: /* Again: Real vs. Imaginary */

Talk:Real number

2008-08-16T20:30:14Z

DiEb: /* The "real" world */

Metric system

2008-08-15T20:24:54Z

DiEb: /* Volume */ previous definition was just wrong (see reference)

The [[metric system]] is a system of measurement developed during the [[French Revolution's]] anti-religious [[The Cult of Reason|Cult of Reason]]<ref>However, according to researcher Pat Norton, the system was described a century earlier by Englishman John Wilkins. [http://abc.net.au/news/stories/2007/07/15/1978802.htm Australian Broadcasting Corporation], 15 July 2007 (retrieved 15 July 2007.)</ref> and based on permanent natural standards rather than on royal decrees. [[Joseph Lagrange]] headed the Revolutionary weights-and-measures committee that developed it and set its initial definitions.

Traditional systems of measurement all used standard lengths of parts of the human body, or standard distances that a human might walk. The metric system, by contrast, initially used the [[earth]] itself as the ultimate standard.

In 1878 the [[United States]] signed the treaty called the ''Convention du Mètre.'' Since then, ''both'' the metric system and the [[United States customary units]] (miles, inches, pounds, etc.) have had legal status in the US. US Customary units are in fact defined by reference to metric units; that is, legally, an inch is ''defined'' to be exactly 25.4 millimeters.

In international scientific usage, the ''Convention du Mètre'' is superseded by the conventions of the [[International System of Units]]. However, many of the conventions of the metric system survive, including definitions of units that are not, strictly speaking, SI.

== Definitions ==
All measurement systems must start with one basic unit. For the metric system, this unit is the ''meter''. Every other named unit of the metric system had a definition that depended upon it.

=== Prefixes ===
All base units of the metric system have ''prefixes'' that multiply, or divide, the unit by various factors of ten (or alternatively, a thousand). They are:<ref>Wile, Dr. Jay L. ''Exploring Creation With Chemistry''. Apologia Educational Ministries, Inc. 1998</ref><ref>[http://lamar.colostate.edu/~hillger/common.html Common metric prefixes]</ref>

{| class="wikitable"
|-
! The prefix:
! Abbreviated:
! Multiplies the unit by:
|-
| atto-
| a
| 10-18
|-
| femto-
| f
| 10-15
|-
| pico-
| p
| 10-12
|-
| nano-
| n
| 10-9
|-
| micro- ([[Greek]] ''small'')
| μ
| 10-6
|-
| milli- ([[Latin]] ''one thousand'')
| m
| 10-3
|-
| centi- (Latin ''one hundred'')
| c
| 10-2
|-
| deci- (Latin ''ten'')
| d
| 10-1
|-
| deka- (Greek ''ten'')
| da
| 101
|-
| hecto- (Greek ''one hundred'')
| H
| 102
|-
| kilo- (Greek ''one thousand'')
| k
| 103
|-
| mega- (Greek ''large'')
| M
| 106
|-
| giga- (Latin ''gigantic'')
| G
| 109
|-
| tera- (Greek ''wonder'')
| T
| 1012
|}

Other, related systems (see below) use the same prefixes to multiply their units.

=== Length ===
The base unit of length, and originally the unit from which all other definitions flowed, is the [[meter]]. The original definition was "one ten-millionth the distance from the equator to the north pole." Lagrange's surveyors computed this distance (probably using a variation on Eratosthenes' tropical-shadow experiment) and inscribed it on a metal bar that they kept at the weights-and-measures headquarters building in Sèvres, [[France]].

But this artifact, being made of metal, tended to expand and contract. Centuries later, when the scientific community realized what had happened, the task of re-standardizing all meter sticks to the equator-to-pole-times-10-7 definition was judged too onerous. Therefore the meter was redefined. At first it was defined as 1,650,763.73 wavelengths of the orange-red line in the excitation spectrum of [[krypton]]-86. Today it is defined as 1⁄299,792,458th of a light-second.

The meter in all its multiples replaces all other units of measurement, long and short. Long distance is measured in the meter multiplied by a thousand--the kilometer. Short distances are measured in centimeters--or millimeters, or micrometers, or however small one needs to measure them.

=== Volume ===
The base unit of volume is derived from the meter: it's the ''cubic meter'' [m3]. Traditionally, the ''liter''<ref>http://www.bipm.org/en/si/si_brochure/chapter4/table6.html</ref> is used. 1 liter is defined as 10-3 m3 = 1 dm3.

=== Mass ===
The base unit of mass is the ''gram''. Originally a gram was the mass of one cubic centimeter (see above), then called a ''milliliter'', of [[water]]. The Lagrange committee fashioned an artifact exactly one thousand times as heavy as this--the volume, in short, of one liter of water--and stored this permanently at Sèvres.

[[Joseph Lagrange|Lagrange]]'s committee also proposed a unit of mass called the ''grave'', to be equal to one thousand grams. But the [[French Revolution|Revolutionary]] government rejected this name as sounding too much like the [[German]] word ''graf'', translated as ''comte'' or [[count]] in French. Therefore, the artifact at Sèvres would be called a kilogram instead--and that artifact has that name to this day.

Mass never changes. But in addition to the prototype meter changing its length over time, scientists later realized that water has different densities at different [[temperature|temperatures]].

Today, the standard [[kilogram]] is kept at the International Bureau of Standards headquarters in Sèvres. It remains today the only generally accepted unit of measurement based on an artifact. But the definition of the liter has now changed, and is no longer dependent on the length of the meter. Today, one liter is defined as the volume of one kilogram of water at four degrees [[Celsius]]. One milliliter is simply one-thousandth of this, and one milliliter of water at this temperature weighs one gram. But to say that one liter is "one thousand cubic centimeters" is no longer correct, simply because the definition of length has changed, while that of mass has not.

== Relation to other systems of measurement ==
The [[International System of Units]] borrows some definitions and concepts from the metric system--most notably the meter and the kilogram--and adds other definitions of other quantities, most notably [[time]], [[force]], [[amount of substance]], plane [[angle]], solid angle, and [[luminous intensity]]. The liter ''is not'' the unit of volume in that system, however, but liter is accepted for use with the SI.

In addition to the International System (or in French, ''Systèm International'', or SI for short), scientists have invented other systems of measurement that include units of time, force, [[energy]], and electric charge. The most popular are the meter-kilogram-second and centimeter-gram-second systems.

Various governing authorities have established conversion ratios to assist people in converting between the metric system and other, more traditional units of measurement. The definitions of US Customary units in the United States is an example of this.

== References ==
<references/>

== See Also ==
* [http://creationwiki.org/Metric_system Metric system] by [[CreationWiki]]
* [http://www.essex1.com/people/speer/metric.html Metric system] by Essex1.com
* [http://lamar.colostate.edu/~hillger/ The United States Metric Association]

[[Category:Science]]
[[Category:Units of Measurement]]

Talk:Real number

2008-08-15T19:41:54Z

DiEb:

User talk:Tolkiendil

2008-08-14T20:07:40Z

DiEb: /* Compulsory Education in Germany */ new section

{{welcome}}

== Compulsory Education in Germany ==

I'm afraid your ''tiny correction'' is wrong: The compulsory education in [[Germany]] takes ten years (''Hauptschule'', ''Realschule''). Pupils may go on for two or three years (''Gymnasium'') - according to the ''Land'' they live in (and as always, it's a little bit more complicated ) --[[User:DiEb|DiEb]] 16:07, 14 August 2008 (EDT)

Werner Heisenberg

2008-08-14T19:44:50Z

DiEb:

[[Image:200.jpg|right|frame]]
'''Werner Karl Heisenberg''' (December 5, 1901 – February 1, 1976) was a [[German]] physicist and [[Nobel]] laureate (1932), most credited with the discovery of [[quantum mechanics]], in 1925. Heisenberg proposed that it is impossible to assign to an electron a position in space at a given time or follow it precisely in its orbit. Heisenberg thereby challenged the model of subatomic particles that was based on planetary orbits. Instead, Heisenberg suggested that mechanical quantities like position and velocity are better represented, not by ordinary [[commutative]] [[number]]s, but by non-commutative mathematical objects called [[matrix|matrices]].

Heisenberg formulated a famous principle that now bears his name: The [[Heisenberg Uncertainty Principle]] (1927), one of the central principles of modern [[physics]]. This principle states that the determination of the position and momentum of a mobile particle always contains errors which, when multiplied, cannot be less than the quantum constant h. These errors are negligible in most of ordinary life but not only significant in the study of subatomic particles. Quantum theory is essential for a valid description of molecules including pharmaceuticals as well as for the description of physical properties (electrical conductivity, magnetism, ...) of solids, in particular semiconductors. Some modern equipment, such as computer chips, lasers or the scanning tunneling microscope, take advantage of effects, which can only be understood through quantum theory.

The need to give up the certainty of simple mechanical and other models and to introduce the uncertainty principle in order to understand and describe some physical effects also inspired to question the certainty in other fields.

==World War II==
During world war II Werner Heisenberg worked on developing the atomic bomb for the Nazi party. After the defeat of the Nazi regime, the American government offered Werner Heisenberg a position at Princeton University instead of charging him with war crime, which he accepted.

== Awards ==

''Apart from many medals and prizes, Heisenberg received an honorary doctorate of the University of Bruxelles, of the Technological University Karlsruhe, and of the University of Budapest (1964); he is also recipient of the Order of Merit of Bavaria, and the Grand Cross for Federal Services with Star (Germany).'' [http://nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-bio.html]

{{DEFAULTSORT:Heisenberg, Werner}}
[[Category:Physicists]]
[[Category:German People]]

Talk:Imperial Diet

2008-08-14T18:28:14Z

DiEb: New page: ==Central Authority== What do you mean by ''With the collapse of central authority during the Thirty Years' War''? There wasn't much of a central authority before 1618... --~~~~

==Central Authority==
What do you mean by ''With the collapse of central authority during the Thirty Years' War''? There wasn't much of a central authority before 1618... --[[User:DiEb|DiEb]] 14:28, 14 August 2008 (EDT)

Imperial Diet

2008-08-14T18:26:19Z

DiEb: 1489, not 1849...

Talk:Frederick the Great

2008-08-14T18:22:39Z

DiEb:

==King in Prussia==
F. was ''King in Prussia'' until 1772, when he declared himself as ''King of Prussia''. There is quite a bit of history hidden behind these prepositions --[[User:DiEb|DiEb]] 11:45, 14 August 2008 (EDT)

:And what is that history, pray tell? I only know that "King of Prussia" is a town in [[Pennsylvania]]. -[[User:AnitaK|AnitaK]] 12:37, 14 August 2008 (EDT)

::Fredericks's granddad - also a Frederick - was the ''markgraf'' (margrave/marquis) of Brandenburg, one of the prince-electors of the German empire. Though this was only one step below the emperor, he aspired to be a king, but this wasn't easy inside the German empire: the only kingdom there was Bohemia. Some of his fellow electors were kings, but of territories '''outside''' of the German empire: the prince-elector of Saxony was for a while king of Poland and so on...
::So, Frederick had to persuade (i.e., bribe) quite a few parties (e.g., the emperor) to get his title. But he couldn't be the ''King of Brandenburg'' (that's the area around Berlin, btw.), as Brandenburg lies inside the German empire. The only areal outside of the empire which Frederick owned was East Prussia, ... the problem: here, the king of Poland was his liege. More money to pay...
::So, he strove to be ''King of Prussia''... But he owned only East Prussia, West Prussia was an integral part of Poland. The compromise: He became ''King in Prussia''. In 1772, Poland was divided - again, and not for the last time - and the king in Prussia got the rest of Prussia, declaring himself the king of Prussia...

::The hole affair was very costly - and somewhat disputed: the Holy See - for instance - didn't acknowledge the title for some hundred years and went on to address its letters to the ''marquis de Brandenbourg''...
::BTW, I don't know anything about this town in Pennsylvania --[[User:DiEb|DiEb]] 14:22, 14 August 2008 (EDT)

Talk:Max Planck

2008-08-14T16:01:37Z

DiEb:

his name is max plank, not planck.[[User:Jaques|Jaques]] 23:50, 12 May 2007 (EDT)
::Max Planck is correct!
--[[User:DiEb|DiEb]] 11:58, 14 August 2008 (EDT)

==Quanta==
"Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator ''''could emit'''' energy in discrete units, or quanta."

Better:
Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator ''''would emit'''' energy in discrete units, or quanta.

or even:
Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator ''''emits'''' energy in discrete units, or quanta.

--[[User:DiEb|DiEb]] 12:01, 14 August 2008 (EDT)

Talk:Max Planck

2008-08-14T15:58:07Z

DiEb:

his name is max plank, not planck.[[User:Jaques|Jaques]] 23:50, 12 May 2007 (EDT)
Max Planck is correct! --[[User:DiEb|DiEb]] 11:58, 14 August 2008 (EDT)

Max Planck

2008-08-14T15:57:26Z

DiEb:

[[Image:Max planck.jpg|right|thumb|150px|Max Planck]]
'''Max Planck''' (Kiel April 23, 1858 - Göttingen October 4, 1947) was a great [[Germany|German]] physicist. He advanced the subjects of [[thermodynamics]], [[thermoelectricity]], and the theory of [[dilute solutions]], and was one of the founders of [[quantum mechanics]].

Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator could emit energy in discrete units, or quanta. The energy emitted by a resonator of frequency '''''ν''''' is '''''hν''''', where "'''''h'''''" is a universal constant that is now called Planck's constant. This explanation resolved important discrepancies between theory and observations, and helped lay a foundation for the development of [[quantum mechanics]].

Planck's research in quantum theory was published in the ''Annalen der Physik'', and his work was summarized in the books ''Thermodynamik'' (Thermodynamics) (1897) and ''Theorie der Wärmestrahlung'' (Theory of heat radiation) (1906).<ref> [http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-bio.html Biography]</ref>

Planck was awarded the [[Nobel Prize]] for physics in 1918.<ref>The Earth and Its Peoples A Global History, Bulliet et al, 2005</ref>

Planck's son was executed in 1945 for his part in an unsuccessful attempt to assassinate [[Adolf Hitler|Hitler]] in 1944.

==References==

<References/>

{{DEFAULTSORT: Planck, Max}}
[[category:physicists]]
[[Category:German People]]

Max Planck

2008-08-14T15:56:32Z

DiEb: April 23, 1858 – October 4, 1947

[[Image:Max planck.jpg|right|thumb|150px|Max Planck]]
'''Max Planck''' (Kiel April 23, 1858 - Göttingen October 4, 1947) was a great [[Germany|German]] physicist. He advanced the subjects of [[thermodynamics]], [[thermoelectricity]], and the theory of [[dilute solutions]], and was one of the founders of [[quantum mechanics]].

Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator could emit energy in discrete units, or quanta. The energy emitted by a resonator of frequency '''''ν''''' is '''''hν''''', where "'''''h'''''" is a universal constant that is now called Planck's constant. This explanation resolved important discrepancies between theory and observations, and helped lay a foundation for the development of [[quantum mechanics]].

Planck's research in quantum theory was published in the ''Annalen der Physik'', and his work was summarized in the books ''Thermodynamik'' (Thermodynamics) (1897) and ''Theorie der Wärmestrahlung'' (Theory of heat radiation) (1906).<ref> [http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-bio.html Biography]</ref>

Planck was award the [[Nobel Prize]] for physics in 1918.<ref>The Earth and Its Peoples A Global History, Bulliet et al, 2005</ref>

Planck's son was executed in 1945 for his part in an unsuccessful attempt to assassinate [[Adolf Hitler|Hitler]] in 1944.

==References==

<References/>

{{DEFAULTSORT: Planck, Max}}
[[category:physicists]]
[[Category:German People]]

Max Planck

2008-08-14T15:54:59Z

DiEb: Category:German People

[[Image:Max planck.jpg|right|thumb|150px|Max Planck]]
'''Max Planck''' (Kiel 1858 - Göttingen 1947) was a great [[Germany|German]] physicist. He advanced the subjects of [[thermodynamics]], [[thermoelectricity]], and the theory of [[dilute solutions]], and was one of the founders of [[quantum mechanics]].

Planck's revolutionary discovery, one of the most influential in all of science, was that a resonator could emit energy in discrete units, or quanta. The energy emitted by a resonator of frequency '''''ν''''' is '''''hν''''', where "'''''h'''''" is a universal constant that is now called Planck's constant. This explanation resolved important discrepancies between theory and observations, and helped lay a foundation for the development of [[quantum mechanics]].

Planck's research in quantum theory was published in the ''Annalen der Physik'', and his work was summarized in the books ''Thermodynamik'' (Thermodynamics) (1897) and ''Theorie der Wärmestrahlung'' (Theory of heat radiation) (1906).<ref> [http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-bio.html Biography]</ref>

Planck was award the [[Nobel Prize]] for physics in 1918.<ref>The Earth and Its Peoples A Global History, Bulliet et al, 2005</ref>

Planck's son was executed in 1945 for his part in an unsuccessful attempt to assassinate [[Adolf Hitler|Hitler]] in 1944.

==References==

<References/>

{{DEFAULTSORT: Planck, Max}}
[[category:physicists]]
[[Category:German People]]

Martin Luther

2008-08-14T15:47:02Z

DiEb: Category:German People

{{Christianity}}
'''Martin Luther''' (November 10, 1483 - February 18,1546) was the leader of the German [[Reformation]].
He was a Monk and Theologian whose writing sparked the [[Protestant Reformation]] which led to the founding of the [[Lutheran Church]] as an alternative to the [[Roman Catholic Church]].

In 1505, Martin Luther was a law student caught outside in the middle of a thunderstorm. A bolt of [[lightning]] struck the ground near him. Terrified, he cried out to St. Anne for protection and promised to become a [[monk]] if he survived the storm. He did survive, and then left law school to join an Augustinian monastery.

At the monastery, Martin Luther felt plagued by the [[devil]] and began to realize how corrupt the [[Catholic church]] had become, being especially disenchanted during a trip to [[Rome]]. He began to form his own opinions about the [[Bible]].

Martin Luther founded [[Protestantism]] on October 31, 1517, when he nailed [[the 95 Theses]] or issues to the door of his [[Wittenberg Church]] in what is now Germany. The source of Luther's complaint was a project to rebuild [[St. Peter's Basilica]] in Rome, for which church officials were offering [[indulgences]] in exchange for donations. Martin Luther objected and produced a list of 95 complaints to spark public debate.

Here were two of Luther's most important Theses:

*Thesis 82a: ''Scilicet. Cur Papa non evacuat purgatorium propter sanctissimam charitatem et summam animarum necessitatem ut causam omnium iustissimam, Si infinitas animas redimit propter pecuniam funestissimam ad structuram Basilice ut causam levissimam?''

*Thesis 82: ''Why does not the pope empty purgatory, for the sake of holy love and of the dire need of the souls that are there, if he redeems an infinite number of souls for the sake of miserable money with which to build a Church? The former reasons would be most just; the latter is most trivial.''

*Thesis 86a: ''Item. Cur Papa, cuius opes hodie sunt opulentissimis Crassis crassiores, non de suis pecuniis magis quam pauperum fidelium struit unam tantummodo Basilicam sancti Petri?''

*Thesis 86: ''Why does not the pope, whose wealth is to-day greater than the riches of the richest, build just this one church of St. Peter with his own money, rather than with the money of poor believers?''

[[Pope Leo X]] spent his time on the beauty flourishing during the [[Renaissance]], and was a patron of the great artist [[Raphael]]. Pope Leo X probably never recognized the significance of Martin Luther and his followers. Some historians feel that if the pope had responded more seriously, then the subsequent schism in Germany would not have occurred.

But German nobles did not like the flow of money to Rome, and they backed Martin Luther in order to cut off the donations by Germans to support Rome.

In 1520, Pope Leo X announced the papal bull [[Exsurge Domine]], which required Luther to withdraw 41 of his 95 Theses or be [[excommunicated]]. Luther refused and was then excommunicated by the Catholic Church.

It was then up to [[German]] authorities to take any legal action against Luther. The 21-year-old Holy Roman Emperor [[Charles V King of Spain|Charles V]] ordered him to stand trial before an assembly (a "[[Diet]]") of estates of the [[Holy Roman Empire]] that met in Worms, a small town in what is now Germany. This famous assembly is therefore known as the "[[Diet of Worms]]." The verdict (the "[[Edict of Worms]]") was "guilty". Luther was declared an outlaw who should be arrested.

But Luther had already left the trial before the verdict was rendered. He hid at Wartburg Castle at Eisenach, where [[Frederick of Saxony|Frederick the Wise]], Elector of Saxony, protected him. Luther took the pseudonym Junker Jorg (Nobleman George), grew a wide beard and dressed like a knight. While in hiding he translated the [[New Testament]] from Greek into German in 1522. (By 1534, Luther had translated all of the [[Old Testament]] from Hebrew to German also.)

Luther translated Romans 3:28 by adding an extra German word for "alone" (alleine or alleyn) after the phrase: "justified by faith": "So halten wir nun dafür, daß der Mensch gerecht werde ohne des Gesetzes Werke, alleyn durch den Glauben." Luther believed that man is justified by faith alone, and that salvation comes only from faith. The Catholic Church (and [[Eastern Orthodox Church]]) taught that man is justified by faith, good works and confession to the Church authorities.

A remarkable invention awaited Luther's German translation of the [[Bible]]: in 1447 the German [[Johannes Gutenberg]] had already invented the movable-type [[Printing Press|printing press]]. This enabled Luther's translation to be printed for the public in September 1522. The Christian world would never be the same again. (The [[Gutenberg Bible]], which put the [[Latin Vulgate]] in book format, had been printed in 1455 and it is the oldest existing book in the West.)

Charles V tried to stop the spread of [[Lutheranism]]. In 1544, Charles V sent armies against the German princes, but the [[Peace of Augsburg]] in 1555 gave each German ruler the right to choose the religion for his state. The princes in southern Germany selected Roman Catholicism, but the princes in northern Germany chose Lutheranism. After [[World War II]], when Germany split into West and East Germany, the West German part contained the predominantly Roman Catholic regions and the East German part consisted of the mostly Lutheran regions.

Martin Luther continued to write, and taught at a German university for his financial support. Luther's writings in defense of his religious views were often harsh and passionate. After first reaching out to the Jews, when they did not embrace his Christianity, in 1543 he published a tract entitled ''On the Jews and Their Lies'' which called for the burning of [[synagogue]]s and Jewish schools, the destruction of their prayer books, their homes razed and their money and property confiscated.<ref>[http://www.humanitas-international.org/showcase/chronography/documents/luther-jews.htm Text of Luther's ''On the Jews and Their Lies]</ref><ref>[http://www.jewishvirtuallibrary.org/jsource/anti-semitism/Luther_on_Jews.html The Jewish Virtual Library]</ref>

Historians debate the impact Luther's writings may have had on German thought leading into the Nazi Holocaust of World War II. Luther harshly criticized many peoples in addition to Jews; a pamphlet of his in 1545, the year before he died, was entitled "Against the Roman Papacy an Institution of the Devil,". It is also recorded that at a round table discussion he said he believed in burning witches. Some of Luther’s writings seem quite vulgar by today's standards, but much of it was also in response to vulgar accusations against his religious views.

Lutheranism has only 70 million adherents today, which is only about 7% of the Catholic population. But nearly all Christians, even Catholics, have sung a famous hymn written by Luther: "A Mighty Fortress is Our God." Moreover, hundreds of millions of Christians worldwide agree with Luther that justification is by faith alone and belong to Protestant congregations that first sprung into being because of Luther's stand.

Luther passed away in February, 1546, having lived his latter years in poor health. He had many children, some of whom preceded him in death, by the former nun [[Katharina von Bora]].

==References==
<references/>

{{DEFAULTSORT:Luther, Martin}}

[[Category:Religious People]]
[[Category:German People]]

Talk:Frederick the Great

2008-08-14T15:45:56Z

DiEb: New page: ==King in Prussia== F. was ''King in Prussia'' until 1772, when he declared himself as ''King of Prussia''. There is quite a bit of history hidden behind these prepositions --~~~~

Frederick the Great

2008-08-14T15:42:01Z

DiEb:

'''Frederick the Great''' aka ''Frederick II'' (German: ''Friedrich der Große'' or ''Friedrich II.'') (January 24, 1712 – August 17, 1786) was king of [[Prussia]] from 1740 to 1786 after taking the throne after the death of his father, [[Frederick William I]]. His childhood had been spent in rigorous military training and education, an education he would put to good use.

He increased and modernized the Prussian [[army]] and turned Prussia into a world power. He fought to oppose [[Austria|Austrian]] ambitions, and earned a great reputation as a military commander in the [[War of the Austrian Succession]] (1740-1748). The war netted him Silesia and some other smaller gains.

Frederick was involved in a huge conflict with the [[Seven Years War]] from 1756 - 1763, fighting multiple opponents who had larger populations. His only ally was [[Great Britain]] and one of his enemies was [[France]]. The war crossed continents where in the American colonies it was called the [[French and Indian Wars]].

In 1772 Frederick participated in the first partition of [[Poland]]. By the time he died in 1786, he had doubled the area of his country.

== Sources ==

* http://www.hyperhistory.com/online_n2/people_n2/persons6_n2/frederick.html
* http://www.abcgallery.com/list/2001nov16.html

[[Category:Military Commanders]]
[[Category:German People]]
[[Category:Heads of State]]

John William Strutt, 3rd Baron Rayleigh

2008-08-13T15:11:54Z

DiEb: Dates of Life

'''John William Strutt, 3rd Baron Rayleigh''' (November 12, 1842 – June 30, 1919) was a [[United Kingdom|British]] physicist, commonly referred to as '''Lord Rayleigh''' in scientific circles. He specialised in wave theory and discovered the phenomenon now known as Rayleigh scattering and predicted the existence of surface waves that are known as Rayleigh waves. In collaboration with William Ramsay he discovered the element [[argon]], which led to him being awarded [[Nobel Prize]] for Physics in 1904.

{{DEFAULTSORT:Strutt, John William, 3rd Baron Rayleigh}}
[[Category: Physicists]]
[[Category:Nobel laureates in Physics]]

Prime number

2008-08-11T13:00:32Z

DiEb:

A '''prime number''' is a [[natural number]] that is divisible by only 1 and itself. The only even prime number is 2. There is no upper limit to the quantity of primes, but there is no known formula for deriving the nth prime. [[Leonhard Euler]] once commented: <blockquote>
''Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.''
</blockquote>

The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the [[natural logarithm]] (base ''[[e]]'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref> This is a formulation of a more general statement known as the [[prime number theorem]].

==The Prime Numbers==
The smallest prime numbers are 2, 3, 5, 7, 11, 13...

It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):
<math>\sum_{p\le x}p^{-1}\approx\log\log x</math>

To construct a table of all prime numbers less than ''n'', you would use a method called the [[Sieve of Eratosthenes]]. Simply write down an ordered list of all counting numbers greater than 1. Beginning with 2*2 = 4, cross out every second number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8</s> 9 <s>10</s> 11 ...

then beginning with 3*3 = 9 cross out every third number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8 9 10</s> 11 ...

Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.

==Alternative Definition==
Mathematicians often prefer the following definition, which, for integers, is equivalent to the one stated above: <blockquote>''A prime number is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

Or in symbolic notation:

<math>p \in \mathbb{Z} \quad prime :\Leftrightarrow p \not\in \{-1, 1\} \wedge (\forall a,b \in \mathbb{Z}: p \vert ab \Rightarrow p \vert a \vee p \vert b) </math>

The advantage of this formulation is that it can be generalized on other structures which allow for addition and multiplication, i.e., [[Ring_(mathematics)|ring]]s. The earlier definition involves the notion of [[irreducibility]].

==Unique Factorization==
According to the [[Fundamental Theorem of Arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.

This means that every integer larger than 1, can be expressed as a product of one or more primes in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

Finding the prime factors for large numbers can take considerable time (millions of years) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be [[NP-complete]].

==Primality testing==
To determine whether a number is prime, you can use the trial division method, provided the number you are testing is not too large. Let ''N'' be the number being tested and ''s'' be its square root. Divide ''N'' by each number in the prime table, beginning with 2. If there is no remainder, stop--N is composite. Otherwise, test again with the next largest prime in the table. If the next largest prime is greater than ''s'', stop--''N'' is prime.

For larger values of ''N'', a method based on Fermat's theorem can be used, though it usually requires a computer. Start with the number 1, and double it ''N-1'' times. Divide this number by ''N'', keeping only the remainder. If the remainder is greater than 1, ''N'' is composite. Unfortunately, the converse is not true--if the remainder is 1, ''N'' is ''probably'' prime, but not necessarily. Algorithms which improve on this method exist, and primality testing is today an active subject of mathematical research.

==The Largest Known Prime==

The largest known prime is ''232582657-1''; it was discovered by the '''Great Internet Mersenne Prime Search''' ('''GIMPS''')<ref>http://www.mersenne.org/</ref>, a distributed computing project launched by George Woltman in early 1996.<ref>http://primes.utm.edu/largest.html</ref>

==Mersenne Prime==
A '''Mersenne prime''' is a prime number ''M'' that is of the form ''M = 2n − 1'', where ''n'' is some [[prime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>. As of April, 2007, only 44 Mersenne Primes have been found. The largest known prime ''232582657-1'', is a Mersenne Prime. The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

==Riemann hypothesis==
Most mathematicians believe a proof of the [[Riemann hypothesis]] is the best hope for discovering some pattern in the distribution of prime numbers. ''The Clay Mathematics Institute'' has offered a one million dollar prize for a proof of the Riemann hypothesis.<ref>http://www.claymath.org/millennium/Riemann_Hypothesis/</ref>

==Goldbach's conjecture==

Goldbach's conjecture asserts that every even [[integer]] greater than two is the sum of two [[prime]] numbers. For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

This conjecture is one of the oldest unsettled statements in mathematics. For centuries mathematicians have attempted to prove this statement, without success. The conjecture is known to hold for numbers less than 3 <math>\cdot</math>1017.

==Cryptography==
The security of [[public-key cryptography]] schemes such as [[RSA]] depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such cryptographic schemes could become vulnerable to attack.

==References==
<references/>

==See Also==

* [[Composite number]]

==External Links==
:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]
:[http://primes.utm.edu/ Prime Number resource Archieve]
[[Category:Mathematics]]

Prime number

2008-08-10T15:31:31Z

DiEb: alternative definition (prime vs. irreducible)

A '''prime number''' is a [[natural number]] that is divisible by only 1 and itself. The only even prime number is 2. There is no upper limit to the quantity of primes, but there is no known formula for deriving the nth prime. [[Leonhard Euler]] once commented: <blockquote>
''Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.''
</blockquote>

The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the natural logarithm (base ''e'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref> This is a formulation of a more general statement known as the [[prime number theorem]].

==The Prime Numbers==
The smallest prime numbers are 2, 3, 5, 7, 11, 13...

It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):
<math>\sum_{p\le x}p^{-1}\approx\log\log x</math>

To construct a table of all prime numbers less than ''n'', you would use a method called the [[Sieve of Eratosthenes]]. Simply write down an ordered list of all counting numbers greater than 1. Beginning with 2*2 = 4, cross out every second number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8</s> 9 <s>10</s> 11 ...

then beginning with 3*3 = 9 cross out every third number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8 9 10</s> 11 ...

Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.

==Alternative Definition==
Mathematicians prefer the following definition, which is - for integers - equivalent to the one stated above:
<blockquote>''A prime number is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

Or in symbolic notation:

<math>p \in \mathbb{Z} \quad prime :\Leftrightarrow p \not\in \{-1, 1\} \wedge (\forall a,b \in \mathbb{Z}: p \vert ab \Rightarrow p \vert a \vee p \vert b) </math>

The advantage of this formulation is that it can be generalized on other structures which allow for addition and multiplication, i.e., rings.
==Unique Factorization==
According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.

This means that every integer larger than 1, can be expressed as a product of one or more [[prime|primes]] in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

Finding the prime factors for large numbers can take considerable time (millions of years) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be NP-complete.

==Primality testing==
To determine whether a number is prime, you can use the trial division method, provided the number you are testing is not too large. Let ''N'' be the number being tested and ''s'' be its square root. Divide ''N'' by each number in the prime table, beginning with 2. If there is no remainder, stop--N is composite. Otherwise, test again with the next largest prime in the table. If the next largest prime is greater than ''s'', stop--''N'' is prime.

For larger values of ''N'', a method based on Fermat's theorem can be used, though it usually requires a computer. Start with the number 1, and double it ''N-1'' times. Divide this number by ''N'', keeping only the remainder. If the remainder is greater than 1, ''N'' is composite. Unfortunately, the converse is not true--if the remainder is 1, ''N'' is ''probably'' prime, but not necessarily. Algorithms which improve on this method exist, and primality testing is today an active subject of mathematical research.

==The Largest Known Prime==

The largest known prime is ''232582657-1''; it was discovered by the '''Great Internet Mersenne Prime Search''' ('''GIMPS''')<ref>http://www.mersenne.org/</ref>, a distributed computing project launched by George Woltman in early 1996.<ref>http://primes.utm.edu/largest.html</ref>

==Mersenne Prime==
A '''Mersenne prime''' is a prime number ''M'' that is of the form ''M = 2n − 1'', where ''n'' is some [[prime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>. As of April, 2007, only 44 Mersenne Primes have been found. The largest known prime ''232582657-1'', is a Mersenne Prime. The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

==Riemann hypothesis==
Most mathematicians believe a proof of the [[Riemann hypothesis]] is the best hope for discovering some pattern in the distribution of prime numbers. ''The Clay Mathematics Institute'' has offered a one million dollar prize for a proof of the Riemann hypothesis.<ref>http://www.claymath.org/millennium/Riemann_Hypothesis/</ref>

==Goldbach's conjecture==

Goldbach's conjecture asserts that every even [[integer]] greater than two is the sum of two [[prime]] numbers. For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

This conjecture is one of the oldest unsettled statements in mathematics. For centuries mathematicians have attempted to prove this statement, without success. The conjecture is known to hold for numbers less than 3 <math>\cdot</math>1017.

==Cryptography==
The security of [[public-key cryptography]] schemes such as [[RSA]] depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such cryptographic schemes could become vulnerable to attack.

==References==
<references/>

==See Also==

* [[Composite number]]

==External Links==
:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]
:[http://primes.utm.edu/ Prime Number resource Archieve]
[[Category:Mathematics]]

Combinatorics

2008-08-08T12:37:14Z

DiEb:

'''Combinatorics''' is a field of [[mathematics]] devoted to the study of counting elements in a set, and mathematical relations that describe their properties. This field is "[[discrete mathematics|discrete]]", meaning it deals with non-[[continuous]] properties. Combinatorics is very useful in the field of discrete [[probability]] theory.

==Combinatorial Functions==
Many functions used in combinatorics deal with how to count or arrange numbers. These rules are all based on the ''basic principle of counting'', which states that for any two groups of objects of size ''m'' and ''n'', there are ''m • n'' possible combinations.<ref name="ross">Ross, Sheldon. "A First Course in Probability 7e". Pearson Education: 2006.</ref> For example, If you have 10 different marbles and 8 different cups, there are 80 (10 • 8) ways to put a marble into a cup.

===Permutations===
A ''permutation'' is a certain way to order a group of objects. For example, ''bac'' is one permutations of the letters ''a'', ''b'', and ''c''. For three objects it is easy to see that there are 6 total permutations by simply writing out all of the combinations (''abc'', ''acb'', ''bac'', ''bca'', ''cab'', and ''cba'').

For larger groups of objects it is harder, if not impossible, to write out all combinations. However a simple formula can be derived from the basic principle of counting, which states that, for a group of ''n'' objects, there are ''n!'' ("n [[factorial]]") possible ways to order those objects. So for our above example, we can calculate that there are 6 possible ways (3 • 2 • 1) to order three objects.

For groups of indistinguishable objects, you must divide the total number of redundant ways to permute the objects:

:<math>\frac {n!}{n_1! n_2! \cdot \cdot \cdot n_r!}</math>, where ''n'' is the total number of objects, and ''n1'' through ''nr'' are the total number of each type of objects within the group.

For example, the letters ''p'' and ''e'' in the word "pepper" are indistinguishable from one another. There are 6 total letters, which are comprised of 3 p's, 2 e's, and 1 ''r''. The total number of permutations would thus be 6! / (3 ! 2! 1!), or 60.

===Combinations===
It is often useful to find out how many groups of ''r'' objects can be created out of a group of ''n'' objects. This can be calculated from a function called "choose" (read aloud as "n choose r"):

:<math>\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{(n-r)!r!}</math>

For example, take a race which has 10 competitors. There are 10 choose 3 different combinations of medal winners (that is, racers who get either 1st, 2nd, or 3rd place), which is a total of 10! / (10 - 3)! • 3! = 10! / 7! • 3! = 120 different combinations.

[[category: mathematics]]

Fourier series

2008-08-08T12:33:51Z

DiEb:

'''Fourier series''' express a piecewise continuous, bounded, periodic function as a linear combination of [[orthogonal]] [[sine]] and [[cosine]] functions.

The Fourier [[series (mathematics)|series]] of a function ''f(t)'' is of the form:
:<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math>
where, ''n'' is an integer and
:<math> \omega_n = n\frac{2\pi}{T}</math>
:<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math>
:<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math>

Fourier series can be generalized to [[Fourier transformation]]s for non-perodic, piecewise continuous, [[square integrable function]]s.

[[category: mathematics]]

Prime number

2008-08-07T22:33:40Z

DiEb: there is no uniform distribution on the integers

A '''prime number''' is a [[natural number]] that is divisible by only 1 and itself. The only even prime number is 2. There is no upper limit to the quantity of primes, but there is no known formula for deriving the nth prime. [[Leonhard Euler]] once commented: <blockquote>
''Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.''
</blockquote>

The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the natural logarithm (base ''e'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref> This is a formulation of a more general statement known as the [[prime number theorem]].

==The Prime Numbers==
The smallest prime numbers are 2, 3, 5, 7, 11, 13...

It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):
<math>\sum_{p\le x}p^{-1}\approx\log\log x</math>

To construct a table of all prime numbers less than ''n'', you would use a method called the [[Sieve of Eratosthenes]]. Simply write down an ordered list of all counting numbers greater than 1. Beginning with 2*2 = 4, cross out every second number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8</s> 9 <s>10</s> 11 ...

then beginning with 3*3 = 9 cross out every third number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8 9 10</s> 11 ...

Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.

==Unique Factorization==
According to the [[fundamental theorem of arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique factorization into prime numbers.

This means that every integer larger than 1, can be expressed as a product of one or more [[prime|primes]] in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

Finding the prime factors for large numbers can take considerable time (millions of years) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be NP-complete.

==Primality testing==
To determine whether a number is prime, you can use the trial division method, provided the number you are testing is not too large. Let ''N'' be the number being tested and ''s'' be its square root. Divide ''N'' by each number in the prime table, beginning with 2. If there is no remainder, stop--N is composite. Otherwise, test again with the next largest prime in the table. If the next largest prime is greater than ''s'', stop--''N'' is prime.

For larger values of ''N'', a method based on Fermat's theorem can be used, though it usually requires a computer. Start with the number 1, and double it ''N-1'' times. Divide this number by ''N'', keeping only the remainder. If the remainder is greater than 1, ''N'' is composite. Unfortunately, the converse is not true--if the remainder is 1, ''N'' is ''probably'' prime, but not necessarily. Algorithms which improve on this method exist, and primality testing is today an active subject of mathematical research.

==The Largest Known Prime==

The largest known prime is ''232582657-1''; it was discovered by the '''Great Internet Mersenne Prime Search''' ('''GIMPS''')<ref>http://www.mersenne.org/</ref>, a distributed computing project launched by George Woltman in early 1996.<ref>http://primes.utm.edu/largest.html</ref>

==Mersenne Prime==
A '''Mersenne prime''' is a prime number ''M'' that is of the form ''M = 2n − 1'', where ''n'' is some [[prime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>. As of April, 2007, only 44 Mersenne Primes have been found. The largest known prime ''232582657-1'', is a Mersenne Prime. The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

==Riemann hypothesis==
Most mathematicians believe a proof of the [[Riemann hypothesis]] is the best hope for discovering some pattern in the distribution of prime numbers. ''The Clay Mathematics Institute'' has offered a one million dollar prize for a proof of the Riemann hypothesis.<ref>http://www.claymath.org/millennium/Riemann_Hypothesis/</ref>

==Goldbach's conjecture==

Goldbach's conjecture asserts that every even [[integer]] greater than two is the sum of two [[prime]] numbers. For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

This conjecture is one of the oldest unsettled statements in mathematics. For centuries mathematicians have attempted to prove this statement, without success. The conjecture is known to hold for numbers less than 3 <math>\cdot</math>1017.

==Cryptography==
The security of [[public-key cryptography]] schemes such as [[RSA]] depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such cryptographic schemes could become vulnerable to attack.

==References==
<references/>

==See Also==

* [[Composite number]]

==External Links==
:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]
:[http://primes.utm.edu/ Prime Number resource Archieve]
[[Category:Mathematics]]

Limit (mathematics)

2008-08-07T22:14:17Z

DiEb: I like your changes, but an "infinitely small ball" is not intuitive :-)

The concept of the '''limit''' is a cornerstone of the mathematical branch of [[analysis]] and captures the intuitive concept of "approaching a value".

==Limit of a Sequence==
Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a</math>, i.e.,
<math>
\lim_{n\to \infty} = a,
</math>
if for any <math>\varepsilon > 0</math>, there exists a number <math>N</math>, such that
<math>
|a_n - a| < \varepsilon</math> for every <math>n > N</math>. Intuitively, this means that if you take an [[interval]] as small as you like, [[center]]ed at the limit point, then most - i.e., all but a finite number - of the points of the sequence are in that interval.

==Limit of a Function==
Let <math>f(x)</math> be a real valued [[function]] in one [[variable]]. We say that

<math>
\lim_{x\to p} f(x) = L
</math>

if for any error <math>\varepsilon</math>, we can find a sufficiently small [[neighborhood]] of <math>p</math> so that <math>f(x)</math> is within <math>\varepsilon</math> of the value <math>L</math> for any <math>x\neq p</math> in that neighborhood. One says that the limit of ''f(x)'' at ''p'' exists and is equal to ''L''.

The standard, though more verbose, way of saying this is that for any <math>\varepsilon > 0</math> there exists a sufficiently small <math>\delta > 0</math> such that <math>|f(x)-L| <\varepsilon</math> whenever <math>0<|x-p|<\delta</math>.

Intuitively, this means that as the variable <math>x</math> approaches the value <math>p</math>, the function <math>f(x)</math> tends to the value <math>L</math>.

[[category:mathematics]]

Continuum

2008-08-07T22:04:18Z

DiEb:

Mathematically, '''continuum''' can refer to the [[Real_line|real line]], its [[Cardinality|cardinality]] <math>\aleph_1</math>, or any [[Continuous|continuous]] [[Connected_(topology)|connected]] [[Dense_subset|dense]] [[Linear_order|linear order]]. Generally, when mathematicians say "the continuum", they are referring to one of the first two possibilities (context clarifies which one). The [[Continuum_hypothesis|Continuum Hypothesis]] conjectures that there is no set of cardinality bigger than that of the [[natural number]]s <math>\aleph_0</math>, but smaller than the cardinality of the set of the real numbers, <math>\aleph_1 = 2^{\aleph_0}</math>.

The continuum is called so because it was the first (and most prominent) [[Continuous|continuous]] set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order. Since the [[Complex_number|complex numbers]] add the [[Imaginary_number|imaginary number]], i, to the real line, this is one reason they have no natural linear order.

[[Category:Mathematics]]

Ideal Gas

2008-08-07T21:50:19Z

DiEb: Insertet text of the "ideal gas law"

An '''Ideal Gas''' is the hypothetically perfect embodiment of a gas in which the particles (atoms or molecules) in the gas are spherical, identical, have no volume, and experience no intermolecular forces between them. All collisions between the particles or the particles and the container are perfectly elastic.

Since this is just a model, real gases only obey the [[ideal gas law]] approximately, not perfectly. Generally, the ideal gas assumption is accurate for unreactive gases at high temperature and/or low pressure. A good rule of thumb is that the assumption can be used above room temperature and below 1 atmosphere of pressure.

[[category:physics]]

Talk:Radioactivity

2008-08-06T19:28:11Z

DiEb: Undo revision 496381 by Jinxmchue (Talk) Have a look a my talk page (user talk:DiEb:DiEb)

Implications of [[Quantum mechanics]] should go here, i.e., the "random" nature of radioactive decay. --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 14:57, 8 July 2008 (EDT)

I deleted the page, because it refers to "[[particles]]" in a misleading way. Also, I understand from high school physics that there two kinds of radioactivity:
#Electromagnetic: X-rays, etc.
#Sub-atomic particles, like neutrons; see [[nuclear radiation]]

We need a simple, basic article that explains how your dentist's assistant makes [[X-ray]] pictures of your teeth - and whether it's as dangerous to your reproductive organs as taking a plane trip from NY to LA. It should also explain the [[nuclear reaction]] which generates electricity - and why the US is safer in this than the Soviets were; see [[Chernobyl]]. The basis for [[atomic weapons]] should also be explained.

Other topics include [[background radiation]], [[cosmic rays]], and the medical use of radiation, as in cancer therapy. --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 16:32, 8 July 2008 (EDT)

: Xrays are neither radioactivity nor produced by radioactivity. An Xray is produced in a cathode ray tube (akin to a traditional television). A gamma ray is produced when a proton is converted into a neutron. This sends a neutrino and a positron out of the nucleus. The positron immediately interacts with an electron and they annihilate each other. This produces two 511 keV photons which are in the Gamma ray range of the spectrum. This form of radiation is used in positron emission tomography.
: In essence, all radioactivity is the emission of a particle of some sort or another. Any photons or electromagnetic waves (depending on how you want to think about it) is the result of an antiparticle annihilating.
: It is also important to to confuse radiation with radioactivity. The two are separate and distinct concepts. --[[User:Rutm|Rutm]] 18:03, 8 July 2008 (EDT)

::Oak Ridge Labs defines [[radiation]] as "A general term referring to particles (e.g., electrons) and electromagnetic waves (e.g., radio waves) that carry energy." So in what way are radiation and [[radioactivity]] separate or distinct? --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 18:33, 8 July 2008 (EDT)

I dont think it is the release of particles actually. Is it not matter(particles) converted into energy via E=MC2? The radioactive element Radium for example releases small amounts of radiation by releasing the energy contained within its mass.
[[User:JJacob|JJacob]] 18:10, 8 July 2008 (EDT)
: Radium releases alpha particles (2 protons and 2 neutrons). Beta radiation are protons. A chart of the decay chain of uranium can be see at [http://www.nuenergy.org/alt/radium.htm here]. The decay product of Radium is radon. Specifically, radium has 88 protons and 138 neutrons and becomes radon with 86 protons and 136 neutrons -- two less protons and two less neutrons. There is also a small amount of energy released in the form of kinetic energy (heat) that was previously holding the alpha particle in the nucleus causing the atom to move. It is this heat/kinetic energy that is captured in a fission power plant. You can read more about the exact nature of the decay of radium [http://www.hps.org/publicinformation/ate/q2549.html here]. --[[User:Rutm|Rutm]] 18:21, 8 July 2008 (EDT)

Thanks for the reading material. I just finished reading a very good book called "E=MC2" by David Bodanis. Check it out, it is a fantastic read.
God bless!
[[User:JJacob|JJacob]] 18:25, 8 July 2008 (EDT)

Clearly I have not made the proper distinction between [[radiation]] and [[radioactivity]]. Is one a subset of the other? If not, how are they related? --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 18:27, 8 July 2008 (EDT)

: Radiation is the product of any body that changes from a higher energy state to a lower energy state. For example, when you run a current through a piece of metal it causes the electrons to move to a higher energy state. As they return to a lower energy state, they emit a photon. This is radiation and how a light bulb works. Čerenkov radiation is produced when a charged body slows down to the speed of light in a different medium. In the case of Čerenkov radiation, a proton is slowed down to 0.75 C (the speed of light in water) and gives off photons. A slower proton is at a lower energy state than a fast one. This concept of energy state also applies to the nucleus of an atom.
: Radiation can be divided into two types - ionizing and non-ionizing. The distinction is how they interact with matter. Non-ionizing radiation are things like microwaves, radio waves, viable light. Ionizing radiation are subatomic particles and high energy photons. This type of radiation can knock off an electron from an atom or molecule leaving an ion. Xrays and even ultraviolet can be ionizing, but they are not produced by a radioactive substance.
: A substance that is radioactive (a radionuclide) emits ionizing radiation naturally. Uranium, radium, radon, carbon-14, etc... For fun, carbon-11, nitrogen-13, oxygen-15 , and fluorine-18 are the radionuclides used in PET scans - radioactive substances are sometimes more familiar elements than the classic big heavy atoms. The distinction is that a substance used to produce x-rays in an x-ray tube is not naturally producing it. Not all ionizing radiation is from a radioactive substance. Gama rays, for example can be formed with supernova. Xrays are given off in the jets from a black hole as electrons are wiped around magnetic fields. --[[User:Rutm|Rutm]] 18:50, 8 July 2008 (EDT)

== McGraw-Hill definition -- radioactivity vs. radiation ==

The McGraw-Hill Children's Dictionary (ISBN 1-57768-298-X) is a resource that I would judge to be aimed at 10-year-olds, at least 6 years younger than what I think CP's target audience should be. The blurb on the back cover indicates a target audience of "elementary school through middle school". I believe that means 8- to 13-year-olds.

Their entry for radioactivity is:
:The giving off of energy as a result of the decay of unstable atoms.
::''Uranium has a high level of radioactivity.''

Their entry for radiation is:
:The waves of energy sent out by sources of heat or light, or by a radioactive material.
::''She wore a hat to protect her skin from the radiation of the sun.''
::''Many people are concerned about radiation from nuclear waste.''

Talk:Radioactivity

2008-08-06T18:44:34Z

DiEb: Undo revision 496359 by Jinxmchue (Talk) Entry was relevant, especially on the talk page