Conservapedia - User contributions [en]

Aspirin

2008-07-03T21:49:49Z

Phillips101:

'''Aspirin''' originated as a [[herbal]] remedy, a bitter powder now called [[salicin]] which was extracted from the bark of the [[willow]] tree in ancient Greek times (however this raw form often induced extreme stomach pains as a side effect). [http://www.howstuffworks.com/aspirin.htm] First synthesized as acetylsalicylic acid {{years ago|1899}} years ago this wonder drug has been an unmixed blessing to patients all over the world, relieving aches, pains and fever.

[[Category:Medicine]]

Quadratic equation

2008-07-03T18:54:02Z

Phillips101:

A '''quadratic equation''' can take two forms. The general formula, written as a function of <math>x</math>, <math>y=f(x)</math>, is:

:<math>y = ax^2 + bx + c</math>

The graph of a quadratic equation is a [[parabola]], one of the [[conic section]]s. (In some cases, the parabola collapses, most obviously when <math>a = 0</math>)

The points where this curve crosses the y axis are represented by the second form of the equation:

:<math>ax^2 + bx + c = 0</math>

These are solved for using the [[quadratic formula]], which will not only solve for [[real number|real]] roots, but result in the [[imaginary number|imaginary]] roots if the parabola does not actually cross the y axis (this is when <math>4ac</math> is greater than <math>b^2</math>).

Quadratic equations are very important in calculating the motion of bodies under constant [[acceleration]], i.e., [[gravity]] (when close to the earth's surface).

The [[derivative]] of a quadratic equation is a simple [[linear function]]:

:<math>\frac{dy}{dx} = 2ax + b</math>

[[category:mathematics]]

Quadratic formula

2008-07-03T18:52:00Z

Phillips101:

{{move|quadratic formula}}

The '''quadratic formula''' is used to simplify the process of solving a [[quadratic equation]].

First, the quadratic equation must be reduced to this format:

:<math>ax^2+bx+c=0\!</math>

Then the coefficients ''a'', ''b'', and ''c'' can be substituted in the formula to find the solutions:

:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

You can prove the formula the following way:

:<math>ax^2+bx+c=0\!</math>

:<math>x^2+\frac{b}{a}x+\frac{c}{a}=0\!</math>

:<math>(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!</math>

:<math>(x+\frac{b}{2a})^2=(\frac{b}{2a})^2-\frac{c}{a}\!</math>

:<math>(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\!</math>

:<math>x+\frac{b}{2a}=\frac{\pm \sqrt {b^2-4ac}}{2a}\!</math>

:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

This method of deriving the formula is done via [[completing the square]].

You can assert that the formula is correct by substituting the formula in place of '''x''' in <math>ax^2+bx+c=0\!</math> and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.

[[Category:Mathematics]]

Quadratic formula

2008-07-03T18:50:32Z

Phillips101:

{{move|quadratic formula}}

The '''quadratic formula''' is used to simplify the process of solving a [[quadratic equation]].

First, the quadratic equation must be reduced to this format:

:<math>ax^2+bx+c=0\!</math>

Then the coefficients ''a'', ''b'', and ''c'' can be substituted in the formula to find the solutions:

:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

You can prove the formula the following way:

:<math>ax^2+bx+c=0\!</math>

:<math>x^2+\frac{b}{a}x+\frac{c}{a}=0\!</math>

:<math>(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!</math>

:<math>(x+\frac{b}{2a})^2=(\frac{b}{2a})^2-\frac{c}{a}\!</math>

:<math>(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\!</math>

:<math>x+\frac{b}{2a}=\frac{\pm \sqrt {b^2-4ac}}{2a}\!</math>

:<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

This method of deriving the formula is done via the method of [[completing the square]].

You can assert that the formula is correct by substituting the formula in place of '''x''' in <math>ax^2+bx+c=0\!</math> and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.

[[Category:Mathematics]]

Complex number

2008-07-03T18:38:00Z

Phillips101:

A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.

Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). This representation is known as the Argand diagram.

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the [[real numbers]].

Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]].

===Polar notation===
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]
: <math>\theta=\arctan\frac{b}{a}</math> is the phase
If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the complex plane.
===Complex Numbers as Matrices===
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
:[a -b]
:[b a],
with <math>a+bi</math> mapping as a function f to the above matrix.
We can see that F and F' are isomorphic because:
The function f is clearly 1-to-1 and onto,
<math>f(x+y)=f(x)+f(y)</math>,
and <math>f(x*y)=f(x)*f(y)</math>.

===In popular culture===
In [[Yvgeny Zamyatin]]'s satirical novel ''[[We (novel)|We]]'', the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.

==See Also==
*[[Arithmetic with complex number]]

[[Category:Mathematics]]
[[category:complex analysis]]

Complex number

2008-07-03T18:37:33Z

Phillips101:

Complex number

2008-07-03T18:34:24Z

Phillips101:

A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.

Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). This representation is known as the Argand diagram.

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the [[real numbers]].

Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]].

===Polar notation===
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]
: <math>\theta=\arctan\frac{b}{a}</math> is the phase
If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the complex plane.
===Complex Numbers as Matrices===
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
:[a b]
:[-b a],
with <math>a+bi</math> mapping as a function f to the above matrix.
We can see that F and F' are isomorphic because:
The function f is clearly 1-to-1 and onto,
<math>f(x+y)=f(x)+f(y)</math>,
and <math>f(x*y)=f(x)*f(y)</math>.

===In popular culture===
In [[Yvgeny Zamyatin]]'s satirical novel ''[[We (novel)|We]]'', the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.

==See Also==
*[[Arithmetic with complex number]]

[[Category:Mathematics]]
[[category:complex analysis]]

Imaginary number

2008-07-03T18:29:53Z

Phillips101:

An '''imaginary number''' in mathematics is any number that contains the imaginary unit, defined as, <math>i^{2} = -1</math>

An imaginary number is of the form, <math>k i</math>, where ''k'' is a [[real number]].

For example <math>\sqrt{-1}</math> has imaginary representation of

:<math>\sqrt{-1}= i</math>.

When a imaginary number is added to real number, they form a [[complex number]].

The analysis of imaginary numbers forms the basis for the field of [[mathematics]] known as
[[complex analysis]].

[[category:mathematics]]
[[category:complex analysis]]

Pi Day

2008-07-02T17:26:42Z

Phillips101:

'''Pi day''' is a [[holiday | holiday]] celebrating [[mathematics]].

Pi Day, a holiday celebrating the mathematical constant [[Pi]] is observed on March 14. Celebrations often involve eating pie, cookies or some other spherical delicacy. The date comes from the first three digits of pi (3/14) (in american notation) with some people beginning their celebration at 1:59:26.5 (the following digits (3.14159265)). Many Elementary and Middle School teachers use March 14 to introduce their students to the concept of Pi

Pi Approximation day is a similar holiday, celebrated on July 22 (from the approximation 22/7).

[[Category:Mathematics]]

Pi

2008-07-02T17:24:44Z

Phillips101:

'''Pi''' is the sixteenth letter of the [[Greek]] [[alphabet]] and is used in its lower case form (<big><math>\pi</math></big>) to represent the mathematical constant of the same name which is defined as ''the ratio of the circumference of a circle to its diameter''. It is an important number and appears in many [[mathematical]] and [[physics|physical]] formulae.

The value of <big><math>\pi</math></big> is an [[irrational number]]; which means that it cannot be fully expressed as a fraction or a decimal (regardless of the number of digits used).

The value of <big><math>\pi</math></big> is approximately 3.14159 in decimal. This value is precise enough for almost all ordinary purposes; it can, for example, be used to calculate the circumference of the Earth with an error of only about 110 feet.

For rough purposes, the fraction 22/7 (= 3.14285...) is sometimes used.

==History==
To some extent, the progress of mathematics—or at least of computation—can be gauged by the progress in the number of digits to which <big><math>\pi</math></big> has been calculated.

Some ancients expressed pi by using fractional approximations. Papyrus of Ahmes, dated c. 1650 B.C., shows that ancient Egyptians had value 3 1/6 = 3.167). The Babylonian value from the same era was 3 1/8 = 3.125<ref>Boyer, A History of Mathematics, 2nd Edition</ref>. Both these values are accurate to within 1 percent. Note that the value 22/7 (3 1/7) is still used today.

[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of pi.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref>
He said it was between 223/71 and 22/7. This is ten times better than the Egyptian and Babylonian values: within 0.04% of pi.

In 1873, Abraham Shanks spent twenty years calculating <big><math>\pi</math></big> to 707 places, but made a mistake in his calculation and only 527 of them were correct. When electronic computers were developed, <big><math>\pi</math></big> was soon calculated to tens of thousands, millions, and billions of places. As of 2002, the record is held by Yasumasa Kanada of Tokyo University at 1,241,100,000,000 digits. That result was never printed out.

==Recreational use==
Memorizing <big><math>\pi</math></big> is a challenge that appeals to some people. Mnemonics have been devised. Counting the letters in the phrase "Now I want a drink—alcoholic, of course" gives <big><math>\pi</math></big> to seven places (which is more than enough for all ordinary purposes). Numerous other mnemonics of this kind have been devised; in 1995, Michael Keith wrote one entitled [http://users.aol.com/s6sj7gt/mikerav.htm Near a Raven] which simultaneously parodies [[Edgar Allen Poe]]'s poem ''The Raven,'' while encoding <big><math>\pi</math></big> to 740 places.

March 14 marks [[Pi Day]], a holiday on which the mathematical constant is celebrated. The date comes from the first three digits of pi; some people begin their celebration at 1:59 pm, derived from the following three digits.

Pi Approximation day is a similar holiday, celebrated on July 22 (from the approximation 22/7). <ref>[http://www.usatoday.com/tech/science/mathscience/2007-03-14-pi-day_N.htm USA Today (3/14/2007) - Pi-day]</ref>

Pi is approximately:

{{quotebox|
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628
0348253421170679821480865132823066470938446095505822317253594081284811174502841027...}}

== Greek Language Usage ==
This letter's name is pronounced the same as its equivalent in English (P) and has the same sound value.

==Does the Bible attempt to define pi?==
Virtually all serious students of the [[Bible]] say no. Still, critics frequently claim that the Bible contains an incorrect value for pi<ref>http://www.cygnus-study.com/writings/allowableerror.shtml</ref>, and the question is raised frequently enough to earn mention in the [[Skeptics Annotated Bible]].

The claim is based on a verse in the [[I Kings|first book of Kings]]:
{{bible quote|He made the Sea of cast metal, circular in shape, measuring ten [[cubit]]s from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.|book=1_Kings|chap=7|verses=23|version=NIV}}
The critics say that this amounts to the Bible claiming that the value of pi is 3.

There are a number of assumptions involved in making this claim, and if any one of the assumptions is wrong, the claim is false.
The assumptions are:
* The value in the Bible is ''wrong'', rather than just imprecise. However, to the nearest [[whole number]], the value is correct,<ref name="math" /> and it was quite common at the time to round numbers.<ref name="jph">Holding, J.P.</ref> For instance the value of 10 could actually be any value between 9.5 and 10.5. If it was to the lower end of that range, then the equation works.
* That both the [[diameter]] and the [[circumference]] are measuring the same edges. It's possible, even if unlikely, that the diameter is an outside measurement and the circumference is an inside measurement. A straight calculation doesn't allow for the thickness of the sides.<ref name="jph" />
* That both the diameter and the circumference are measuring the same part of the object. The object is also described as having an outward-turned rim. The easiest places to measure the diameter would be across the wider rim, and the easiest place to measure the circumference would be around the body below the rim.<ref>Grigg, 1995.</ref>

The claim is correct however because it describes the circumference as something it is not; the basic definition of incorrect. However, some stick by the calim that rounding pi to 3 is not "incorrect," an "error" or "wrong" and it perfectly acceptable in some circumstances.<ref>http://mathforum.org/library/drmath/view/52573.html</ref> The creation of a "sea of cast metal" by human beings in ancient times without modern construction tools and measuring equipment is apparantly understood as one of those circumstances.

==Bibliography==
* Grigg, Russell, [http://creationontheweb.com/content/view/1731/ Does the Bible say pi equals 3.0?] ''Creation'' 17(2):24–25, March 1995.
* Holding, James Patrick, [http://www.tektonics.org/lp/piwrong.html Pi Gets In Your Eye] (Tektonics).
* Peterson & Rick, [http://mathforum.org/library/drmath/view/52573.html Rounding Pi] 1st June(?), 1999 (The Math Forum).

==See Also==
*[[Pi day]]
*[http://yacas.sourceforge.net/Algochapter5.html#c5s5 Calculation of pi with computers]

==References==
{{reflist}}

[[Category:Mathematics]]

E

2008-07-02T17:18:41Z

Phillips101: /* Formulae for e */

{{lowercase}}
'''''e''''' is a useful [[mathematical]] constant which is a [[transcendental]] number approximately equal to 2.718281828459045 . ''e'' can be used in [[logarithm]]s as the base, called a [[natural logarithm]]. ''e'' is named for [[Swiss]] mathematician [[Leonhard Euler]], though he did not discover the constant.

It has some remarkable properties that its exponential to any [[real number]] <math>x</math> is:

:<math>\frac{d}{dx}e^x = e^x.</math>

The exponetial function is the [[eigenfunction]] of the derivative operator.

==Calculations for ''e''==
*With limits - <math>e=\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x</math><br /><br />
*With infinite series - <math>e=\sum_{n=0}^{\infty}\frac{1}{n!}</math>

[[category:mathematics]]

Roid rage

2008-07-02T17:16:44Z

Phillips101:

"Roid rage," a play on the unrelated problem of "[[road rage]]", is the violent behavior that can be triggered through use of [[anabolic steroids]]. Anabolic steroids have been used by professional wrestlers, bodybuilders, football and baseball players and other athletes in a misguided attempt to enhance their muscle growth and strength [citation needed].

Roid rage can cause "paranoia, depression and explosive outbursts." It was suspected as a possible cause in the murders-suicide by the professional wrestler Chris Benoit.<ref>http://www.msnbc.msn.com/id/19424899/</ref>

== References ==

<references/>

[[category:psychology]]