Conservapedia - User contributions [en]

Talk:Algebra

2014-02-25T08:06:37Z

AmyD:

'''This page on Algebra is totally inadequate.'''

See the example entry below from http://mathworld.wolfram.com/Algebra.html

----
<quotation>
The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwarizmi about algebraic methods. In modern usage, algebra has several meanings.

One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of mathematics is often referred to as abstract algebra.

'''The word "algebra" can also refer to the "school algebra" generally taught in American middle and high schools. This includes the solution of polynomial equations in one or more variables, and basic properties of functions and graphs. Mathematicians call this subject "arithmetic," reserving the word "algebra" for the more advanced aspects of the subject.'''

Finally, the word is used in a third way, not as a subject area but as a particular type of algebraic structure. Formally, an algebra is a vector space V over a field F with a multiplication. The multiplication must be distributive and, for every f in F and x,y in V must satisfy
f(xy)==(fx)y==x(fy).

An algebra is sometimes implicitly assumed to be associative or to possess a multiplicative identity.

Examples of algebras include the algebra of real numbers, vectors and matrices, tensors, complex numbers, and quaternions. (Note that linear algebra, which is the study of linear sets of equations and their transformation properties, is not an algebra in the formal sense of the word.) Other more exotic algebras that have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to entirely unenlightening names which are commonly used by algebraists without further explanation or elaboration.

SEE ALSO: Abstract Algebra, Alternative Algebra, Associative Algebra, Banach Algebra, Boolean Algebra, Borel Sigma-Algebra, C-*-Algebra, Cayley Algebra, Clifford Algebra, Commutative Algebra, Derivation Algebra, Exterior Algebra, Fundamental Theorem of Algebra, Graded Algebra, Hecke Algebra, Heyting Algebra, Homological Algebra, Hopf Algebra, Jordan Algebra, Lie Algebra, Linear Algebra, Measure Algebra, Nonassociative Algebra, Power Associative Algebra, Quaternion, Robbins Algebra, Schur Algebra, Semisimple Algebra, Sigma-Algebra, Simple Algebra, Steenrod Algebra, Umbral Algebra, von Neumann Algebra.

</quotation>

----
As you can see by this, the entry on Algebra appears to be deficient in any number of areas, but in particular, the author of the original seems to be mistaken in assuming that algebra is merely that element which is emboldened in the text above.

--[[User:CatWatcher|CatWatcher]] 13:32, 6 April 2007 (EDT)

:It's not a mistake; it's the context of this encyclopedia. We meant "school algebra".

:After that has been explained, then you are welcome to write about advanced kinds of algebra. --[[User:Ed Poor|Ed Poor]] <sup>[[User talk:Ed Poor|Talk]]</sup> 20:07, 3 September 2008 (EDT)

I realize that it's tacky to argue with someone who has been banned, but the "improvement" suggested by CatWatcher is '''so''' not going to happen. The educational articles here are aimed at teenagers. People who want world-class sophistication can go to the mathworld site mentioned above. Oh. And he left out Grassmann algebra, though perhaps he meant for Exterior algebra to include that. [[User:SamHB|SamHB]] 23:47, 1 December 2010 (EST)

Can anyone help with the formatting? I don't know how to represent mixed numbers or the plus-minus sign in wiki formatting. [[User:AmyD|AmyD]] 03:06, 25 February 2014 (EST)

Algebra

2014-02-25T08:04:35Z

AmyD: /* Factoring */

{{Math-e}}
'''Algebra''' is a major branch of [[mathematics]] that analyzes the relationships between quantities or items. In higher math the principal fields of algebra are [[linear algebra]], which focuses on matrices, and [[abstract algebra]], which includes [[group theory]].<ref>The name algebra comes from the [[Arabic]] word ''al jebr'', which means reduction or "reunion of broken parts" [http://dictionary.reference.com/browse/algebra Algebra] as mentioned in a book ''Hisab al-jabr w'al-muqabala'' translated as ''Science of the Reunion and the Opposition''. This text was written in in about 830 AD by [[Muhammad ibn Mūsā al-Khwārizmī| Mohammad ibn-Musa al-Khwarizmi]] of [[Baghdad]]. See the [http://www.sjsu.edu/depts/Museum/alkhwa.html Biography of Al-Khwarizmi]</ref>

==Elementary Algebra==

While algebra is a very wide-ranging and advanced topic in theoretical mathematics, the word also refers to a very important topic in middle-school and high-school mathematics. This topic, which is sometimes known as "elementary algebra" to distinguish it from the more advanced branches, is the handling of arithmetical expressions and operations involving quantities that aren't known numbers, but are symbols like x, y, or z. These symbols are generally referred to as "variables" or "unknowns".

Whereas, in elementary arithmetic a typical question might be:
::What is the value of 3+4?
in elementary algebra a typical question might be:
::If x+4=7, what is the value of x?

Of course this example is so simple as to be practically pointless—why would someone ask a question this way?—the use of algebra opens up sophisticated ways of solving sophisticated problems.

Before continuing, let's see how we actually solved that. We subtracted 4 from each side of the equation. On the left side, subtracting 4 canceled the "+4", so we got "x+4-4", which is, of course, just x. On the right side, we got 7-4, which is 3. We used the principle that
::You can apply mathematical operations to something, even if it contains unknowns like "x".
The goal was to get an equation that has just "x" on the left side, with some number on the right. In the original problem, x was on the left side, but hidden with some other mathematical operations. Our job was to strip off those operations, one at a time. Here's a more complicated example:
::<math>\left(\frac{\sqrt{x+31}}{3} + 1\right) \times 6 = 18\,</math>
We undo the multiplication, by dividing both sides by 6:
::<math>\frac{\sqrt{x+31}}{3} + 1 = 3\,</math>
We undo the addition:
::<math>\frac{\sqrt{x+31}}{3} = 2\,</math>
We undo the division:
::<math>\sqrt{x+31} = 6\,</math>
We undo the square root:
::<math>x+31 = 36\,</math>
We undo the addition:
::<math>x = 5\,</math>

As one gets more skillful at these kinds of manipulations, one can handle more complex problems. A typical case involves an equation in which the unknown appears in more than one place. After stripping away as many operations as we can, we might be left with something like:
::<math>3 \times x + 4 \times x = 56\,</math>
At this point we use the ''distributive law'', along with the principle that
::The various laws of mathematics (commutative, associative, distributive) work even when the items appearing in the expressions are unknowns.
The distributive law tells us that
::<math>3 \times x + 4 \times x = (3+4) \times x\,</math>
so we get
::<math>(3+4) \times x = 56\,</math>
and hence x=8. You can plug x=8 into <math>3 \times x + 4 \times x\,</math> and see that the result is 24+32, or 56.

==Polynomial Manipulations==
As one's proficiency increases, one can handle increasingly complicated equations. The next step involves things like this:
::<math>\frac{x^2 - 8 x + 15}{x-3} = 7\,</math>
This can be solved if one knows that
::<math>(x-3) \times (x-5) = x^2 - 8 x + 15\,</math>
so
::<math>\frac{x^2 - 8 x + 15}{x-3} = x-5\,</math>
from which one gets x=12.

This was just a case of the principle that you can perform arithmetic operations even when the expression contain unknowns. In this case we were actually applying the distributive law 3 times.
::<math>(x-3) \times (x-5) = (x) \times (x-5) + (-3) \times (x-5) = (x) \times (x) + (x) \times (-5) + (-3) \times (x) + (-3) \times (-5)\,</math>
When we multiply the sum of two things times the sum of two things, we get the sum of four things. This is an extremely common operation, and there is a mnemonic for it—"FOIL"—which stands for "first, outside, inside, last". We multiply the first item in each parenthesized expression; that's x times x. We multiply the "outside" items; that's x times -5. The "inside" items are -3 times x, and the "last" items are -3 times -5.

This same principle applies when there are more than two items in each factor; it just doesn't have a useful mnemonic word. Each term in the first factor (first parenthesized expression) get multiplied by each term in the second factor, and all the products are added up.

==Quadratic equations==
A quadratic equation is a type of equation in which the degree of x is exactly 2. Usually, these equations have two possible values of x. An example of a quadratic equation would be:
::<math>x^2 - 5x + 6 = 0\,</math>

There are several ways to solve quadratic equations. Let's look at three of the most common and use each to solve the above equation.

===Factoring===
Factoring is a method of solving quadratic equations that involves factoring the quadratic expression into smaller parts, and solving each part individually. It's often the fastest way to solve quadratics, if you can "see" the factors easily. In the case of our above equation, we can indeed factor it:
::<math>x^2 - 5x + 6 = (x - 2)(x - 3)\,</math>

Substituting the factored expression into the original equation yields:
::<math>(x - 2)(x - 3) = 0\,</math>

The key here is to recognize that if multiplying two or more numbers gives a result of zero, at least one of those numbers must itself equal zero. (It is impossible for there to be two nonzero numbers whose product is zero.) So the above equation can be split into two equations:
::<math>x - 2 = 0\,</math>
::<math>x - 3 = 0\,</math>

Thus x must equal either 2 or 3. There is no way to tell which is the "real" value of x, so the actual answer is written as:
::<math>x = 2, 3\,</math>

This is just shorthand for "x is either 2 or 3."

===Completing the square===
Completing the square is a method of solving quadratic equations that involves turning the left-hand side of the equation into a perfect square trinomial and then taking the square root of both sides. In our original equation:
::<math>x^2 - 5x + 6 = 0\,</math>

the left-hand side is not a perfect square trinomial. That is to say, if we tried to take the square root of it, we couldn't simplify it any further, so that wouldn't help us in solving the equation. We need to change the equation to make it a perfect square.

The key here is the coefficient of the linear term, which is the number that's multiplied by the x that isn't squared. For us, that's 5. We divide that number by 2 and then square it, which yields
::<math>(5/2)^2 = 25/4 = 6 + 1/4\,</math>

So if we add 1/4 to both sides, the left side will be a perfect square trinomial. Thus our new equation is:
::<math>x^2 - 5x + 6 + 1/4 = 1/4\,</math>

Or, equivalently:
::<math>(x - 5/2)^2 = 1/4\,</math>

We can now take the square root:
::<math>x - 5/2 = +- 1/2\,</math>

The "+-" indicates that the 1/2 is either positive or negative. The result is thus
::<math>x = 5/2 +- 1/2\,</math>

Since 5/2 + 1/2 = 3, and 5/2 - 1/2 = 2, this is equivalent to the result obtained through factoring.

==Notes and references==
<references/>

[[Category:Algebra]]
[[Category:Mathematics]]

Algebra

2014-02-25T07:54:28Z

AmyD:

{{Math-e}}
'''Algebra''' is a major branch of [[mathematics]] that analyzes the relationships between quantities or items. In higher math the principal fields of algebra are [[linear algebra]], which focuses on matrices, and [[abstract algebra]], which includes [[group theory]].<ref>The name algebra comes from the [[Arabic]] word ''al jebr'', which means reduction or "reunion of broken parts" [http://dictionary.reference.com/browse/algebra Algebra] as mentioned in a book ''Hisab al-jabr w'al-muqabala'' translated as ''Science of the Reunion and the Opposition''. This text was written in in about 830 AD by [[Muhammad ibn Mūsā al-Khwārizmī| Mohammad ibn-Musa al-Khwarizmi]] of [[Baghdad]]. See the [http://www.sjsu.edu/depts/Museum/alkhwa.html Biography of Al-Khwarizmi]</ref>

==Elementary Algebra==

While algebra is a very wide-ranging and advanced topic in theoretical mathematics, the word also refers to a very important topic in middle-school and high-school mathematics. This topic, which is sometimes known as "elementary algebra" to distinguish it from the more advanced branches, is the handling of arithmetical expressions and operations involving quantities that aren't known numbers, but are symbols like x, y, or z. These symbols are generally referred to as "variables" or "unknowns".

Whereas, in elementary arithmetic a typical question might be:
::What is the value of 3+4?
in elementary algebra a typical question might be:
::If x+4=7, what is the value of x?

Of course this example is so simple as to be practically pointless—why would someone ask a question this way?—the use of algebra opens up sophisticated ways of solving sophisticated problems.

Before continuing, let's see how we actually solved that. We subtracted 4 from each side of the equation. On the left side, subtracting 4 canceled the "+4", so we got "x+4-4", which is, of course, just x. On the right side, we got 7-4, which is 3. We used the principle that
::You can apply mathematical operations to something, even if it contains unknowns like "x".
The goal was to get an equation that has just "x" on the left side, with some number on the right. In the original problem, x was on the left side, but hidden with some other mathematical operations. Our job was to strip off those operations, one at a time. Here's a more complicated example:
::<math>\left(\frac{\sqrt{x+31}}{3} + 1\right) \times 6 = 18\,</math>
We undo the multiplication, by dividing both sides by 6:
::<math>\frac{\sqrt{x+31}}{3} + 1 = 3\,</math>
We undo the addition:
::<math>\frac{\sqrt{x+31}}{3} = 2\,</math>
We undo the division:
::<math>\sqrt{x+31} = 6\,</math>
We undo the square root:
::<math>x+31 = 36\,</math>
We undo the addition:
::<math>x = 5\,</math>

As one gets more skillful at these kinds of manipulations, one can handle more complex problems. A typical case involves an equation in which the unknown appears in more than one place. After stripping away as many operations as we can, we might be left with something like:
::<math>3 \times x + 4 \times x = 56\,</math>
At this point we use the ''distributive law'', along with the principle that
::The various laws of mathematics (commutative, associative, distributive) work even when the items appearing in the expressions are unknowns.
The distributive law tells us that
::<math>3 \times x + 4 \times x = (3+4) \times x\,</math>
so we get
::<math>(3+4) \times x = 56\,</math>
and hence x=8. You can plug x=8 into <math>3 \times x + 4 \times x\,</math> and see that the result is 24+32, or 56.

==Polynomial Manipulations==
As one's proficiency increases, one can handle increasingly complicated equations. The next step involves things like this:
::<math>\frac{x^2 - 8 x + 15}{x-3} = 7\,</math>
This can be solved if one knows that
::<math>(x-3) \times (x-5) = x^2 - 8 x + 15\,</math>
so
::<math>\frac{x^2 - 8 x + 15}{x-3} = x-5\,</math>
from which one gets x=12.

This was just a case of the principle that you can perform arithmetic operations even when the expression contain unknowns. In this case we were actually applying the distributive law 3 times.
::<math>(x-3) \times (x-5) = (x) \times (x-5) + (-3) \times (x-5) = (x) \times (x) + (x) \times (-5) + (-3) \times (x) + (-3) \times (-5)\,</math>
When we multiply the sum of two things times the sum of two things, we get the sum of four things. This is an extremely common operation, and there is a mnemonic for it—"FOIL"—which stands for "first, outside, inside, last". We multiply the first item in each parenthesized expression; that's x times x. We multiply the "outside" items; that's x times -5. The "inside" items are -3 times x, and the "last" items are -3 times -5.

This same principle applies when there are more than two items in each factor; it just doesn't have a useful mnemonic word. Each term in the first factor (first parenthesized expression) get multiplied by each term in the second factor, and all the products are added up.

==Quadratic equations==
A quadratic equation is a type of equation in which the degree of x is exactly 2. Usually, these equations have two possible values of x. An example of a quadratic equation would be:
::<math>x^2 - 5x + 6 = 0\,</math>

There are several ways to solve quadratic equations. Let's look at three of the most common and use each to solve the above equation.

===Factoring===
Factoring is a method of solving quadratic equations that involves factoring the quadratic expression into smaller parts, and solving each part individually. It's often the fastest way to solve quadratics, if you can "see" the factors easily. In the case of our above equation, we can indeed factor it:
::<math>x^2 - 5x + 6 = (x - 2)(x - 3)\,</math>

Substituting the factored expression into the original equation yields:
::<math>(x - 2)(x - 3) = 0\,</math>

The key here is to recognize that if multiplying two or more numbers gives a result of zero, at least one of those numbers must itself equal zero. (It is impossible for there to be two nonzero numbers whose product is zero.) So the above equation can be split into two equations:
::<math>x - 2 = 0\,</math>
::<math>x - 3 = 0\,</math>

Thus x must equal either 2 or 3. There is no way to tell which is the "real" value of x, so the actual answer is written as:
::<math>x = 2, 3\,</math>

This is just shorthand for "x is either 2 or 3."

==Notes and references==
<references/>

[[Category:Algebra]]
[[Category:Mathematics]]

Algebra

2014-02-25T07:38:54Z

AmyD:

Mathematics

2014-02-25T07:31:52Z

AmyD: /* Branches of Mathematics */

'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems. The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>

Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.

==Symbols, Equations, and Theories==

Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two [[function]]s.

A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.

==Pure and Applied Mathematics==
[[Applied mathematics]] concerns the use of mathematical methods for practical purposes. [[Pure mathematics]] involves reasoning about abstract structures.
====Applied mathematics====
Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as [[physics]], [[chemistry]], [[medicine]], and [[biology]], as well as [[engineering]], [[mechanics]] and [[technology]]. [[Economics]] and [[information theory]] also uses applied mathematics. Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.

====Pure mathematics====
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.

==Branches of Mathematics==

====[[Arithmetic]]====
Arithmetic is the study of combination of [[number]]s. Its basic operations are addition, subtraction, multiplication and division.

====[[Algebra]]====
Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in [[number theory]], but they can also can be more general structures such as [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an [[additive inverse]] and division is considered to be multiplication by a [[multiplicative inverse]]. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of [[algebraic structures]] such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].

====[[Analysis]]====
Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[calculus]], including the calculus of several variables, [[vector calculus]] and [[tensor calculus]]. Also included is [[numerical analysis]], the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include [[real analysis]] and [[complex analysis]].

====[[Chaos Theory]]====
Chaos theory is the study of systems that are highly sensitive to initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

====[[Combinatorics]]====
Combinatorics is the study of situations in which elements of a set or sets are combined or permuted in various ways. An example of a combinatorics problem would be "in a group of six men and four women, how many possible ways are there to choose three men and two women?" Derangements are another concept in combinatorics. A derangement is a re-ordering of a set so that no element ends up where it was originally; a derangement problem typically asks how many arrangements are possible for a given set, meeting given conditions.

====[[Game Theory]]====
Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

====[[Geometry]]====
Geometry is the study of shapes and special relationships. It was defined by [[Felix Klein]] as the study of [[invariant]]s under [[Group (mathematics)|group]]s of [[transformation]]s. For example, the [[Euclidean transformation]]s are [[translation]], [[rotation]] and [[reflection]]. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in [[Euclidean geometry]]. Other types of transformations, such as the [[affine transformation]]s, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

====[[Logic]]====
Logic is the study of [[reasoning]]. It examines general forms that [[argument]]s can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.

====[[Set theory]]====
Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since all of mathematics can be expressed in terms of [[set]]s. Sets are defined by a collection of [[axiom]]s called the [[Zermelo-Fraenkel]] axioms. One of the axioms, the [[Axiom of Choice]], has been the subject of much discussion

====[[Probability and Statistics]]====
Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.

====[[Topology]]====
Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with [[Leonard Euler]]'s consideration of the [[Königsberg Bridges Problem]], which also introduced [[Graph Theory]]. [[Beck's map of the London Underground]] in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. [[Differential geometry]] is a specialized field of its own.

====[[Trigonometry]]====
In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, senoidal functions are used to describe oscillatory motion and waves.

== References ==

<references/>
[[Category:Mathematics]]

Mathematics

2014-02-25T07:15:50Z

AmyD: /* Topology */

'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems. The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>

Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.

==Symbols, Equations, and Theories==

Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two [[function]]s.

A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.

==Pure and Applied Mathematics==
[[Applied mathematics]] concerns the use of mathematical methods for practical purposes. [[Pure mathematics]] involves reasoning about abstract structures.
====Applied mathematics====
Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as [[physics]], [[chemistry]], [[medicine]], and [[biology]], as well as [[engineering]], [[mechanics]] and [[technology]]. [[Economics]] and [[information theory]] also uses applied mathematics. Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.

====Pure mathematics====
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.

==Branches of Mathematics==

====[[Arithmetic]]====
Arithmetic is the study of combination of [[number]]s. Its basic operations are addition, subtraction, multiplication and division.

====[[Algebra]]====
Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in [[number theory]], but they can also can be more general structures such as [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an [[additive inverse]] and division is considered to be multiplication by a [[multiplicative inverse]]. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of [[algebraic structures]] such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].

====[[Analysis]]====
Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[calculus]], including the calculus of several variables, [[vector calculus]] and [[tensor calculus]]. Also included is [[numerical analysis]], the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include [[real analysis]] and [[complex analysis]].

====[[Chaos Theory]]====
Chaos theory is the study of systems that are highly sensitive to its initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

====[[Game Theory]]====
Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

====[[Geometry]]====
Geometry is the study of shapes and special relationships. It was defined by [[Felix Klein]] as the study of [[invariant]]s under [[Group (mathematics)|group]]s of [[transformation]]s. For example, the [[Euclidean transformation]]s are [[translation]], [[rotation]] and [[reflection]]. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in [[Euclidean geometry]]. Other types of transformations, such as the [[affine transformation]]s, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

====[[Logic]]====
Logic is the study of [[reasoning]]. It examines general forms that [[argument]]s can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.

====[[Set theory]]====
Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since all of mathematics can be expressed in terms of [[set]]s. Sets are defined by a collection of [[axiom]]s called the [[Zermelo-Fraenkel]] axioms. One of the axioms, the [[Axiom of Choice]], has been the subject of much discussion

====[[Probability and Statistics]]====
Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.

====[[Topology]]====
Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with [[Leonard Euler]]'s consideration of the [[Königsberg Bridges Problem]], which also introduced [[Graph Theory]]. [[Beck's map of the London Underground]] in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. [[Differential geometry]] is a specialized field of its own.

====[[Trigonometry]]====
In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, senoidal functions are used to describe oscillatory motion and waves.

== References ==

<references/>
[[Category:Mathematics]]

Mathematics

2014-02-25T07:13:11Z

AmyD: /* Branches of Mathematics */

'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems. The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>

Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.

==Symbols, Equations, and Theories==

Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two [[function]]s.

A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.

==Pure and Applied Mathematics==
[[Applied mathematics]] concerns the use of mathematical methods for practical purposes. [[Pure mathematics]] involves reasoning about abstract structures.
====Applied mathematics====
Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as [[physics]], [[chemistry]], [[medicine]], and [[biology]], as well as [[engineering]], [[mechanics]] and [[technology]]. [[Economics]] and [[information theory]] also uses applied mathematics. Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.

====Pure mathematics====
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.

==Branches of Mathematics==

====[[Arithmetic]]====
Arithmetic is the study of combination of [[number]]s. Its basic operations are addition, subtraction, multiplication and division.

====[[Algebra]]====
Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in [[number theory]], but they can also can be more general structures such as [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an [[additive inverse]] and division is considered to be multiplication by a [[multiplicative inverse]]. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of [[algebraic structures]] such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].

====[[Analysis]]====
Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[calculus]], including the calculus of several variables, [[vector calculus]] and [[tensor calculus]]. Also included is [[numerical analysis]], the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include [[real analysis]] and [[complex analysis]].

====[[Chaos Theory]]====
Chaos theory is the study of systems that are highly sensitive to its initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

====[[Game Theory]]====
Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

====[[Geometry]]====
Geometry is the study of shapes and special relationships. It was defined by [[Felix Klein]] as the study of [[invariant]]s under [[Group (mathematics)|group]]s of [[transformation]]s. For example, the [[Euclidean transformation]]s are [[translation]], [[rotation]] and [[reflection]]. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in [[Euclidean geometry]]. Other types of transformations, such as the [[affine transformation]]s, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

====[[Logic]]====
Logic is the study of [[reasoning]]. It examines general forms that [[argument]]s can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.

====[[Set theory]]====
Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since all of mathematics can be expressed in terms of [[set]]s. Sets are defined by a collection of [[axiom]]s called the [[Zermelo-Fraenkel]] axioms. One of the axioms, the [[Axiom of Choice]], has been the subject of much discussion

====[[Probability and Statistics]]====
Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.

====[[Topology]]====
It is the study of special properties that are preserved under continuous deformation of objects. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with [[Leonard Euler]]'s consideration of the [[Königsberg Bridges Problem]], which also introduced [[Graph Theory]]. [[Beck's map of the London Underground]] in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. [[Differential geometry]] is a specialized field of its own.

====[[Trigonometry]]====
In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, senoidal functions are used to describe oscillatory motion and waves.

== References ==

<references/>
[[Category:Mathematics]]

User talk:AmyD

2014-02-23T02:04:13Z

AmyD: Created page with "Hello Conservapedia! I'm a math student from the University of Nevada, Las Vegas. I was drawn to the idea of an online encyclopedia and educational resource that didn't toe the..."

Hello Conservapedia! I'm a math student from the University of Nevada, Las Vegas. I was drawn to the idea of an online encyclopedia and educational resource that didn't toe the scientific "party line" on evolution. I hope to contribute positively to Conservapedia's math articles.