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This portal is for the academic discipline of mathematics. For related portals of logic and statistics, please see portals: mathematics, logic, and statistics.

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Mathematics, from the Greek: μαθηματικά or mathēmatiká, is the study of quantities (numbers) and their operations, interrelations, combinations, generalizations, and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. It evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

There are approximately 20733 mathematical articles in Wikipedia.

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e is the unique number such that the slope of y=e^x (blue curve) is exactly 1 when x=0 (illustrated by the red tangent line). For comparison, the curves y=2^x (dotted curve) and y=4^x (dashed curve) are shown.

The mathematical constant e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:

The sum of the infinite series

$\begin{align} e & = \sum_{n = 0}^\infty \frac{1}{n!} \\ & = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \\ \end{align}$

where n! is the factorial of n.
The global maximum of the function

$f(x) = x^{1 \over x}.$
The limit:

$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$

The number e is also the base of the natural logarithm. Since e is transcendental, and therefore irrational, its value can not be given exactly. The numerical value of e truncated to 20 decimal places is 2.71828 18284 59045 23536.

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Image credit: Dick Lyon

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In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or parallel postulate is equivalent to:

Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line (see 1).

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).

And hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line (see 3).

These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.

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More mathematics categories

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...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
...that every natural number can be written as the sum of four squares?
...that the largest known prime number is over 12 million digits long?
...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
...that it is unknown whether π and e are algebraically independent?
...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

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