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Portal:Mathematics

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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

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Exp derivative at 0.svg
e is the unique number such that the slope of y=ex (blue curve) is exactly 1 when x=0 (illustrated by the red tangent line). For comparison, the curves y=2x (dotted curve) and y=4x (dashed curve) are shown.
Image credit: Dick Lyon

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:

  1. 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.
  2. The global maximizer of the function
     f(x) = x^{1 \over x}.
  3. 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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graph of an increasing curve showing cumulative share of income earned versus cumulative share of people from lowest to highest income
Credit: BenFrantzDale

A Lorenz curve shows the distribution of income in a population by plotting the percentage y of total income that is earned by the bottom x percent of households. It is usually plotted with a diagonal line (reflecting a hypothetical "equal" distribution of incomes) for comparison. An example of a cumulative distribution function, the curve was developed by economist Max O. Lorenz in 1905 to describe income inequality. A derived quantity is the Gini coefficient, first published in 1912 by Corrado Gini, which is the ratio of the area between the diagonal line and the Lorenz curve (area A in this graph) to the area under the diagonal line (the sum of A and B); higher Gini coefficients reflect more income inequality. See also the Pareto principle and power law.

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