number (mathematics)

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Symbol used in counting or measuring. In mathematics, there are various kinds of number. The everyday number system is the decimal (‘proceeding by tens’) system, using the base ten.

Real numbers include all rational numbers (integers, or whole numbers, and fractions) and irrational numbers (those not expressible as fractions). Complex numbers include the real and imaginary numbers (real-number multiples of the square root of -1). The binary number system, used in computers, has two as its base. The natural numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, give a counting system that, in the decimal number system, continues 10, 11, 12, 13, and so on. These are whole numbers (integers), with fractions represented as, for example, 1/4, 1/2, 3/4, or as decimal fractions (0.25, 0.5, 0.75). They are also rational numbers.

Irrational numbers cannot be represented in this way and require symbols, such as √2, π, and e. They can be expressed numerically only as the (inexact) approximations 1.414, 3.142, and 2.718 (to three places of decimals), respectively. The symbols π and e are also examples of transcendental numbers, because they (unlike √2) cannot be derived by solving a polynomial equation (an equation with one variable quantity) with rational coefficients (multiplying factors). Complex numbers, which include the real numbers as well as imaginary numbers, take the general form a + bi, where i = √-1 (that is, i² = -1), and a is the real part and bi the imaginary part.

Evolution of number systems
The ancient Egyptians, Greeks, Romans, and Babylonians all evolved number systems, although none had a zero, which was introduced from India by way of Arab mathematicians in about the 8th century and allowed a place-value system to be devised on which the decimal system is based. Other number systems have since evolved and have found applications. For example, numbers to base two (binary numbers), using only 0 and 1, are commonly used in digital computers to represent the two-state ‘on’ or ‘off’ pulses of electricity. Binary numbers were first developed by German mathematician Gottfried Leibniz in the late 17th century.

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