Real Numbers

Real Numbers, class of numbers comprising all positive and negative numbers, together with zero. Real numbers include the rational numbers. Rational numbers comprise all numbers that are equal to the quotient (result of dividing one number by another) of two integers, which are the positive and negative whole numbers: 1, 2, 3…, -1, -2, -3…, and 0. Thus the numbers Š, 7 (7/1), and –1.2 (-6/5) are rational numbers. In addition to rational numbers, real numbers include irrational numbers. Irrational numbers are numbers such as the square root of two (Ã), pi (p), and e that are not the quotient of any two integers. The real numbers are a subset of the complex numbers, which also include the set of imaginary numbers—numbers that are a multiple of i, where i is the square root of –1—as well as numbers that are a combination of real and imaginary numbers, such as 2 + 3i.

Real numbers can all be written as decimal numbers. The decimals may have a definite termination point (such as 5 or 3.427), endlessly repeat in a pattern (such as 2.12121…), or continue forever with no pattern (3.14159265…).

The idea of real numbers arose when ancient Greek mathematicians encountered difficulties with using only rational numbers. They discovered, for example, that à is not rational. The numbers p and e are often encountered in geometry and physics (p occurs in the equations for the area and perimeter of a circle, for instance). The recognition of these important 'irrational' numbers resulted in the creation of the set of real numbers.