Numbers, Real

A real number line is a familiar way to picture various sets of numbers. For example, the divisions marked on a number line show the integers, which are the counting numbers {1, 2, 3,…}, with their opposites {−1, −2, −3,…}, with the number 0, which divides the positive numbers on the line from the negative numbers.

But what other numbers are on a real number line? One could make marks for all the fractions, such as , and so forth, as well as marks for all the decimal fractions, such as 0.1, −0.01, 0.0000001, and so on. Any number that can be written as the ratio of two integers (such as , , and so on), where the divisor is not 0, is called a rational number, and all the rational numbers are on a real number line.

Are there any other kinds of numbers on a real number line in addition to the integers and rational numbers? What about , which is approximately 1.4142? Because the decimal equivalent for never ends and never repeats, it is known as an irrational number. The set of real numbers consists of the integers and the rational numbers as well as the irrational numbers. Every real number corresponds to exactly one point on a real number line, and every point on a number line corresponds to exactly one real number.

Are there any "unreal" numbers? That is, are there any numbers that are not on a real number line? The set of real numbers is infinitely large, and one might think that it contains all numbers, but that is not so. For example, the solution to the equation x² = −1 does not lie on a real number line. The solution to that equation lies on another number line called an imaginary number line, which is usually drawn at right angles to a real number line.

There are numbers, called complex numbers, that are the sum of a real number and an imaginary number and are not found on either a real or an imaginary number line. These complex numbers are found in an area called the complex plane. So both imaginary and complex numbers are "unreal," so to speak, because they do not lie on a real number line.

The set of real numbers has several interesting properties. For example, when any two real numbers are added, subtracted, multiplied, or divided (excluding division by zero), the result is always a real number. Therefore, the set of real numbers is called "closed" for these four operations.

Similarly, the real numbers have the commutative, associative, and distributive properties. The real numbers also have an identity element for addition (0) and for multiplication (1) and inverse elements for all four operations. These properties, taken all together, are called the field properties, and the real numbers thus make up a field, mathematically speaking.