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...