Irrational numbers are numbers that are neither whole numbers (like 2, 0, or -3) nor ratios of whole numbers. Irrational numbers are real numbers in the sense that they appear in measurements of geometric objects--for example, the number pi (), which is the ratio of the circumference of a circle to the length of its diameter, is an irrational number. However, irrational numbers cannot be represented as decimals, unlike rational numbers, which can be expressed either as finite decimals or as infinite decimals that eventually follow a repeating pattern. For instance, the decimal 6.412121212... is equal to 6348/990. By contrast, irrational numbers have infinitely long decimal expansions that never form a repeating pattern. Thus, the number pi can never be written down exactly in decimal form, it can only be approximated, by decimals such as 3.14159.

Irrational numbers were discovered in the school of Pythagoras, a great Greek mathematician who founded a...