How To Tell That A Number Is Rational

A rational number is, as the name implies, any number that can be expressed as a ratio, or fraction. The number 6 is rational number because it can be expressed as 6/1, though this would be unusual. 4.5 is a rational number, as it can be represented as 9/2.

Many important numbers in mathematics, however, are irrational, and cannot be written as ratios. These include pi, or π, which is the ratio of the circumference of a circle to its diameter and is equal to 3.141592654...; and the square root of 5, equal to 2.236067977... The trailing dots indicate an infinite, non-repeating series of digits to the right of the decimal point.

A number of methods exist for determining whether a number is rational.

Any number that can be written as a fraction or a ratio is a rational number. The product of any two rational numbers is therefore a rational number, because it too may be expressed as a fraction. For example, 5/7 and 13/120 are both rational numbers, and their product, 65/840, is also a rational number. (65/140 reduces to 13/28, but this is not vital for present purposes.)

This is less trivial than it may seem, because it is easy to forget that whole numbers (... −3, −2, −1, 0, 1, 2, and so on) can be written as fractions with a denominator of 1, e.g., −3/1, −2/1, and so on.

Importantly, some numbers that contain an infinite sequence of numbers to the right of a decimal sign are rational; the key is that this must include a repeating sequence. For example

\(0.444444... = \frac{4}{9} \text{ and } 0.285714285714... = \frac{2}{7}\)

The repeating segment is often signified by a bar over the repeating portion:

\(0.444444... = 0.\bar{4} \text{ and } 0.285714285714... = 0.\overline{285714}\)

Most numbers that are expressed as square roots are irrational numbers. The exceptions are so-called perfect squares, which are the squares of whole numbers (0² = 0, 1² = 1, 2² = 4, 3² = 9, 4² = 16, e.t.c.).