Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Irrational Numbers

While every rational number p/q is equivalent to a terminating or repeating decimal, not all decimals are equivalent to a fraction. Decimals that cannot be represented as a ratio of integers, p/q, are called irrational. For example, the following decimal numbers have structures that make them non-repeating and non-terminating:

· 0.01001000100001000001 …

· 0.21221222122221222221 …

· 0.12123123412345123456 …

The most “famous” irrational number, p, has been computed to millions of decimal places. Since a decimal with this many places is effectively unusable, calculations involving p are based on approximations like 3.14159 … While this approximation is probably sufficient for most applications in school mathematics, one should never confuse approximations of p with the “real thing”, which is neither terminating nor repeating. As a matter of practicality, may elementary textbooks approximate p with the fraction 22/7, which is equivalent to the repeating decimal . Unfortunately, use of this approximation leads some students to mistakenly conclude that p is a rational number.

Other common irrational numbers include the square roots of many counting numbers. For example, Like p, the decimal representations of these irrational numbers can never be written out completely. Any attempt to write them as decimals must be viewed as an approximation. For that reason, many people prefer to represent irrational numbers with symbols rather than approximations (e.g., p rather than 3.14159 …) whenever possible.

It has been know since the time of Euclid that the square roots of many integers are irrational. The following proof that is irrational begins by assuming the opposite of what we hope to prove. Assume that is rational. Therefore is equivalent to a ratio p/q in which the integers p and q have no common factors (i.e., p/q is reduced to lowest terms). The following proof demonstrates how this assumption leads to contradictions of other known facts. This is sufficient grounds to say that the assumption is false, therefore is irrational. This approach is called indirect proof.