Math CentralQuandaries & Queries


Question from Peter, a student:

How do you take a random, non-terminating, non-repeating decimal into a fraction?

Hi Peter.

A fraction is a ratio of finite integers. Whenever you have a ratio of integers, the corresponding decimal value either terminates or repeats in no more digits than the size of the denominator. For example, 3/7
repeats in 7 or less digits: 0.428571428571428571.... in fact, it repeats in 6 digits. 1/25 repeats or terminates in less than 25 digits: 1/25 = 0.04 (terminates).

Thus if a decimal NEVER repeats and NEVER terminates, it cannot be represented as a ratio of two integers. So we call such number irrational (not a ratio). An example of this is the number π. The decimal equivalent of π never repeats and never terminates (here's a link to the first 70 billion digits). Other common examples are any square root except for roots of perfect squares and the number φ which is the Golden Ratio.

Stephen La Rocque.

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