RATIONAL NUMBERS AND THEIR DECIMAL EXPANSION

Rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion.

In this section, we are going to consider a rational number, say hat a/b ≠ 0, and explore exactly when the decimal expansion of a/b is terminating and when it is non-terminating repeating or recurring.

We can do so by considering several examples.

Consider the following rational numbers.

0.375

0.104

0.0875

23.3408

We can express the above rational numbers whose denominators are powers of 10.

Let us try and cancel the common factors between the numerator and denominator and see what we get.

Do you see any pattern?

It appears that, we have converted a real number whose decimal expansion terminates to a rational number of the form a/b where 'a' and 'b' are relatively prime, and the prime factors of the denominator (that is b) has only powers of 2 or powers of 5 or both.

We should expect the denominator to look like this, since powers of 10 can only have powers of 2 and 5 as factors.

Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form a/b, where the prime factors of 'b' is of the form 2ⁿ5^m, where 'n' and 'm' are some non-negative integers.

Let us write the results formally :

From the discussion above, we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.

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