## Decimal Representation Of Rational Numbers

**Theorem:
**

Let be a rational number such that and prime factorization of q is of the form where m, n are non-negative integers then x has a decimal representation which terminates.

For example :

**Theorem:
**

Let be a rational number such that and prime factorization of q is not of the form , where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

For example :

## Solved Examples Based on Decimal Representation Of Rational Numbers

### Question:

Without actually calculating, state whether the following rational numbers have a terminating or non-terminating repeating decimal expansion.

(i) (ii) (iii)

Hint: If the denominator is of the form for some non negative integer m and n, then rational number has terminating decimal otherwise non terminating.

### Solution:

(i)

Since which is not of the form .

\ It has non terminating decimal representation.

(ii)

Since q = which is of the form .

\ It has a terminating decimal representation.

(iii)

Since is not of the form . It has a non-terminating decimal representation.

### Question:

What can you say about the prime factorization of the denominators of the following rationales:

(i) 36.12345 (ii)

### Solution:

(i) Since 36.12345 has terminating decimal expansion. So, its denominator is of the form where m, n are non-negative integers.

(ii) Since has non terminating repeating decimal expansion. So, its denominator has factors other than 2 or 5.