# Decimal Expansion Of Rational Numbers

Before going into a representation of the decimal expansion of rational numbers, let us understand what rational numbers are. Any number that can be represented in the form of p/q, such that p and q are integers and q ≠ 0 are known as rational numbers. So when these numbers have been simplified further, they result in decimals. Let us learn how to expand such decimals here.

**Examples:** \( 6 , -8.1,\frac{4}{5}\) etc. are all examples of rational numbers.

## How to Expand Rational Numbers in Decimals?

The real numbers which are recurring or **terminating** in nature are generally rational numbers.

For example, consider the number 33.33333……. It is a rational number as it can be represented in the form of 100/3. It can be seen that the decimal part .333…… is the **non-terminating repeating** part, i .e. it is a recurring decimal number.

Also the terminating decimals such as 0.375, 0.6 etc. which satisfy the condition of being rational (\(0.375\) = \(\frac{3}{2^3}\) ,\(0.6\) = \(\frac{3}{5}\)).

Consider any decimal number. For e.g. 0.567. It can be written as 567/1000 or \(\frac{567}{10^3}\) . Similarly, the numbers 0.6689,0.032 and .45 can be written as \(\frac{6689}{10^4}\) ,\(\frac{32}{10^3}\) and \(\frac{45}{10^2}\) respectively in fractional form.

Thus, it can be seen that any decimal number can be represented as a fraction which has denominator in powers of 10. We know that prime factors of 10 are 2 and 5, it can be concluded that any decimal rational number can be easily represented in the form of \(\frac{p}{q}\), such that p and q are integers and the prime factorization of q is of the form \(2^x~ 5^y\), where x and y are non-negative integers.

This statement gives rise to a very important theorem.

### Theorems

**Theorem 1:** If m be any rational number whose decimal expansion is terminating in nature, then m can be expressed in form of \(\frac{p}{q}\), where p and q are co-primes and the prime factorization of q is of the form \(2^x~ 5^y\), where x and y are non-negative integers.

The converse of this theorem is also true and it can be stated as follows:

**Theorem 2:** If m is a rational number, which can be represented as the ratio of two integers i.e. \(\frac{p}{q}\) and the prime factorization of q takes the form \(2^x~ 5^y\), where x and y are non-negative integers then, then it can be said that m has a decimal expansion which is terminating.

Consider the following examples:

- \(\frac{7}{8}\) = \(\frac{7}{2^3}\) = \(\frac{7~×~5^3}{2^3~×~5^3}\) = \(\frac{875}{10^3}\)
- \(\frac{3}{80}\) = \(\frac{3}{2^4~×~5}\) = \(\frac{3~×~5^3}{2^4~×~5^4}\) = \(\frac{375}{10^4}\)

Moving on, to decimal expansion of rational numbers which are recurring, the following theorem can be stated:

**Theorem 3:** If m is a rational number, which can be represented as the ratio of two integers i.e. \(\frac{p}{q}\) and the prime factorization of q does not takes the form \(2^x~ 5^y\), where x and y are non-negative integers. Then, it can be said that m has a decimal expansion which is non-terminating repeating (recurring).

Consider the following examples:

- \(\frac{1}{6}\) = \(0.1666….\) = \(0.1\overline{6}\)
- \(\frac{7}{12}\) = \(0.58333…\) = \(0.58\overline{3} \)
- \(\frac{9}{11}\) = \(0.8181…\) = \(0.\overline{81}\)

### Rational Number to decimal Examples

**Case 1: Remainder equal to zero**

**Example: Find the decimal expansion of 3/6.**

Here, the quotient is 0.5 and the remainder is 0. Rational number 3/6 results in a terminating decimal.

**Case 2: Remainder not equal to zero*** *

**Example: Express 5/13 in decimal form***.*

Here, the quotient is 0.384615384 and the remainder is not zero. Notice that the number…384 after the decimal is repeating. Hence, 5/13 gives us a non-terminating recurring decimal expansion. And this can be written as 5/13=

**A rational number gives either terminating or non-terminating recurring decimal expansion. **Thus, we can say that a number whose decimal expansion is terminating or non-terminating recurring is rational.* *

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