Before going into representation of Decimal number let us understand what rational numbers are.

Any number that can be represented in the form of \(\frac{p}{q}\)

**Examples:** \( 6 , -8.1,\frac{4}{5}\)

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

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For example, consider the number 33.33333……. It is a rational number as it can be represented in form of \(\frac{100}{3}\)**non-terminating repeating** part i.e.it is recurring decimal number.

Also the terminating decimals such as 0.375, 0.6 etc. which satisfy the condition of being rational (\(0.25\)

Consider any decimal number. For e.g. 0.567. It can be written as \(\frac{567}{1000}\)

Thus, it can be seen that any decimal number can be represented as a fraction which has denominator in powers of 10. As it is known 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 \(\frac{p}{q}\)

This statement gives rise to a very important theorem, which is as follows:

**Theorem 1:** If m be any rational number whose decimal expansion is terminating in nature, then m can be expressed in form of \(\frac{p}{q}\)

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. \(\frac{p}{q}\)

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. \(\frac{p}{q}\)

Consider the following examples:

- \(\frac{1}{6}\)
= \(0.1666….\) = \(0.16\) ̇ - \(\frac{7}{12}\)
= \(0.58333…\) = \(0.583 \) - \(\frac{9}{11}\)
= \(0.8181…\) = \(0.\overline{81}\) <

Rational Number to decimal:

**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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