Decimal Expansion Of Rational Numbers

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

Any number that can be represented in the form of \(\frac{p}{q}\) , such that \(p\) and \(q\) are integers and \(q ≠ 0\) are known as rational numbers.

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

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

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Decimal - Expansion Of Rational Numbers

For example, consider the number 33.33333……. It is a rational number as it can be represented in form of \(\frac{100}{3}\). It can be seen that the decimal part .333…… is the 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 (\(0.25\) = \(\frac{3}{2^3}\) ,\(0.6\) = \(\frac{3}{5}\)) are rational numbers.

Consider any decimal number. For e.g. 0.567. It can be written as \(\frac{567}{1000}\) 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. 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 \(\frac{p}{q}\) , 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, 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 \(\frac{p}{q}\), 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, it can be said that x has a decimal expansion which is terminating.

Consider the following examples:

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

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 x has a decimal expansion which is non-terminating repeating (recurring).

Consider the following examples:

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

Rational Number to decimal:

Case 1: Remainder equal to zero

Example: Find the decimal expansion of 3/6.

Rational numbers

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.

Rational numbers

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=recurring 

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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Practise This Question

A rational number pq has a terminating decimal expansion if prime factorisation of q has: