Decimal Expansion of Rational Numbers

Decimal Expansion of Rational Numbers (\[\frac{x}{y}\])

Before we try and understand what decimal numbers are, let us first decode the rational numbers decimal expansion. Rational numbers are any numbers that can be represented in the form of j / k where j and k are integers and k ≠ 0. When you try and simplify rational numbers, it results in decimals. Let us look into how the rational numbers are converted into decimal numbers. Some of the examples of rational numbers are 8, -7.1, \[\frac{5}{6}\], and many more are all the examples of rational numbers. 

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Different Types of Decimal Numbers

Decimal numbers are of two types-

  1. Terminating

  2. Non- terminating, which is further classified as repeating and non-repeating.

Recurring or terminating real numbers are rational numbers. Consider the number 33.33333…. which is a rational number that can also be represented in the form of 100 / 3 and 0.3333… is the recurring or the non-terminating repeating part of the number. Decimals like 0.7896, 0.043 and .69 can be written as 7896 / 104, 43 / 103 and 69 / 10respectively as fractions. Therefore any decimal number can be represented as fractions that have denominators in powers of 10 whose prime factors are 2.5. It can thus be concluded that any decimal rational numbers can be represented in the form of j / k such that j and k are integers and the prime factorization of k is in the form of 89y, where x and y are non-negative integers.


How to Expand Rational Numbers in Decimals?

Rational numbers are the numbers that usually are recurring or terminating by nature. 

Example: Expand \[\overline{5.31}\].

Solution: \[\overline{5.31}\] =  5 + 0.31 + 0.0031 + 0.000031 + 0.00000031 + 0.0000000031 + 0.000000000031 + . . .

\[\overline{5.31}\] =  5 + 31 x 10-2 + 31 x 10-4 + 31 x 10-6 + 31 x 10-8 + 31 x 10-10 + . . . . . .

= 5 + \[\sum_{k=1}^{\infty}\] \[(31.10^{-2})\]\[(10^{-2})^{k-1}\]

= 5 + \[\frac{(31\times 10^{-2})}{1-10^{-2}}\]

= 5 + \[\frac{31}{99}\]

= \[\frac{485+31}{99}\]

= \[\frac{526}{99}\]


To understand better, let’s take another example. Consider a number 33.3333……This number is considered as a rational number as it can be represented 100 / 3. The recurring part .333 is the decimal part that never terminates. Some examples of terminating decimals are 6.7456, 38.34, 23.015, etc. These numbers satisfy the conditions for being a rational number. Consider one of the examples: 6.7456. This can be written as 67456 / 10000 or \[\frac{67456}{10^{4}}\].  In the same manner, you can write the other examples as 3834 / 100 or \[\frac{3834}{10^{2}}\], 23015 / 1000, or \[\frac{23.015}{10^{3}}\] in the fractional form. 


Hence, you can see that the decimal numbers can be represented as shown above. The denominator is in the powers of 10. Since the prime factors of 10 are 5 and 2, you can represent any decimal rational number as a fraction in the form of j / k. Here, j and k can also be represented in the form 2x 5y and x and y are positive integers. With this, it gives rise to theorems. 


Theorem 1: If m is any rational number by nature whose decimal expansion is terminating, it can be expressed in the form of j / k, where j and k are co-primes and the prime factorization of k is of the form 25y, where x and y are non-negative integers.


The converse of this theorem holds good and can be stated as follows - 


Theorem 2: If m is a rational number which can be represented in j / k form and the prime factorization of k is 25y, where x and y are non-negative integers, then x is said to have a terminating decimal expansion.     


Let’s understand the following examples to understand better.

  1. \[\frac{3}{4}\] = \[\frac{3}{2^{2}}\] = \[\frac{3\times 5^{2}}{2^{2}\times 5^{2}}\] = \[\frac{75}{10^{4}}\]

  2. \[\frac{5}{8}\] = \[\frac{5}{2^{3}}\] = \[\frac{5\times 5^{3}}{2^{3}\times 5^{3}}\] = \[\frac{125}{10^{3}}\]


The following theorem can be stated for the decimal expansion of rational numbers that are recurring.


Theorem 3: If n is a rational number which can be represented as the ratio of two integers j / k and the prime factorization of k does not take the form of 25y, where x and y are non-negative integers, it can be said that x has a decimal expansion which is repeating. 


To understand better, check these rational number to decimal examples below: 

  1. \[\frac{3}{7}\]= 0.4285714285714. . . . . . . . . . = 0.43.

  2. \[\frac{7}{13}\]= 0.538461538461. . . . . . . . . . = 0.54.

  3. \[\frac{9}{13}\]= 0.692307692307. . . . . . . . . . . = 0.69.


Solved Examples

Below are the examples of how to expand rational numbers in decimals.


Case 1: When the remainder is equal to zero. 


Question 1: Find the decimal expansion of 4 / 8



0 . 5

      8

4 . 0

4 . 0

    0

Here, the remainder of 4 / 8 = 0 and the quotient is 0.5 When 4 is divided by 8, it has a terminating decimal. 


Case 2: When the remainder is not equal to zero.


Question 2: Find the decimal expansion of 7 / 12


0 . 5 8 3 3 3 3 3 

            12

7 . 0 0 0 0 0 0 0 

6   0


1   0 0

     9 6


    0 4 0

        3 6


        0 4 0

           3 6


          0 4 0

              3 6


              0 4 0

                 3 6


                0 4 0

                    3 6


                    0 4                 


Here, the remainder of 7 / 12 = 04 and the quotient is 0.583333 When 7 is divided by 12, it has a recurring decimal.

FAQ (Frequently Asked Questions)

1) What are the Theorems of Decimal Number Expansion of the Rational Number?

Theorem 1: If m is any rational number by nature whose decimal expansion is terminating, it can be expressed in the form of j / k, where j and k are co-primes and the prime factorization of k is of the form 2x  5y, where x and y are non-negative integers.


The converse of this theorem holds good and can be stated as follows:

Theorem 2: If m is a rational number which can be represented in j / k form and the prime factorization of k is 2x  5y, where x and y are non-negative integers, then x is said to have a terminating decimal expansion.     


Theorem 3: If n is a rational number which can be represented as the ratio of two integers j / k and the prime factorization of k does not take the form of 2x  5y, where x and y are non-negative integers, it can be said that x has a decimal expansion which is repeating.