DECIMAL REPRESENTATION OF RATIONAL NUMBERS

Let us consider the rational number p/q, where both p and q are integers and q  ≠  0.

We can get the decimal representation of the rational number p/q by long division division.

When we divide p by q using long division method either the remainder becomes zero or the remainder never becomes zero and we get a repeating string of remainders.

Case 1 (Remainder = 0) :

Let us express 7/16  in decimal form. Then 7/16  =  0.4375.

In this example, we observe that the remainder becomes zero after a few steps.

Also the decimal expansion of 7/16 terminates.

Similarly, using long division method we can express the following rational numbers in decimal form as

1/2 = 0.5

7/5 = 1.5

-8/25 = -0.32

In the above examples, the decimal expansion terminates or ends after a finite number of steps.

Key Concept (Terminating Decimal) :

When the decimal expansion of p/q, q ≠ 0 terminates (i,e., comes to an end), the decimal expansion is called terminating.

Case 2 ( Remainder ≠ 0) :

Does every rational number has a terminating decimal expansion?

Before answering the question, let us express 5/11 and 7/6 in decimal form. 

Thus, the decimal expansion of a rational number need not terminate.

In the above examples, we observe that the remainders never become zero. Also we note that the remainders repeat after some steps. So, we have a repeating (recurring) block of digits in the quotient.

Key Concept (Non-terminating and Recurring) :

In the decimal expansion of p/q, q ≠ 0 when the remainder never becomes zero, we have a repeating (recurring) block of digits in the quotient. In this case, the decimal expansion is called non-terminating and recurring.

To simplify the notation, we place a bar over the first block of the repeating (recurring) part and omit the remaining blocks.

So, we can write the expansion of 5/11 and 7/6 as follows..

The following table shows decimal representation of the reciprocals of the first ten natural numbers. We know that the reciprocal of a number n is 1/n. Obviously, the reciprocals of natural numbers are rational numbers.

Thus we see that,

A rational number can be expressed by either a terminating or a non-terminating and recurring (repeating) decimal expansion.

The converse of this statement is also true. 

That is, if the decimal expansion of a number is terminating or non-terminating and recurring (repeating), then the number is a rational number.

Practice Questions

Express the following rational numbers as decimal numbers.

1)  3/4

2)  5/8

3)  9/16

4)  7/25

5)  47/99

6)  1/999

7)  26/45

8)  27/110

9)  2/3

10)  14/9

Answers

1)  3/4  =  0.75

2)  5/8  =  0.625

3)  9/16  =  0.5625

4)  7/25  =  0.0.28

5)  47/99  =  0.474747.........

6)  1/999  =  0.001001001........

7)  26/45  =  0.577777........

8)  27/110  =  0.2454545.........

9)  2/3  =  0.6666........

10)  14/9  =  1.5555..........

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