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.

**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.

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.

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

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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