Decimal Representation of Rational Numbers

Decimal Representation of Rational Numbers:

1) Write the rational number `7/4` in decimal form.

(1) 7 = 7.0 = 7.000 (Any number of zeros can be added after the fractional part.)

(2) 1 is the quotient and 3 the remainder after dividing 7 by 4. Now we place a decimal point after the integer 1. Writing the 0 from the dividend after the remainder 3, we divide 30 by 4. As the quotient, we get now is fractional, we write 7 after the decimal point. Again we bring down the next 0 from the dividend and complete the division.

2) Write `2 1/5` in decimal form. 

We shall find the decimal form of `2 1/5 = 11/5` in three different ways. 

Find the decimal form of `1/5`.

∴ `1/5` = 0.2

3) Write the number `2/11` in decimal form.

∴ `2/11` = 0.1818.......

`2/11 = 0.bar18`

4) Work out the decimal form of `5/6`.

`5/6` = 0.833...

`∴ 5/6 = 0.8dot3`

Here, a single digit or a group of digits occurs repeatedly on the right of the decimal point. This type of decimal form of a rational number is called the recurring decimal form.

If in a decimal fraction, a single-digit occurs repeatedly on the right of the decimal the point, we put a point above it as shown here. `5/6 = 0.8dot3` and if a group of digits occurs repeatedly, we show it with a horizontal line above the digits. Thus, `2/11 = 0.bar18`.