Decimal Representation of rational numbers

Decimal Representation of rational numbers

1. Number System

> Decimal Representation of Rational Numbers

Maths Learning Centre, Jalandhar

2. Every rational number? can be expressed as a decimal.

On dividing p by q, two possibilities arise 3) 2.0 (.666. .
(i) The remainder becomes zero and the division concludes 5) 13 (2.6 18 _
after a finite number of steps. In this case, the decimal —10 20
expansion obtained also terminates or ends. 30 —18

y : ‘ ' 30 20
(ii) The remainder never becomes zero and a repeating string
of remainders is obtained. x _=18_

2

In this case, we get a digit or a block of digits repeatinginthe
decimal expansion.

3. Examples of terminating decimals

(5 = 0.25 (i) 2 = 0.625
4)1.0(.25 8) 5.0 (.625
8 | -48
20
e _=16_
— 40
* -40

x

4. Examples of repeating (or recurring) decimals

In a recurring decimal, we place a bar over the first block of
the repeating digits and omit the other repeating blocks.
11) 3.0 (.2727...

—22
80
-77
30
—22
80
TT
3

3 = 0.2727... = 0.27.
a

5. l = 2142857142857 ... = 2.142857 7) 15 (2.14285714...

7
-14
10
7,
7 30
(nts 28
se 20
-14
60
-56
40
-35
50
=
10
-7
30
er
2

6. Length of period: Repeated number of decimal places in a
rational number is called the length of its period.

The number of entries in the repeating string of remainders
is less than the divisor. .
2 =
3 0.6 (7

only one number 6 repeats itself and the divisor is 3. NE

15 _——

z= 2.142857
a set of 6 digits, namely 142857, repeats itself and the
divisor is 7.

7. Find the decimal expansion of. Can you predict what the
decimal expansions of ,=,2,2are, without actually doing
the long division? If so, how?

4 ———
7” 0.1428571428 ... = 0.142857.

2 = 2x2 = 2x 0.142857 = 0.285714;
$= 3x2 = 3 x 0.142857 = 0.428571;
$= 4x2 = 4x 0.142857 = 0.571428;

$= 5x2 = 5 x 0.142857 = 0.714285.

7) 1.0 (.14285714...
=7
30
28
20
-14
60
56
40
-35
50
—49
10
7
30
—28
2