Class 9 Maths Number Systems | Real numbers in Decimal form |

Real numbers (both rational & irrational) can be represented in decimal form. Eg: 10/3, 7/8, √2 can be represented in decimal form. There are 3 different cases for rational number conversion to decimal form.

Rational Number

- Case 1: The remainder becomes 0. E.g. : 7/8
- Case 2: The remainder never becomes 0 with repetition. E.g.: 10/3

Irrational Number

- Case 3:The remainder never becomes 0 with NO repetition E.g.: √2

**Case 1**: Rational Number, remainder becomes 0. The decimal expansion ends or terminates after finite number of steps. We call it terminating decimals. E.g.: ½ = 0.5 , 7/8= 0.875 , 5/2 = 2.5 etc

**Case 2:** Rational Number, The remainder never becomes 0 with repetition. E.g.: 10/3 = 3.3333333333333…… & 1/7 = 0.142857142857142857142857….

We call it non terminating recurring decimals.

We can also convert non terminating recurring decimals to rational number form.

E.g.: 0.3333333…… can be converted to p/q rational number form.

Let x= 0.33333333…. (eq 1)

Multiplying eq1 by 10 on both side, we get

10x = 3.333333333…… (eq 2)

Subtracting Eq 1 from Eq 2 we get,

9x = 3

Or x = 1/3

Thus 0.3333333…… = 1/3

**Case 3**: Irrational number, Remainder never becomes 0 with NO repetition E.g.: √2

We have seen that decimal expansion of a rational number is either terminating or non-terminating repeating. Thus we can also say that, if the number is either terminating or non-terminating repeating, then it is Rational number. Number whose decimal expansion is Non terminating, non-repeating is Irrational number.

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