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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