Theorem1.4:

 Let x = p / q be a rational number, such that the prime factorization of q is of the form 2n5 m, where n and m are non-negative integers. Then x has a decimal expansion which terminates.

DO THIS
Write the denominator of the following rational numbers in 2n5 m form where n and m are non-negative integers and then write them in their decimal form

(i) 3 / 4

(ii) 7 / 25

(iii) 51 / 64

(iv) 14 / 25

(v) 80 / 100

1.2.2 NON-TERMINATING, RECURRING DECIMALS IN RATIONAL NUMBERS

Let us now consider rational numbers whose decimal expansions are non-terminating and recurring.

Let us look at the decimal conversion of 1/7.

1/7= 0.1428571428571 ..... which is a non-terminating and recurring decimal.Notice, the block of digits '142857' is repeating in the quotient.

Notice that the denominator i.e., 7 can't be written in the form 2n5 m

DO THIS
Write the following rational numbers in the decimal form and find out the block of repeating digits in the quotient.

(i) 1 / 3

(ii) 2 / 7

(iii) 5 / 11

(iv) 10 / 13


From the 'Do this' exercise and from the example taken above, we can formally state as below:

Theorem-1.5 : Let x = p/q be a rational number, such that the prime factorization of q is not of the form 2n5 m , where n and m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

From the above discussion, we can conclude that the decimal form of every rational number is either terminating or non-terminating repeating.





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