**REVISITING RATIONAL NUMBERS AND THEIR DECIMAL EXPANSIONS :
**

We have already studied in the previous class that rational numbers have either a terminating decimal repeating decimal expansion. We have to consider a rational number as (where ) as terminating or non-terminating repeating (or recurring) decimal expansion.

e.g.

**(A)** (i) 0.0527 =

(ii) 26.12489 =

**(B) **(i) 0.0875 =

(ii) 23.3408 =

Now, we have converted a real number whose decimal expansion terminates into a rational number of the form. , where p and q are coprime, and prime factorization of denominator

(i.e., q) has only power of 2 or power of 5 or both. We should also understand that the denominator is a power of 10 and can be only prime factors 2 and 5.

Now, we get that this real number is a rational number of the form , where the prime factorization of q is of the form 2^{n}, 5^{m}, and n, m are some non-negative integers.

**Theorem : **Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form , where p and q are coprime, and the prime factorization of q is of the form 2^{n}5^{m}, where n, m are non-negative integers.

Let be a rational number in the lowest form such that the prime factorization of b is of the form . Where n and m are non-negative integers.

*Example*:

(i)

(ii)

(iii)

**Theorem : **Let be a rational number, such that the prime factorization of q is of the form 2^{n}5^{m}, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

**Example:
**

In this case, denominator 7 is not of the form 2^{n}5^{m}. So, from above mentioned theorems, will not have a terminating decimal expansion. Hence, 0 (zero) will not be remainder and the remainders will start repeating after a certain stage. We have a block of digits, 142857, repeating in the quotient of .

Hence, it is is true that any rational number is not converted by the above theorems.

**Theorem :**Let be a rational number, such that the prime factorization of q is not of the form 2^{n}5^{m}, where n, m are non-negative integers. Then, x has a decimal expansion which is not-terminating repeating (recurring).

Hence, we can say that the decimal expansion of every rational number is either terminating or non-terminating repeating (recurring).