Question

Decimal representation of a rational number cannot be:
A. terminating
B. non-terminating
C. non-terminating and repeating
D. non-terminating and non-repeating

Answer 1

Hint: Consider examples of terminating, non-terminating decimals and find out how the rational numbers are represented in decimal form. Then choose the options which cannot be formed in decimal representation.

Complete step-by-step answer:
We know that the whole numbers written in the form of the ratio of the whole number is known as rational numbers. The rational numbers can be positive, negative or zero. So a rational number can be of the form \[{}^{p}/{}_{q}\], where p and q are integers and \[q\ne 0.\]

The rational number can be also expressed as decimal fraction. So when we convert a rational number to a decimal number, it can be either terminating or non-terminating.

Terminating decimals are those which come to an end after a few repetitions after a decimal point. Example- 0.5, 0.75, 123.456 etc. whereas non-terminating decimals keep on continuing after decimal point.

Example \[\pi =3.141592653....\] it keeps on continuing after the decimal point.

If a rational number can be expressed in the form \[\dfrac{p}{{{2}^{n}}\times {{5}^{m}}}\]

then the rational number will be terminating decimals.

E.g.:- \[{}^{5}/{}_{8}=\dfrac{5}{{{2}^{3}}\times {{5}^{0}}}\] so terminating.

\[\dfrac{9}{1280}=\dfrac{9}{{{2}^{8}}\times {{5}^{1}}}\] terminating; \[\dfrac{4}{45}=\dfrac{4}{{{3}^{2}}\times {{5}^{1}}}\] non-terminating.

Examples of rational numbers to non-terminating decimals,

\[{}^{1}/{}_{3}=0.3333.....\] non-terminating

\[{}^{1}/{}_{7}=0.142851.....\] non-terminating

Hence from all this we can conclude that decimal representation of a rational number can be terminating, non-terminating and non-terminating and repeating. But a decimal representation cannot be non-terminating, non-repeating.

Option D is the correct answer.

Note: It can be easily understood by taking a few examples alone.

\[{}^{1}/{}_{2}=0.5\], this is an example of a terminating decimal.

\[{}^{1}/{}_{3}=0.3333....\], this is an example of non-terminating repeating numbers.

\[{}^{1}/{}_{7}=0.142851....\], this is an example of non-terminating decimal.

But we can’t find an example for non-terminating non-repeating decimal representatives.