Question

Write whether the rational number \[\dfrac{64}{455}\] will have a terminating decimal expansion or a non-terminating repeating decimal expansion \[\]

Answer 1

Hint: Any fraction can be known as terminating or non terminating by dividing the numerator by denominator. If the result has finite decimal places, then it is called terminating decimal. If it has infinite decimal places then it is called non-terminating decimal.

Complete step-by-step solution:
Given rational number is \[\dfrac{64}{455}\] .
Let us consider this rational number as \[x\].
It is given by \[x=\dfrac{64}{455}\]
We need to divide \[64\] with $455$ so that we can obtain the value of x.
\[\]\[x=0.1406593406593407...\]
Hence the value of a given rational number is a non-terminating number. It is also a non-repeating decimal value. The decimal does not have terminal decimal expansion.
Additional Information: Real numbers are of two types. Rational numbers and irrational numbers. Irrational numbers are those which cannot be expressed as the \[\dfrac{p}{q}\] . whereas rational numbers are those numbers which can be represented as the \[\dfrac{p}{q}\] . rational numbers can be divided into terminating and non-terminating decimals. We can define terminating decimals as the numbers which have finite decimal places. Non -terminating decimals are those which have infinite decimal places.

Note: The given expression is a non-terminating decimal. But we have two types of non-terminating decimal expansions. They are repetitive and non-repetitive. Repetitive decimal expansions are defined as a particular set of decimal values kept on repeating. Non-repeating decimal expansions are those whose decimal points do not repeat and keep on changing. However, non-terminating decimals are not to be confused with irrational numbers.