Answer
See explanation
Work Step by Step
Consider the following rationals with terminating decimal expansions:
(i) $\dfrac{1}{2}=0.5=\dfrac{5}{10}$
(ii)$\dfrac{1}{5}=0.2=\dfrac{2}{10}$
(iii)$\dfrac{1}{4}=0.25=\dfrac{25}{100}$
In general, any terminating decimal number can be re-written in the quotient of the integers such that the denominator is of the form $10^a$, where $a$ is a non-negative integer.
Now, suppose the rational number is of the form:
$\dfrac{a}{2^n\cdot 5^m}$, where $n,m$ some non-negative integers.
If $n$ is smaller than $m$, then, multiplying and dividing by $2^{m-n}$, we get $\dfrac{a}{2^n\cdot 5^m}=\dfrac{a\cdot 2^{m-n}}{2^m\cdot5^m }=\dfrac{a\cdot 2^{m-n}}{10^m}$, which will be a terminating decimal number.
If $m$ is smaller than $n$, then, multiplying and dividing by $5^{n-m}$, we get $\dfrac{a}{2^n\cdot 5^m}=\dfrac{a\cdot 5^{n-m}}{2^n\cdot5^n }=\dfrac{a\cdot 5^{n-m}}{10^n}$, which will be a terminating decimal number.
Thus, rational numbers have the terminating decimal expansion if the factorization of the denominator is of the form $2^n\cdot 5^m$, where $n,m$ are non-negative integers.
A rational number is a number that can be written as a fraction of two integers. When converting a rational number to decimal form, its decimal either terminates or repeats. By examining the denominator of a fraction in simplest form, one can often predict the behavior of the decimal. Fractions whose denominators contain only the prime factors 2 and 5 will always have terminating decimals. This is because the base of our number system is 10, which is 2×5, and only multiples of these primes divide evenly into powers of 10. Fractions with denominators containing other prime factors, such as 3, 7, or 11, will produce repeating decimals, since these numbers cannot evenly divide powers of 10. Observing the denominator, therefore, provides a simple and reliable way to know whether a fraction’s decimal representation will terminate or repeat.
