Question

A terminating decimal is : \[\]
A. Natural number \[\]
B. Whole number\[\]
C.A rational number\[\]
D. An integer \[\]

Answer 1

Hint: We recall the decimal numbers, the terminating decimal where digits are terminated after a decimal point, and the recurring decimal where digits are repeated infinitely many times. We recall that recurring decimal can be converted into rational numbers and vice versa.\[\]

Complete step by step answer:
We know that the decimal number is always represented with two parts: the whole number part before the decimal point and the fractional part after the decimal point. For example in 10.12 the whole number part is 10 and the fractional part is 12.\[\]

We call a decimal number terminating decimal if it has a terminating non-zero digit in the fractional part, for example in 10.12 the digit 2 is the terminating digit and hence 10.12 is a terminating decimal. If the decimal number does not have a terminating digit; for example in 10.122222... it is called a non-terminating decimal because 2 repeats itself infinite times. The terminating decimals with repletion of digits are called recurring decimals. \[\]

We know that every rational number is in the form of $\dfrac{p}{q}$ where $p$ is an integer and $q$ is a non-zero integer. If we divide $p$ by $q$ either we can get a terminating decimal or a recurring decimal. So every terminating decimal can be represented as a rational number. If there are $m$ digits before the decimal point ${{w}_{m}}{{w}_{m-1}}...{{w}_{1}}$ as the whole number part and the decimal number is terminated after $n$ digits from the decimal point with ${{f}_{0}}{{f}_{1}}...{{f}_{n}}$ as the fractional part then we can write in rational number as

\[{{w}_{m}}{{w}_{m-1}}...{{w}_{1}}\cdot {{f}_{0}}{{f}_{1}}...{{f}_{n}}=\dfrac{{{w}_{m}}{{w}_{m-1}}...{{w}_{1}}{{f}_{0}}{{f}_{1}}...{{f}_{n}}}{100...\left( \text{ }n\text{ zeroes} \right)}\]
So every terminating decimal is a rational number. \[\]

Note:
 We note that every terminating decimal is a rational number but every rational number may not be a terminating decimal. The natural numbers, whole numbers and integers are always terminated before the decimal point and do not have fractional parts. The non-terminating decimal numbers without repetition of digits are irrational numbers.