Question

How can a repeating decimal be a rational number?

Answer 1

Hint: In this question, we need to show a repeating decimal is a rational number. First, we will discuss the repeating decimal and rational numbers. We know that any fraction is a rational number. Hence, we will consider a repeating decimal and convert it into the form of a fraction using the algebraic method. Hence, proving the given statement, a repeating decimal is a rational number.

Complete step by step answer:
First, let us see what repeating decimal is,
Repeating decimals are those numbers that keep on repeating the same value after the decimal point. These numbers are also known as Recurring numbers.
We also have non-repeating decimals, i.e.,
Non-repeating rational numbers are those, which do not repeat their value after the decimal point. They are also known as non-terminating and non-recurring numbers.
The common definition of a rational number that is known is that any number that can be written in fraction form is a rational number.
Therefore, let us now consider a repeating decimal and determine whether it can be written in the form of a fraction.
Now, let us consider $ x = $ $ 1.222.... $
Now, let us multiply and divide $ 1.222.... $ by $ 10 $ and $ 100 $ respectively, we have,
$\Rightarrow$ $ x = 1.222 \ldots \times \dfrac{{10}}{{10}} $
$\Rightarrow$ $ x = \dfrac{{12.222 \ldots }}{{10}} $
$\Rightarrow$ $ 10x = 12.222 \ldots $ $ \to \left( 1 \right) $
$\Rightarrow$ $ x = 1.222 \ldots \times \dfrac{{100}}{{100}} $
 $ x = \dfrac{{122.222 \ldots }}{{100}} $
$\Rightarrow$ $ 100x = 122.222 \ldots $ $ \to \left( 2 \right) $
Now, subtract equation $ \left( 1 \right) $ from $ \left( 2 \right) $ , we have,
 $ 100x - 10x = 122.222 \ldots - 12.222 \ldots $
 $ 90x = 110 $
$\Rightarrow$ $ x = \dfrac{{110}}{{90}} $
Therefore, $ x = \dfrac{{11}}{9} $.
Hence, from the above example we can say that a repeating decimal can be written as a fraction using algebraic methods, thus we can say any repeating decimal is a rational number.

Note:
It is important to note here that the repeating decimal can be a rational number; also we can say that an integer can be written as a fraction simply by giving it a denominator of one, so any integer is a rational number. Then, we can also say that a terminating decimal can be written as a fraction by multiplying and dividing the decimal according to the place values of the decimal, i.e., if we have a decimal in tenth place then we will multiply and divide the decimal by $ 10 $ similarly for the other place values.