Yes, repeating decimals are always rational numbers. A rational number is defined as any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
A repeating decimal consists of digits that repeat indefinitely after a certain point. For example, the decimal 0.333… can be expressed as the fraction 1/3. Here’s how you can understand it:
Let’s consider the number x = 0.333…
Multiplying both sides by 10 gives us:
10x = 3.333…
Then, if we subtract the original equation from this new one, we get:
10x – x = 3.333… – 0.333…
Which simplifies to:
9x = 3
Dividing both sides by 9 results in:
x = 1/3
This illustrates that the repeating decimal 0.333… can be expressed as the rational number 1/3.
Similarly, any repeating decimal can be converted into a fraction, which confirms that all repeating decimals are indeed rational numbers.