Why are Non-terminating, Repeating Decimals Rational

Last night, I received a Facebook message from a Grade 8 student asking why non-terminating repeating decimals are rational. I am posting the answer here for reference.

Rational numbers is closed under addition. That is, if we add two rational numbers, we are guaranteed that the sum is also a rational number. The proof of this is quite easy, so I leave it as an exercise for advanced high school students.

Before discussing non-terminating decimals, let me also note that terminating decimals are rational. I think this is quite obvious because terminating decimals can be converted to fractions (and fractions are rational). For example, 0.842 can be expressed as

\displaystyle\frac{842}{1000}.

Further, terminating decimals can be expressed as sum of fractions. For example, 0.842 can be expressed as

\frac{8}{10} + \frac{4}{100} + \frac{2}{1000}.

Since rational numbers is closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions. 

Now, for non-terminating decimals, a certain strategy is needed to show that it is a rational number. The strategy is to multiply the decimal to powers of 10 and subtract them so that the repeating decimals are eliminated. For example, to show that 0.7345345 \cdots (with 345 repeating indefinitely) is rational, we let x = 0.7345345 \cdots.

Now, 10x = 7.345345 \cdots and 10000x = 7345.345 \cdots .

Now, subtracting both sides of the equations, we have

10000x - 10x = 7345.345 \cdots - 7.345 \cdots

which results to

9990x = 7338.

Now, x =\displaystyle\frac{7338}{9990} which is a fraction. Therefore,

0.7345345 = \frac{7338}{9990}

 is rational.

You may also want to read more examples of converting non-terminating, repeating decimals in Irrational Numbers as Decimals.

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