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Are Repeating Decimals Rational Numbers?


It is a fact that repeating decimals are indeed rational numbers. This means that any repeating decimal number can be expressed in the form of a fraction. Given any repeating decimal number, we can find a fraction of the form \frac{p}{q} (p and q integers) such that the repeating decimal is equal to the rational number \frac{p}{q}. We can understand how to convert a repeating decimal to a fraction by looking at the three examples below.

recurring decimal are rational numbers

Example 1:

Consider the recurring decimal 0.77777\ldots.We can convert it into a fraction as follows:

\begin{align}\text{Let }x=0.77777\ldots \end{align}

We multiply the above equation by 10 (by shifting one decimal place to the right) which gives us that,

\begin{align}10x=7.777777\ldots \end{align}

Subtracting equation (1) from equation (2) we get that,

\begin{align*}10x-x&=7.777777 - 0.777777\ldots \\9x &=7 \text{ (recurring part gets cancelled by subtraction)}\\ x &= \frac{7}{9}\end{align*}

This means that we have expressed the recurring decimal 0.77777\ldots in the form of a rational number 7/9.

Example 2:

We express the repeating decimal 0.545454\ldots as a fraction as follows:

As before we first name the number using a variable.\begin{align}\text{Let }x = 0.545454\ldots \end{align}

Instead of multiplying by 10 as done in the previous example we now multiply by 100. This is because the pattern repeats in blocks of two. If the pattern were repeating in blocks of 3 we would have multiplied by 1000.

\begin{align}100x=54.545454\ldots \end{align}

Subtracting equation (3) from equation (4) we get that,

\begin{align*}100x-x&=54.545454 - 0.545454\ldots \\99x &=54 \\ x &= \frac{54}{99} = \frac{6}{11}\end{align*}

Therefore we conclude that 0.545454\ldots = \frac{6}{11}.

Example 3:

Let us now consider a different case. Consider the repeating decimal 0.8222\ldots. Here, since the pattern does not start repeating immediately after the decimal we multiply the number by 10 and 100 and subtract the two resulting equations to bring it to fractional form.

\begin{align*}\text{Let }x = 0.8222\ldots \end{align*} \begin{align}10x=8.2222\ldots \end{align} \begin{align}100x=82.2222\ldots \end{align}

Subtracting equation (5) from equation (6) we get that,

\begin{align*}100x-10x&=82.2222\ldots - 8.2222\ldots \\90x &=74 \\ x &= \frac{74}{90} = \frac{37}{45}\end{align*}

We have thus written the number 0.8222\ldots in rational form as \frac{37}{45}.

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