# 0.1 Rational numbers  (Page 2/2)

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For example, the rational number $\frac{5}{6}$ can be written in decimal notation as $0,8\stackrel{˙}{3}$ and similarly, the decimal number 0,25 can be written as a rational number as $\frac{1}{4}$ .

## Notation for repeating decimals

You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal.

## Converting terminating decimals into rational numbers

A decimal number has an integer part and a fractional part. For example $10,589$ has an integer part of 10 and a fractional part of $0,589$ because $10+0,589=10,589$ . The fractional part can be written as a rational number, i.e. with a numerator and a denominator that are integers.

Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For example:

• $\frac{1}{10}$ is $0,1$
• $\frac{1}{100}$ is $0,01$

This means that:

$\begin{array}{ccc}\hfill 10,589& =& 10+\frac{5}{10}+\frac{8}{100}+\frac{9}{1000}\hfill \\ & =& 10\frac{589}{1000}\hfill \\ & =& \frac{10589}{1000}\hfill \end{array}$

## Fractions

1. Write the following as fractions:
1. $0,1$
2. $0,12$
3. $0,58$
4. $0,2589$

## Converting repeating decimals into rational numbers

When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example.

If we wish to write $0,\stackrel{˙}{3}$ in the form $\frac{a}{b}$ (where $a$ and $b$ are integers) then we would proceed as follows

$\begin{array}{cccc}\hfill x& =& 0,33333...\hfill & \\ \hfill 10x& =& 3,33333...\hfill & \text{multiply}\phantom{\rule{4.pt}{0ex}}\text{by}\phantom{\rule{4.pt}{0ex}}\text{10}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\text{both}\phantom{\rule{4.pt}{0ex}}\text{sides}\hfill \\ \hfill 9x& =& 3\hfill & \text{(}\text{subtracting}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{second}\phantom{\rule{4.pt}{0ex}}\text{equation}\phantom{\rule{4.pt}{0ex}}\text{from}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{first}\phantom{\rule{4.pt}{0ex}}\text{equation}\text{)}\hfill \\ \hfill x& =& \frac{3}{9}=\frac{1}{3}\hfill & \end{array}$

And another example would be to write $5,\stackrel{˙}{4}\stackrel{˙}{3}\stackrel{˙}{2}$ as a rational fraction.

$\begin{array}{cccc}\hfill x& =& 5,432432432...\hfill & \\ \hfill 1000x& =& 5432,432432432...\hfill & \text{multiply by}\phantom{\rule{4.pt}{0ex}}\text{1000}\phantom{\rule{4.pt}{0ex}}\text{on both sides}\\ \hfill \\ \hfill 999x& =& 5427\hfill & \text{(}\text{subtracting the second equation from the first equation}\text{)}\\ \hfill x& =& \frac{5427}{999}=\frac{201}{37}\hfill & \end{array}$

For the first example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the first example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring.

In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?

The number of zeros is the same as the number of recurring digits.

Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like $\sqrt{2}=1,4142135...$ cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals.

## Repeated decimal notation

1. Write the following using the repeated decimal notation:
1. $0,11111111...$
2. $0,1212121212...$
3. $0,123123123123...$
4. $0,11414541454145...$
2. Write the following in decimal form, using the repeated decimal notation:
1. $\frac{2}{3}$
2. $1\frac{3}{11}$
3. $4\frac{5}{6}$
4. $2\frac{1}{9}$
3. Write the following decimals in fractional form:
1. $0,633\stackrel{˙}{3}$
2. $5,3131\overline{31}$
3. $0,99999\stackrel{˙}{9}$

## Summary

• Real numbers can be either rational or irrational.
• A rational number is any number which can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b\ne 0$
• The following are rational numbers:
1. Fractions with both denominator and numerator as integers.
2. Integers.
3. Decimal numbers that end.
4. Decimal numbers that repeat.

## End of chapter exercises

1. If $a$ is an integer, $b$ is an integer and $c$ is irrational, which of the following are rational numbers?
1. $\frac{5}{6}$
2. $\frac{a}{3}$
3. $\frac{b}{2}$
4. $\frac{1}{c}$
2. Write each decimal as a simple fraction:
1. $0,5$
2. $0,12$
3. $0,6$
4. $1,59$
5. $12,27\stackrel{˙}{7}$
3. Show that the decimal $3,21\stackrel{˙}{1}\stackrel{˙}{8}$ is a rational number.
4. Express $0,7\stackrel{˙}{8}$ as a fraction $\frac{a}{b}$ where $a,b\in \mathbb{Z}$ (show all working).

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