Decimals

Lesson

If we were given $\frac{7}{9}$79 and asked to express it as a decimal, we could enter it into our calculator and get $0.777777777$0.777777777. Our calculator eventually runs out of space, but we know that we have a recurring decimal $0.\overline{7}$0.7 and that the sevens in $0.777777777$0.777777777... go on forever.

Now, what if we were given the recurring decimal $0.0\overline{12}=0.012121212$0.012=0.012121212... and asked to convert it into a fraction? We could type $0.012121212$0.012121212 into our calculator and press the button to convert it into a fraction, but most calculators won't do it, and even if they do, it will give you the fraction for the terminating decimal $0.012121212$0.012121212, NOT for the recurring decimal $0.0\overline{12}$0.012.

Fortunately, we can use what we've learnt about basic algebra to convert any recurring decimal to a fraction. To do this, we use a little trick to get rid of the endless recurring part of the decimal.

Say we have $59.\overline{4}=59.444444444$59.4=59.444444444... and $32.\overline{4}=32.444444444$32.4=32.444444444... .

If we subtract $32.\overline{4}$32.4 from $59.\overline{4}$59.4 then we have:

$59.\overline{4}-32.\overline{4}$59.4−32.4 | $=$= | $59.444444444$59.444444444... |

$=$= | $-32.444444444$−32.444444444.... | |

$=$= | $27$27 |

The recurring part cancels out! This makes sense, since $0.\overline{4}-0.\overline{4}$0.4−0.4 equals nothing.

This fact will come in handy when we are converting recurring decimals to fractions.

Let's learn out conversion method using the fraction $0.0\overline{12}$0.012.

Let $x$`x` equal the recurring decimal you are trying to convert.

$x=0.0\overline{12}$`x`=0.012

Look for the repeating digits in the recurring decimal. In $0.012121212$0.012121212... , they are '$12$12'.

Shift one lot of the repeating digits to the left of the decimal place. Remember that multiplying by $10$10, $100$100, $1000$1000, etc. will shift the digits left across the decimal place by $1$1 place, $2$2 places, $3$3 places, etc.

Therefore, $1000\times0.012121212$1000×0.012121212...$=$=$12.1212121$12.1212121... and so we have a new equation for $x$`x`.

$1000x=12.1212121$1000`x`=12.1212121...

Check your original equation $x=0.012121212$`x`=0.012121212... . Do the repeating digits start on the right? In this case, they don't, so we'll have to make a new equation where they do.

$10x=0.12121212$10`x`=0.12121212...

We have our two equations $1000x=12.1212121$1000`x`=12.1212121... and $10x=0.12121212$10`x`=0.12121212... .

We subtract the corresponding sides of the equations like so.

$1000x-10x$1000x−10x |
$=$= | $12.\overline{12}-0.\overline{12}$12.12−0.12 |

$990x$990x |
$=$= | $12$12 |

$x$x |
$=$= | $\frac{12}{990}$12990 |

$x$x |
$=$= | $\frac{2}{165}$2165 |

And so our recurring decimal $0.0\overline{12}$0.012 is in fact the fraction $\frac{2}{165}$2165.

If you follow the method above, you can convert any recurring decimal into a fraction!

Starting with $x=0.\overline{1}$`x`=0.1, express $x$`x` as a fraction in simplest form.

Starting with $x=0.\overline{216}$`x`=0.216, express $x$`x` as a fraction in simplest form.

Starting with $x=0.6\overline{87}$`x`=0.687, express $x$`x` as a fraction in simplest form.