7.08 Rational and irrational numbers

A rational number is a number which can be written as a fraction where both the numerator and denominator are integers. An irrational number is a number which cannot be written as a fraction of two integers.

We can write any terminating or recurring decimal as a fraction, therefore these are rational numbers. However, decimals which are neither terminating nor recurring are irrational numbers.

You might be familiar with one irrational number already: \pi. Like all other irrational numbers, \pi really does go on forever without repeating itself. We say therefore that it doesn't terminate, or repeat.

Another number that is famously irrational is \sqrt{2}. In fact, the square root of most numbers are irrational. If a root is irrational it is called a surd. The square roots of perfect squares are rational, \sqrt{1},\, \sqrt{4},\,\sqrt{9},\,...

Examples

What if we were given the recurring decimal 0.0\overline{12}=0.012\,121\,212\,... and asked to convert it into a fraction? We could type 0.012\,121\,212 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 0.012\,121\,212, not for the recurring decimal 0.0\overline{12}.

Fortunately, we can use some simple algebra to convert any recurring decimal to a fraction. To do this, we use a nice trick to get rid of the endless recurring part of the decimal.

Examples

Consider one third. This can be written as a fraction \dfrac{1}{3}, and as a decimal we know it repeats forever as 0.333\,333... So if we want to do an exact calculation that includes \dfrac{1}{3}, we should keep it as a fraction throughout the calculation.

If we type 1\div3 into a calculator, and it would show us around 8 or 9 digits on the screen. This is now an approximation. 0.333\,333\,333\,3 is a good approximation of \dfrac{1}{3}, but even this has been rounded to fit on your calculator screen, so it is no longer the exact value.

If we were given \dfrac{7}{9} and asked to express it as a decimal, we could enter it into our calculator and get 0.777\,777\,777. Our calculator eventually runs out of space, but we know that we have a recurring decimal 0.\overline{7} and that the sevens in 0.777\,777\,777\,... go on forever.

To convert from a fraction to a decimal, we can rewrite the fraction as division expression. For example, \dfrac{3}{7} is 3 divided by 7. Then we can use short division, adding extra zeros as required to the numerator.

Examples

Example 3

Write the fraction \dfrac{7}{9} as a recurring decimal.

Worked Solution

Create a strategy

Use short division, and adding extra zeros as required to the numerator.

Apply the idea

Set up the short division including some trailing zeros.

9 does not go into 7 so we put a 0 above the 7 and copy the decimal point to the top.

9 goes into 70 seven times since 9\times 7 =63, so we put a 7 above the first zero and carry the remaining 7 to the next zero.

Again, 9 goes into 70 seven times so we put a 7 above the second zero and carry the remaining 7 to the next zero.

Again, 9 goes into 70 five times so we put a 7 above the third zero. We can see that this pattern will continue forever, so this is a recurring decimal.

\dfrac{7}{9}=0.\overline{7}

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