1.12 Rational and irrational numbers

What is an irrational number?

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: $\pi$π. Like all other irrational numbers, $\pi$π 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 $\sqrt{2}$√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, $\sqrt{1},\sqrt{4},\sqrt{9},\dots$√1,√4,√9,….

Practice question

Question 1

But how can we write irrational numbers as decimals?

Exact forms and approximations

Consider one third. This can be written as a fraction $\frac{1}{3}$13​, and as a decimal we know it repeats forever as $0.333333\dots$0.333333…. So if we want to do an exact calculation that includes $\frac{1}{3}$13​, we should keep it as a fraction throughout the calculation.

If we type $1\div3$1÷​3 into a calculator, and it would show us around $8$8 or $9$9 digits on the screen. This is now an approximation. $0.3333333333$0.3333333333 is a good approximation of $\frac{1}{3}$13​, but even this has been rounded to fit on your calculator screen, so it is no longer the exact value.

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\dots$0.777777777… go on forever.

Now, what if we were given the recurring decimal $0.0\overline{12}=0.012121212\dots$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 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.