Irrational numbers

piAlthough most people think of mathematics as some sort of ‘exact’ science – and who can blame them, because right from the start, they have been taught and have learnt that answers are either ‘right’ or ‘wrong’ -0 there are many ways in which mathematicians struggle with the idea of ‘exactness’.

Take, as an example, the whole concept of irrational numbers. They can’t be written as fractions. They can’t be written as repeating or recurring decimals. They exist only as an infinite string of decimals that would far outnumber the number of atoms in the entire universe!

So how can we think of irrational numbers as an ‘exact’ amount? Let’s be even more crazy. Imagine going into a shop and asking for ‘exactly’ 3 apples. Easy, you get exactly 3 apples. Now go into the same shop and ask for the ‘square root of 3’ apples. The shop assistant – who, of course, has an A* in GCSE Maths because they studied Dr. Zargle’s model answers! – immediately whips out a calculator, takes the square root of 3, and announces that you can have ‘1.732050808…’ of an apple.

But look what those three dots at the end of the calculator display are hiding. This number has no end. And if it has no end it can’t be ‘exact’. In fact the only way to write down the ‘square root of 3’ exactly is

In other words, you can only represent this number symbolically – whereas other numbers, like 3, or 57, or 5628674 etc., you can write as they are – exactly.

Now think of a football. You cannot find the volume, or the surface area of any sphere exactly. Why not? Because in the formula for both the volume and surface area of a sphere is the number ∏

And, guess what? Π was the first irrational number mankind discovered. But mathematicians say they can give an exact value for the volume and surface area of a football. But how? Because the area of circles, the surface areas of spheres, the volumes of spheres, etc., are all given by formulae derived from the Calculus.

The Calculus (something you will study in the 6th Form) uses a method that makes an approximation, gets an answer, makes another approximation using the first answer to give a still better approximation. This goes on and on until a so-called ‘exact’ answer is found.

Now, if, instead of using numbers to do this repeated approximation, we use algebra, the exact answers come out in the form of a formula – which can be used, like a tool, to find the ‘exact’ answer. But there is one problem: when I say ‘exact’, I, and every other maths teacher, am blurring the truth. Why? Because in all these formulae derived from the Calculus, there appears ‘pi’ (∏). And Π is ‘irrational’ – it is a never-ending, never-repeating decimal, and so – if you think about it – how can you write down an exact value of ∏.

In truth an exact value just doesn’t exist (unlike the numbers 7, or 87, or 157689 – these have exact values). π has now been calculated to 12 BILLION decimal places! But there is no pattern, no repeating, no end……no ‘exactness’.

So the next time you kick a football, or throw a basketball or cricket ball, just remember you are playing with something whose exact volume and surface area is a string of decimals that would just keep spilling over from one universe into the next without end! Which, maybe, is another way of saying that an ‘exact’ value for the surface area and volume of a sphere just doesn’t exist!