Infinity and a maths joke? Really? Last year I was given a mug with it on, so it must be good! The imaginary number i says to π “Be rational”. π replies “Be real”. i² says ok. Not a classic, I know, but not a bad gift for a nerdy maths teacher!
π is probably the most famous irrational number, a number that goes on for ever to an infinite number of decimal places, never repeating. Computers can now record pi to literally trillions of decimal places! GCSE students happily recall what an irrational number is, but we rarely encourage them to think through the implications. For example, if you imagine a perfectly spherical planet, and started walking around that planet in steps of exactly one metre, you would never land on the same spot twice! The number one is a rational number, but the circumference of the sphere is based on pi, an irrational number.
Infinity is another word that gets spoken of without ever really thinking about the implications. A common understanding of infinity is to start counting. 1,2,3,4,5 …. and never stop, to keep on counting for ever and ever and ever. A mind blowing maths fact is that if you add all these numbers together, all the way to infinity, they add up to minus a twelfth! I love that fact. Here is a good explanation by some clever people at Nottingham University.
Now, again as every good GCSE student knows, rational numbers are all numbers that can be written down as fractions. Even recurring decimals are all fractions. 0.33333…. is a ⅓. A GCSE question could be to write a number like 0.3636363636…. as a fraction. (The answer is 12/33 btw).
A fraction is simply one counting number over another counting number. In other words all the numbers we used when counting to infinity. So if we now consider all the numbers between 0 and 1, we can argue using fractions, that there must be an infinite number of numbers between 0 and 1. But, in doing so, we have only used rational numbers. We haven’t even touched irrational numbers!
So if by using rational numbers as fractions we can generate an infinite number of numbers between 0 and 1, then if we then make all of these a multiple of π, we’ll have another set of infinite numbers between and 1. π is only one irrational number. There are many others, the most common being roots of non square numbers e.g. √2, √3, √5 etc. In fact, there are an infinite number of irrational numbers. If we then started using these as multiples as well, we can now generate an infinite set of infinite numbers.
In other words, between 0 and 1 there are infinity x infinity number of numbers. The concept of infinity between 0 and 1 is much larger than the concept of infinity of starting to count and never stopping! Mathematicians and scientists call these countable infinity and uncountable infinity. So Buzz Lightyear was right after all. There is an infinity and beyond, except beyond infinity is much closer to home. It is the numbers from 0 to 1.
All this from a quality maths joke on a mug!