On 9th October 2019 I uploaded some interesting facts about the English language that I had read about. Here are some interesting facts about mathematics that I have read about.

A vast number of people have been working on mathematics for a very long time. One would imagine that all the major discoveries must have been made long ago, but that is not the case. Pierre de Fermat proved that 26 is the only number to lie between a square and a cube in the seventeenth century; but that 8 and 9 are the only pair of integer powers next to each other was not proven until 2002.

A prime number is a whole number that cannot be exactly divided by another whole number, such as 7 and 11. Euclid proved more than 2000 years ago that there is an infinite number of these. Twin primes are prime numbers with only one number between them, such as 11 and 13, but so far nobody has ever been able to prove that there is an infinite number of twin primes.

A statement like this that is believed to be true but has never been proved in called a conjecture. In 1769 Leonard Euler made the conjecture that there was no whole- number solution to the equation x^{4} + y^{4} + z^{4} = w^{4}. Two hundred years later Noam Elkies disproved it by finding a solution.

In 2004 I read that if you divide the recurring decimals of 1/7 into three groups of two and add them up you get 99, and if you divide them into two groups of three and add them up you get 999. I wondered whether this was a coincidence, so I used the calculator of my computer to do the same thing with 1/13 and got the same result. The next reciprocal of a prime number I could find with six recurring decimals was 1/41, but it this case I did not get the same result.

I am always excited when I discover an aspect of mathematics that is new to me. One of my most remarkable discoveries is the equation e^{i}^{π} = -1. It is quite easy to remember. You start off with ‘I spy’, take away the ‘s’, add ‘e to the power’ at the beginning and add ‘equals minus one’ at the end. e is 2.71828… π is 3.14159… i is the square route of -1. If you imagine ordinary numbers being in a line with negative numbers to the left, 0 in the middle and positive numbers to the right, i is the same distance above 0 as +1 is to the right. Multiples of 1 are called imaginary numbers and lie on the vertical line through 0 and i. If you add or subtract ordinary numbers to or from imaginary numbers you get complex numbers, which are found to the left and right of the line of imaginary numbers, so forming a plane.

I assumed that these were merely mathematical curiosities of no possible use to anyone, but I later learned that i is used in aerodynamics and quantum theory and that some equations cannot be solved without it.

We have seen that ordinary numbers are 1-dimensional and that complex numbers are 2-dimensional. But that is not all. There are also 4-dimensional numbers called quaternions and 8-dimensional numbers called Cayley numbers. Whether these are useful or merely interesting curiosities I don’t know.

I later read about an even simpler equation, i^{i} = 0.20787… I can’t imagine how anyone managed to work that out.

It is well known that there are 5 regular polyhedra. What is less well known is that there are 6 regular objects in 4 dimensions and 3 in 5 or more dimensions. I can’t imagine how anyone managed to work that out either.