For lots of people, maths is summed up by its famous equations. Many have a tremendous power and allow us to understand the natural world or solve a difficult physical problem. Have you ever thought about whether you have a favourite equation and how you might introduce it to your pupils? Many equations are exciting because they give a glimpse into an advanced mathematical world which would normally remain a mystery. When we look at **E=mc²** we can find out about general relativity and see how such basic quantities as Energy, Mass and the Speed of Light are connected – would this area of physics have ever become as famous without the equation?

Here are five different criteria which you could use to choose your favourite equation.

Some of the oldest equations can be the best. Historians think that Pythagoras’ Theorem was in use 1000 years before the birth of Pythagoras, making it over 4000 years old. It can be understood by a Year 7 pupil and yet it is still used by architects and builders in the 21st Century. Pythagoras of Samos is a figure who is surrounded by legend and intrigue - in around 500BC Pythagoras founded a community which based its beliefs on mathematics, including the idea that the planets move silently according to mathematical equations while producing a silent symphony of music. Nobody also really knows how Pythagoras died – did he refuse to step on a field of beans and escape his attackers as some legends say?

Some equations are just more beautiful than others. The equation which often wins the title of most beautiful is called Euler’s Identity and can be written as:

This is a remarkable equation, as it contains five of the most basic or important mathematical quantities (e, i, π, 1 and 0) and connects them in one elegant equation. What is even more astounding is that this equation hasn’t been invented by anyone – these quantities are simply connected by virtue of the underlying mathematics. But remember – beauty is in the eye of the beholder – which equation is most beautiful to you?

There are many equations which can directly be seen as ‘useful’ and which model phenomena in the real world. As the world grapples with the COVID-19 pandemic, the SIR equations might move to the top of this list. Models based on the SIR equations are used to predict the spread of a disease, and to help governments to decide on how to limit the spread.

The SIR equations are a group of differential equations which model what will happen over time. To understand what they mean you will need to have studied some A-level calculus.

In the most basic model above, β is the rate of infection and λ is the rate of recovery.

Some equations are really surprising. One of these is the Leibniz Formula which states that:

The fact that Pi (π) can be connected to a string of fractions is just amazing! Which equation has the wow factor for you?

Some equations come with a whole back story and drama of their own and Fermat’s Last Theorem has to be top of this list. In 1637 mathematician Pierre De Fermat wrote in a margin: ‘I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.’ Fermat’s Last Theorem is simple – it states that

has no integer solutions for

Did Fermat find a proof? Or was he mistaken or simply teasing us? For over 300 years mathematicians were racing to reproduce this elusive proof and only one succeeded. In 1993 Andrew Wiles revealed that he had found a proof for Fermat’s Last Theorem – this made international news, especially as Wiles had spent six years in secrecy working on his proof. When you are trying to teach your pupils about never giving up think of Andrew Wiles – the legend of mathematical dedication.