and all that Chris Budd Q What is

and all that Chris Budd

Q. What is the greatest mathematical formula ever?

The winner every time The equation that sets the gold standard of mathematical beauty What does this formula mean, and why is it so important?

The number e and how things grow What does 100% annual compound interest mean? Start with £ 100, in one year have £ 200, in two years have £ 400 Start with £x, wait n years, get £y

But, we could PHASE the interest Break up the year into M intervals and make M increases of (100/M)% Start with £ 100, how much do we get? M=1 100% once £ 200 M=2 50% twice £ 225 M=4 25% four times £ 244. 14 M=10 10% ten times £ 259. 37 M=100 1% 100 times £ 270. 48 1000 times £ 271. 69 M=1000 0. 1% As M gets very large these numbers approach 2. 718 times £ 100

If we repeat this phased interest starting with £x for n years we get In general the exponential function tells us how everything changes and grows, from temperatures to populations.

, circles, odd numbers and integrals The Greeks knew that the ratio of the circumference to the diameter of a circle is the same for all circles Archimedes showed that Chinese

Some formulas for pi Leibnitz Euler Ramanujan

Negative numbers and -1 A short history of counting: Early people counted on their fingers Good for counting cows Suppose that someone lends you a cow. But the cow dies How many cows do you have now?

If x is the number of cows, we must solve the equation To solve this we must invent a new type of number, the negative numbers -1, -2, -3, -4, -5 …. These numbers obey rules

An imaginary tale Having invented the negative numbers, do we need any more? How do we solve the equation Invent the new (imaginary) number Complex number i

Euler realised that there was a wonderful link between complex numbers and geometry -b+ia Imaginary a+ib Real Multiplying by i rotates the dashed line by 90 degrees rotates by the angle

And now for the great moment ……. Putting it all together …. Euler’s fabulous formula … Is a rotation in the complex plane

Can derive the result using a Taylor series

Why does Euler’s formula matter Describes things that grow Describes things that oscillate Alternating current Radio/sound wave Quantum mechanical wave packet

We can also combine them Fourier series: sound synthesisers, electronics Fourier transform: Image processing, crystallography, optics, signal analysis

In Conclusion • Euler’s fabulous formula unites all of mathematics in one go • It has countless applications to modern technology • Will there ever be a better formula?