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Chris Budd and all that

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Q. What is the greatest mathematical formula ever?

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The winner every time 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?

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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

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

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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.

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, 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

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Some formulas for pi Leibnitz Euler Ramanujan

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

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-1,-2,-3,-4,-5 …. 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 These numbers obey rules

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

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

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

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Can derive the result using a Taylor series

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

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

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

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