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e, i, and all that Chris Budd Q. What is the greatest mathematical formula ever? 11 2 1 1 1 1 1 1 4 3 5 7 9 11 a b c , n 2 n n n 1 1 z p prime 1 p z n 1 n The winner every time The equation that sets the gold standard of mathematical beauty i e 1 0 i e 1 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 y 1 1 x n 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 2.718281828 e 1 1 1 1 e 1 1 1 2 1 2 3 1 2 3 4 If we repeat this phased interest starting with £x for n years we get y en x ye an 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 C d Archimedes showed that 22 7 A r 2 Chinese 355 113 3.1415926535897932387 Some formulas for pi 1 1 1 1 1 1 4 3 5 7 9 11 2 6 Leibnitz 1 1 1 1 1 2 2 2 2 2 1 2 3 4 5 Euler 8 (4n)!(1103 26390n) 9801 n0 (n!) 4 3964 n 1 dx 2 1 x 2 e Ramanujan x2 / 2 dx 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 x 1 0 To solve this we must invent a new type of number, the negative numbers -1,-2,-3,-4,-5 …. These numbers obey rules 4 (5) 9, 4 (5) 1, 4 5 20 An imaginary tale Having invented the negative numbers, do we need any more? How do we solve the equation x 2 1 Invent the new (imaginary) number Complex number 1 1 11 1 i i i 1 a ib (a ib ) (c id ) ac bd i (ad bc) Euler realised that there was a wonderful link between complex numbers and geometry Imaginary -b+ia a+ib Real i (a ib ) b ia Multiplying by i rotates the dashed line by 90 degrees Multiplying by cos( ) i sin( ) rotates by the angle And now for the great moment ……. Putting it all together …. Euler’s fabulous formula … ei cos( ) i sin( ) e i Is a rotation in the complex plane (180o ) ei 1 (90o ) ei / 2 i 2 Can derive the result using a Taylor series 2 3 4 5 6 x x x x x ex 1 x 2! 3! 4! 5! 6! e i 1 i 2 2! i 3 3! 4 4! i 5 6 5! 6! 3 5 1 i 2! 4! 6! 3! 5! 2 4 6 cos( ) i sin( ) Why does Euler’s formula matter e e e e x i i t i ( x ct ) it e (x) Describes things that grow Describes things that oscillate Alternating current Radio/sound wave Quantum mechanical wave packet We can also combine them u (t ) 2i nt c e n n Fourier series: sound synthesisers, electronics 1 u (t ) 2 it F ( ) e d Fourier transform: Image processing, crystallography, optics, signal analysis f eiy ix 2 ix 2 2 e h d x / e g d x d / 2 In Conclusion i e 1 • 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?