Chris Budd

,, ie and all that

Q. What is the greatest mathematical formula ever?

211

11

1

9

1

7

1

5

1

3

11

4

2, ncba nnn

1

1

1

1

nzzprimep np

The winner every timeThe winner every timeThe equation that sets the gold standard of mathematical The equation that sets the gold standard of mathematical beautybeauty

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1ie

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

xy n11

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)%

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

M=1000 0.1% 1000 times £271.69

Start with £100, how much do we get?

As M gets very large these numbers approach

2.718 times £100

e718281828.2

4321

1

321

1

21

1

1

11e

xey n

aney

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

d

C

2rA

The Greeks knew that the ratio of the circumference to the diameter of a circle is the same for all circles

Archimedes showed that

7

22

113

355

8979323871415926535.3

Chinese

Some formulas for pi

11

1

9

1

7

1

5

1

3

11

4

22222

2

5

1

4

1

3

1

2

1

1

1

6

044396)!(

)263901103()!4(

9801

81

nnn

nn

21 x

dx dxe x

2/22

Leibnitz

Euler

Ramanujan

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

01x

-1,-2,-3,-4,-5 ….

2054,1)5(4,9)5(4

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

An imaginary tale

Having invented the negative numbers, do we need any more?

How do we solve the equation

12 x 11111

1iiInvent the new (imaginary) number

i

)()()( bcadibdacidciba

iba Complex number

Euler realised that there was a wonderful link between complex numbers and geometry

iabibai )(

a+ib

-b+ia

Multiplying by i rotates the dashed line by 90 degrees

)sin()cos( i Multiplying by rotates by the angle

Real

Imaginary

And now for the great moment …….

Putting it all together ….

Euler’s fabulous formula …

ie

ie

e

io

io

2/)90(2

1)180(

)sin()cos( iei

Is a rotation in the complex plane

)sin()cos(

!5!3!6!4!21

!6!5!4!3!21

!6!5!4!3!21

53642

65432

65432

i

i

iiie

xxxxxxe

i

x

Can derive the result using a Taylor series

Why does Euler’s formula matter

xeie

tie

)( ctxie

)(xe ti

Describes things that grow

Describes things that oscillate

Alternating current

Radio/sound wave

Quantum mechanical wave packet

We can also combine them

n

ntinectu 2)(

deFtu ti

)(2

1)(

Fourier series:

sound synthesisers, electronics

Fourier transform:

Image processing,

crystallography, optics,

signal analysis

2// 222 dxdgexdheef xixiyi

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?

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