Download e, i, pi and all that (Yr 12 talk)

e, i, 
and all that
Chris Budd
Q. What is the greatest mathematical formula ever?
11  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
ye
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 n0
(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  11  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 )
it
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 ) 

2i nt
c
e
n
n  
Fourier series:
sound synthesisers,
electronics
1
u (t ) 
2

it
F
(

)
e
d


Fourier transform:
Image processing,
crystallography, optics,
signal analysis
f   eiy 
 ix
2
 ix
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?