Chris Budd and all that. Q. What is the greatest mathematical formula ever?
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Transcript of Chris Budd and all that. Q. What is the greatest mathematical formula ever?
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Chris Budd
,, ie and all that
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Q. What is the greatest mathematical formula ever?
211
11
1
9
1
7
1
5
1
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11
4
2, ncba nnn
1
1
1
1
nzzprimep np
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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
01ie
1ie
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
xy n11
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 £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
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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.
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, 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
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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
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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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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
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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
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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
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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
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)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
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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
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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
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2// 222 dxdgexdheef xixiyi
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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?
1ie