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Mathematics& Proofs

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A formal system of maths begins withand it uses

to prove

AxiomsDeductionTheorems

Euclid

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ScienceTheory

(relative)

Falsify

MathsTheorem(absolute)

Proofs

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Proofs vs Conjecture

Sum of the first odd numbers = n2

e.g.1+3 = 4 22 = 4

TrueBut this is inductive thinking because

you can’t test to infinityParticular GeneralThis is Conjecture

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A more complex example

Does this formula generate the sequence of square numbers 1, 4, 9, etc

N2+n x (n-1) x (n-2) x (n-3) x (n-4)

e.g. n=112+1 x (1-1) x (1-2) x (1-3) x (1-4)

12+1 x (0) x (-1) x (-2) x (-3)=1

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Goldbach’s Conjecture

Every even number is the sum of 2 primes2=1+14=1+36=3+38=5+3

Tested to 100,000,000,000,000But still a conjecture

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Goldbach’s Conjecture

In Maths 100,000,000,000,000is no closer to infinity than

1or 10or 100

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Proven

Sum of the first odd numbers = n2 e.g. 1+3 = 4 22 = 4

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Proofs

There may be several proofs for a single theorem

The most simple proofs are prefered

These are often refered to as ‘elegant’ or ‘beautiful’

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‘Elegant’ Proofs

There are 1,024 people in a knock-out tennis tounament. What is the total number of games that must be played before a champion can be

declared?

First round – 512 gamesSecond round – 256 gamesThird round – 128 gamesForth round – 64 gamesFifth round – 32 gamesSixth round – 16 games

Seventh round – 8 gamesEighth round – 4 gamesNinth round – 2 gamesTenth round – 1 game

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‘Elegant’ Proofs

There are 1,024 people in a knock-out tennis tounament. What is the total number of games that must be played before a champion can be

declared?

There is only 1 winner.The number of games is equal to the

number of losers = 1,023

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‘Elegant’ Proofs

There are 1,024 people in a knock-out tennis tounament. What is the total number of games that must be played before a champion can be

declared?

There is only 1 winner.The number of games is equal to the

number of losers = 1,023

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‘Elegant’ Proofs

If you tie a piece of string around a football, and then you want to add enough string to make it go all the

way around 1 inch from the surface, how much extra string do you need?

Answer = 6 inches

Now imagine you want to do the same around the Earth. How much extra string

will you need?

Hint: = circumference/diameter