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Powerpoint Templates Page 2
A formal system of maths begins withand it uses
to prove
AxiomsDeductionTheorems
Euclid
Powerpoint Templates Page 4
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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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
Powerpoint Templates Page 12
‘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
Powerpoint Templates Page 13
‘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