Powerpoint Templates Page 1 Powerpoint Templates Mathematics & Proofs

Powerpoint Templates Page 2 A formal system of maths begins with and it uses to prove Axioms Deduction Theorems Euclid

Powerpoint Templates Page 3 Science Theory (relative) Falsify Maths Theorem (absolute) Proofs

Powerpoint Templates Page 4 Proofs vs Conjecture Sum of the first odd numbers = n 2 e.g. 1+3 = = 4 True But this is inductive thinking because you can’t test to infinity Particular General This is Conjecture

Powerpoint Templates Page 5 A more complex example Does this formula generate the sequence of square numbers 1, 4, 9, etc N 2 +n x (n-1) x (n-2) x (n-3) x (n-4) e.g. n= x (1-1) x (1-2) x (1-3) x (1-4) x (0) x (-1) x (-2) x (-3)=1

Powerpoint Templates Page 6 Goldbach’s Conjecture Every even number is the sum of 2 primes 2=1+1 4=1+3 6=3+3 8=5+3 Tested to 100,000,000,000,000 But still a conjecture

Powerpoint Templates Page 7 Goldbach’s Conjecture In Maths 100,000,000,000,000 is no closer to infinity than 1 or 10 or 100

Powerpoint Templates Page 8 Proven Sum of the first odd numbers = n2 e.g. 1+3 = 4 22 = 4

Powerpoint Templates Page 9 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’

Powerpoint Templates Page 10 ‘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 games Second round – 256 games Third round – 128 games Forth round – 64 games Fifth round – 32 games Sixth round – 16 games Seventh round – 8 games Eighth round – 4 games Ninth round – 2 games Tenth round – 1 game

Powerpoint Templates Page 11 ‘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