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Elegant Proofs and Mathematical Conjectures

rhona-berg avatar
rhona-berg
13 diapositives · 245 Views · Nov 20, 2014

Delve into the realm of mathematics and proofs, exploring Euclid's formal system with axioms, deductions, theorems, and more. Discover the beauty of elegant proofs and test your understanding with intriguing conjectures.

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Elegant Proofs and Mathematical Conjectures
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  1. Mathematics & Proofs Powerpoint Templates

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

  3. Maths Theorem (absolute) Proofs Science Theory (relative) Falsify

  4. Proofs vs Conjecture Sum of the first odd numbers = n2 e.g. 1+3 = 4 22 = 4 True But this is inductive thinking because you can’t test to infinity Particular General This is Conjecture

  5. 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=1 12+1 x (1-1) x (1-2) x (1-3) x (1-4) 12+1 x (0) x (-1) x (-2) x (-3)=1

  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

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

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

  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’

  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

  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

  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

  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: p = circumference/diameter