Axioms and Theorems

Remember syllogisms?

The Socrates Syllogism All human beings are mortal Socrates is a human being Therefore Socrates is mortal premises conclusion

Mathematical proof Mathematical proof is similar in structure to a syllogism. In maths we start with axioms (“premises”). These are the starting points and basic assumptions. We then use deductive reasoning to reach a conclusion, known in maths as a theorem.

For example, the axioms of arithmetic For any numbers m, n m + n = n + mand mn = nm For any numbers m, n and k (m + n) + k = m + (n + k) and (mn)k = m(nk) For any numbers m, n and k m(n + k) = mn + mk There is a number 0, which has the property that for any number n n + 0 = n There is a number 1 which has the property that for any number n n x 1 = n For every number n, there is a number k such that n + k = 0 For any numbers m, n and k if k ≠ 0 and kn = km, then n = m

A proof

A domino and chessboard problem

Imagine a chessboard has had two opposing corners removed. Can you cover the remaining squares with dominoes? (whose size is equal to two squares)

Impossible? Trial and error seems to show it can’t be done. How can we be sure without trying every possible combination (of millions)?

Mathematical proof Mathematical proof aims to show using axioms and logic that something is true in all circumstances, even if all circumstances cannot be tried. Once proved mathematically, something is true for all time.

Proving that the domino problem is impossible.

A domino can only cover two adjoining squares, so these two adjoining squares MUST be of different colours as no two adjoining squares are the same colour. Covering a black and a white square

Proving that the domino problem is impossible. Therefore the first 30 dominoes (wherever they are put) must cover 30 white squares and 30 black. This MUST leave two black squares uncovered. And since these can’t be together, they cannot be covered by one domino. Therefore it is impossible.

Proof Note we have proved this without having to try every combination, and our logic shows that the proof has to be true for any arrangement of dominoes. Science can NEVER be this certain

Another example The square root of 2 is an irrational number (cannot be written as a fraction) This is a proof by Euclid who used the method of proof by contradiction.

Proof by contradiction This starts by assuming by something is true, and then showing that this cannot be so.

Euclid’s proof that √2 is irrational Euclid started by assuming that √2 is rational i.e. √2 = p/q

Euclid’s proof that √2 is irrational √2 = p/q square both sides 2 = p 2 /q 2 and rearrange 2q 2 = p 2

Euclid’s proof that √2 is irrational 2q 2 = p 2 If you take any number and multiply it by 2 it must be even, this means that p 2 is an even number. If a square is an even number, the original number (p) itself must be even. Therefore p can be written as p= 2m where m is a whole number.

Euclid’s proof that √2 is irrational 2q 2 = p 2 If p= 2m where m is a whole number, 2q 2 = (2m) 2 = 4m 2 Divide both sides by 2 and we get q 2 = 2m 2

Euclid’s proof that √2 is irrational q 2 = 2m 2 By the same argument as before, we know q 2 is even and so q must also be even so can be written as q = 2n where n is a whole number. Going back to the start √2 = p/q = 2m/2n

Euclid’s proof that √2 is irrational √2 = p/q = 2m/2n This can be simplified to √2 = m/n And we are back where we started!

Euclid’s proof that √2 is irrational √2 = m/n This process can be repeated over and over again infinitely and we never get nearer to the simplest fraction. This means that the simplest fraction does not exist, i.e. our original assumption that √2 = p/q is untrue! This shows that √2 is indeed irrational.

Andrew Wiles Euclid’s proof is a very simple one. When Andrew Wiles proved that there are no whole number solutions for the following equation y n + x n = z n for n > 2 his proof was over 100 pages long and only 6 other mathematicians in the world could understand it! YouTube - Proof that Girls are Evil

Homework Find the shortest proof of Pythagoras' theorem (or any other mathematical proof) that you can find and copy it into your TOK books. For Wednesday.