Mathematical Proof

A domino and chessboard problem

A domino and chessboard problem

Imagine a chessboard has had two opposing corners removed. A special prize for the first group to cover the remaining squares with dominoes. They can’t overlap!

Impossible?

Trial and error seems to show it can’t be done.

How can we be sure without trying every possible combination (of millions)?

Proving that the domino problem is impossible.

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

Remember syllogisms?

• 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, nm + n = n + m and 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 km(n + k) = mn + mk

• There is a number 0, which has the property that for any number nn + 0 = n

• There is a number 1 which has the property that for any number nn x 1 = n

• For every number n, there is a number k such thatn + k = 0

• For any numbers m, n and kif k ≠ 0 and kn = km, then n = m

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.

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 = p2/q2

and rearrange

2q2 = p2

Euclid’s proof that √2 is irrational

2q2 = p2

If you take any number and multiply it by 2 it must be even, this means that p2 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

2q2 = p2

If p= 2m where m is a whole number,

2q2 = (2m)2 = 4m2

Divide both sides by 2 and we get

q2 = 2m2

Euclid’s proof that √2 is irrational

q2 = 2m2

By the same argument as before, we know q2 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

yn + xn = zn for n > 2

his proof was over 100 pages long and only 6 other mathematicians in the world could understand it!

http://www.youtube.com/watch?v=kBw_i6tlQfU

Homework

Find the shortest Mathematical proof that you can find and print it out. Bring it to the next lesson so you can stick it in your ToK books.