Theorems and conjectures

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Last lesson - Theorems

• All human beings are mortal • Socrates is a human being• Therefore Socrates is mortal

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.

premises

conclusion

That’s my syllogism!

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.

Example of a proof

Euclid’s proof that 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. This starts by assuming that something is true, and then showing that this cannot be so.

Euclid’s proof that √2 is irrationalAssume that √2 = p/q

square both sides2 = p2/q2

and rearrange2q2 = p2

p2 must be even number which means p itself must be even. Therefore p can be written as p= 2m where m is a whole number.

2q2 = (2m)2 = 4m2

Divide both sides by 2 and we getq2 = 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.

√2 = p/q = 2m/2nThis can be simplified to

√2 = m/n And we are back where we started! 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.

Conjecture

A conjecture is a hypothesis that appears to be true but has not yet been proved to be true.

Goldbach’s conjecture

“Every even number is the sum of two primes”

Goldbach’s conjecture

“Every even number is the sum of two primes”. For example

2 = 1 +1 12 = 7 + 5

4 = 2 + 2 14 = 7 + 7

6 = 3 + 3 16 = 13 + 3

8 = 5 + 3 18 = 13 + 5

10 = 5 + 5 20 = 17 + 3

Goldbach’s conjecture

“Every even number is the sum of two primes”.

This has been tested on computers up to the number 100,000,000,000,000 and has been found true for all these numbers.

But it has not yet been proved true for ALL numbers, hence it is a conjecture.

Kepler’s conjecture

Before we continue……

Before we delve deeper into the mathematical world, we need to do some reading!

Propositions