can someone give me a simple definition of a mathematical proof?

Answer 1

I don't know if there is such a thing as a "definition" of a proof, other than "I know what it is when I see one".

The idea is to use very precise definitions and very precise reasoning to convince the reader that the outcome cannot be anything else than what you have set out to prove (this includes eliminating other possibilities).

You could read the description at
http://en.wikipedia.org/wiki/Mathematical_proof

It will not give you a "satisfying" simple definition, but it may give you different ways in which a proof can be constructed.

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An example:
Statement to prove: There is no limit to prime numbers (they go on forever).

Recipe used: proof by contradiction.

We set up the opposite statement and show that the opposite statement causes a contradiction.

opposite statement: there exists such a thing as a "biggest prime". Let us call it N.
Implication: there are no prime numbers higher than this biggest prime. or: if a number X is bigger than N, it cannot be prime.

If there is a number that is a biggest prime, this means that there is a finite list of prime numbers (meaning: we could list them all if we had enough time... and paper).

We take all primes and multiply them together. We get a product which we call M.

We know a few things about M:

It is an even number (because it has 2 as a factor).
It ends with a 0 (5 is a factor of this even number).
It is much bigger than N (because N is a factor).
It is divisible by all the prime numbers (by construction).

If M is a number, then M+1 is also a number.

Is M+1 divisible by 2? no because it is odd (when we divide it by 2, we get a remainder of 1)
Is M+1 divisible by 3? when we divide it by 3, we get a remainder of 1
Is M+1 divisible by 5? no because it ends with 1; numbers divisible by 5 end with 0 or 5. If we divide M+1 by 5, we get a remainder of 1.
Is M+1 divisible by 7? no... remainder = 1...
...
Is M+1 divisible by N? no. If we divide it by N, we get a remainder of 1.

Therefore, we have that
M+1 is a prime number (impossible, because M+1 is bigger than N, our "biggest prime"), or
M+1 is divisible by a prime not on our list (impossible because we listed all primes).

We have a contradiction.

Therefore, our "opposite statement" cannot be true.

Answer 2

go to www.google.co.uk and type

define: mathematical proof

you should get some definition

hope this helps!