Proof By Contradiction

[hamzah]

I basically want to know the general method of 'proof by condraction' and an explanation behind it, an example I've been given is:

 

Complete the following proof by contradiction to show that (5)^(1/2) is irrational.

 

Assuming that (5)^(1/2) is rational, let (5)^(1/2) = a/b

 

where a and b are integers that have no common factors

Then 5b^2 = a^2 -> 5 is the factor of a^2

[Dave]

The method for proof by contradiction is quite simple. Basically you assume a logical statement, then use a number of operations to manipulate your statement so that it implies something that is not true, hence the original statement must be wrong and hence false.

[hamzah]

Yeah I do understand what is meant by it, but I'm unsure how to, sometimes, proof something by contradiction, such as the above problem (I posted).

[Dave]

Proof by contradiction isn't something you can just learn overnight really. You need to be able to manipulate the statement and make an argument where the end result contradicts your first statement, and in a lot of cases, this is quite hard. I've come to see that contradictions are often used for quite short proofs most of the time, and they provide a quick and easy way to prove statements that are not as approachable by a direct method. This isn't the case for all contradictive proofs though, as obviously quite a few are long winded.

 

I suggest when you look through a proof that uses a contradiction, look at the way the statement goes through one logical step to another and see how the statement is being manipulated. I think it's one of the hardest methods of proof to learn, because mainly you learn completely by example. Hope this helps in some way.

[wolfson]

If we know that 5 goes into a2 then we know that 5 must also be a factor of a (Can you see why?). So, we can set a = 5x for some other integer x. Therefore 5b2 = (5x)2. Can you finish it off from here?

[bloodhound]

once u get into larger numbers. the proof by contradiction that they are irrational of rational gets quite confusing. you have to use the fact that any number can be expressed as multiple or prime numbers. and this combination is unique.

 

i am still quite baffled. have to revise really badly