Proof by Contradiction [IB Math AA HL]

[@folalibi2743]

I have a question about this proof: why do we need to posit that a and b have no common factors? This would mean that the fraction a/b cannot be simplified, but I don't see why we need to posit this. Because this assumption is the key to the entire proof, I would appreciate a bit of explanation. This series of videos is very useful, thank you!

[@rehmmyteon5016]

If root 2 is irrational, root 2 cannot be presented in the form a/b, therefore having no common factors.

[@folalibi2743]

@@rehmmyteon5016 Thanks for your answer! At the same time, I still don't understand why we need to reduce the fraction a/b to the lowest terms, as we do when we specify that "a" and "b" are integers with no common factors ("coprime integers," as they say). By manipulating the negative proposition, we end up showing that "a" and "b" have a common factor of 2. But this becomes a contradiction (and therefore a proof) only because we first specified that a and b had no common factor. So while this step was obviously necessary, I still don't understand why it's valid. Sorry for being so slow!

[@barknozturk6319]

the answer is fairly simple. because you can't use the same letters after simplifying. let's assume that a and b do have common factors. In fact, let's say that a is 9 and b is 6. If you write the two values as a/b, you'll get 9/6. A ratio that could be simplified to 3/2. This is where the issue comes up. You may think that nothing has changed. Well, you can no longer use a and b for the remainder of your work. you have to give different names to 3 and 2 as the numerators and the denominators are not the same values anymore. a/b becomes c/d or whatever. So you did all that work for nothing since you have to prove that c/d is not rational now. Plus, you're operating with letters anyway. You will most likely not require to simplify the letters themselves, since you don't know the actual value. Instead, you may get to a point like this: 2a = 4b, therefore, a = 2b. since you don't know what a and b are, you will just let them be. There will be no need to simplify them. So the goal is to make sure that you take out the possibility of them being simplifiable or divisible so that you can get the most accurate result. Since the question ends up revealing that it's a contradiction due to a and b being coprime, you basically start off by giving that information away to avoid any other outcome.

[@folalibi2743]

@@barknozturk6319 Thank you for your answer! It clarifies everything. :)

@@barknozturk6319 ne güzel anlattın be güzel ib kardeşim