rotund harbor
Can someone explain how proof by contradiction works please. In the images, we are trying to prove that squareroot of 2 is irrational.
I have a question based on the workings. How can we assume that a is even? Is it because 2 * b^2 is always even and so must be a^2. Now if the square of a number is even, does that mean that the squareroot of that number is also even?
Similarly, if 2 divides a, this mean 4 divides a^2?
plain knollBOT
urban valve
we are not assuming that a is even
we start by assuming that sqrt(2) is rational
and using that assumption to show that a has to be even
obsidian summit
the basic idea of a proof by contradiction is to assume the opposite of our statement is true, but then showing said assumption leads to a logical contradiction down the road. this contradiction shows that our assumption was impossible, so the original statement must be true.
rotund harbor
yep I see, can we say the following:
2 * b^2 is always even, this mean a^2 is also even and so must be a?
In general if the square of a number is even, does that mean that the squareroot of that number is also even?
obsidian summit
of course. squaring a number is multiplying it by itself. consider the two cases of squaring an even and squaring an odd number separately. which gives you an even number?
try it!
obsidian summit
you can even prove this algebraically
an even number is 2k, and an odd number is 2j + 1 for some natural j, k
rotund harbor
yep, if we pick up some numbers 2k will always be even and 2j + 1 will always be odd
obsidian summit
squaring an even number gives us 4k^2 = 2(2k)(k), and since k is a natural and there's a constant factor of 2, this must be even
likewise, squaring an odd number gives us (2j+1)^2 = 4j ^2 + 4j + 1 = 2(2j^2 + 2j) + 1, and since j is a natural, the expression inside the brackets must be natural as well, so we can say it's some natural m
then 2(2j^2 + 2j) + 1 = 2m + 1 and we get another odd
rotund harbor
yeah I see make sense now, thanks !!
obsidian summit
anything you stilll wanna know about your original question?
rotund harbor
nope all good for now, thanks !