00:01
For the proof here, we assume by contradiction that there is going to be some rational number r, such that we have r squared is equal to 2 9th.
00:12
Since r is rational, we can always write it in lowest terms, writing r being equal to a over b, where a and b have no common factor.
00:21
So the greatest common denominator or divisor, the greatest common divisor of a and b is going to be 1, and of course b can be 0.
00:28
We can then square both sides, and we get a over b squared.
00:32
That's going to give us a squared over b squared is going to be equal to the two nines.
00:37
And then we can cross multiply here.
00:39
This is going to imply that nine times a squared is equal to two times b squared.
00:46
Now we can show that a is even from here.
00:48
The right side is even, so the left side must be even.
00:51
So this is going to imply that a squared is going to be even.
00:56
And this then implies that a is even.
01:00
Because if a wasn't even, well, then a would be odd, and odd times an odd is always odd.
01:05
We can easily prove that.
01:06
So again, a squared being even implies that a is even.
01:10
If a is even, we can say a is going to be equal to two times some integer...