Let’s suppose, for a moment, that the square root of 2 were rational. Then there would be a pair of numbers a and b such that
We can manipulate that in the following way,
So, if the square root of 2 were rational, we would be able to find integers a and b to solve the above equation. Of course, it isn’t, so we can’t.
But what if it were something very very close to zero? Say the expression came out to e, where e was very very small. In that case …
Looking at the above, if e is very very close to 0 and b is some integer, then e/b will be even closer to 0. Which means that a/b is only very slightly off from the square root of 2.
The question then is how to find such a and b so as to yield very small e.
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