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we rewrote the difference between root(2) and an arbitrary rational number (a/b) as that above expression.
so now we have:
3 - root(2) is roughly equal to 1.59.
that means we only have to consider the rationals that are smaller than 1.59, because if (a/b) > 1.59, (a/b) is obviously not going to be equal to root(2), which roughly equal to 1.41
so we can rearrange and get
root(2) + (a/b) <= 3
multiply by b!!!
(b*root(2)) + a <= 3b
remember our original expression?
here's the trick we're gonna use: what if you showed that it was impossible for the difference between root(2) and _any_ rational number to be zero?
that's abstraction babey!!! and it's the most powerful tool in math
consider an arbitrary rational number. a rational number can be expressed as a ratio of two integers, a/b
Let's take the difference between root(2) and this arbitrary rational number a/b, root(2) - a/b
With a little bit of algebra we get the following. (contd)
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