>>12088629Here's a proof by contradiction why the square root of 2 cannot be rational. There are lots of proofs for numbers like pi, e, the golden ratio and so on, but the proof for sqrt(2) is just a lot more intuitive and simpler and requires less mathematical knowledge to understand. If you're interested in why pi is irrational, I would advise you to look it up for yourself. There are probably tons of videos about it on youtube. Anyway here it goes:
We begin by assuming the opposite, that sqrt(2) is rational
This means that sqrt(2) can be written as the ratio
We note that if is the most simplified way of writing this ratio, we p and q cannot both be even (otherwise they would have the common factor 2)
Since we know the left hand side is even (we're multiplying any random number by 2 there, thus it must be even), so is . From this it follows that p must also be even, because if p were odd, would also be odd (an odd number times an odd number is odd).
We now know p must be even, this means we can write p as . It follows then that:
. But from the same logic we used to prove that p was even, it would follow now that q had to be even as well. But this would then imply that both p and q were even. And this goes against the definition of the most reduced representation of a rational number (p/q). Therefore our assumption must have been false, sqrt(2) can not be represented as a ratio of two integers.
