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Here is my proof:

If you wanted to cut a piece of paper exactly 1.284736 cm, you would probably measure it to the tenths place (1.3) or the hundreths place (1.28). If you wanted to be even more exact, you could keep on going until the ten millionth place.

Now, suppose you wanted to cut this piece of paper exactly the square root of two.

As we all know, the sqrt(2) is approximately 1.414213562. I say "approximately" since this number goes on forever. Therefore, you can never get an EXACT measurement since you always have another number in the decimal that you haven't taken into account.

As I said before, I am probably very wrong about this statement.