what are irrational numbers?

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  1. Irrational numbers are numbers that cannot be written as a fraction. E.g. sqrt(2) cannot be a fraction p/q:

    sqrt(2) = p/q --->

    p^2/q^2 = 2 --->

    p^2 = 2 q^2

    Factor both sides in prime factors. The number p has a certain number of factors, p^2 has double that number, so the left hand side contains an even number of prime factors. By the same reasoning q^2 also contains an even number of prime factors, so 2 q^2 contains an odd number of prime factors as 2 is a prime number. But this means that p^2 can never equal 2 q^2!

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