This is false. It is possible to prove a negative by proving that the converse leads to contradiction.
For example:
You assert there are two integers, a and b, such that a/b is minimal in form and equals the square root of two, sqrt(2).
I say, let us assume that there are, as you say.
By definition of square root, this means (a/b)^2 = 2.
By mathematics, this means a^2/b^2 = 2.
By mathematics, this means a^2 = 2 * b^2.
By mathematics, 2 * b^2 is even. This means a^2 is even as well, and can be expressed as (2*k)^2 for some given k.
By mathematics, (2k)^2 = 2b^2.
By mathematics, 4k^2 = 2b^2.
By mathematics, 2(2k^2) = 2(b^2).
The ability to simplify contradicts your assertion that a/b was in simplest form.
As such, I have proved the negative existential: There exist no integers, a and b, such that a/b is the square root of two. I've also incidentally proven that sqrt(2) is irrational, but that's moot here.
You can prove negative existence.
2007-04-03
09:03:15
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19 answers
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asked by
Anonymous