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Proving What It Is By Proving What It Is Not
Proving What It Is By Proving What It Is Not

This kind of thing is some of what I do in my spare hours. Sad, really... when I think about it.

In situations involving dichotomies you can prove something is a certain thing by proving it can't be the other thing, instead of having to try to prove it is the thing you are asserting. This method runs a bit counterintuitive to people but it is a sound secondary method of logic.

This method, for example, can be used to prove √2 is an irrational number.

Numbers can be rational or irrational. One or the other. A dichotomy.

Numbers that can't be designated by the division of one integer by another are deemed irrational.

A rational number, on the other hand, is any real number that we can represent as a fraction by the division of one positive integer by another, i.e. x∕y, and the numerator and denominator share no factors in common.

So, using this secondary method that I alluded to at the beginning, I now present proof that √2 is an irrational number by trying to prove it is a rational number:

a.       If √2 were rational then √2 = x/y    (as per the rational number description above)

b.      If we square both sides:
2 = x²/y²
therefore
x² = 2(y²)

c.       Analysis: If x were an odd number then would be an odd number because if you square an odd number, the product is odd. But x² = 2(y²) and is any positive integer and is multiplied by 2 here, so that will always result in an even number. So has to be an even number and therefore x has to be even as well.
In summary,
is an even number since it is expressed as 2 times another whole number and x must also be even because if you square an odd number, the product is always an odd number.

d.      So, we now mathematically represent the logic stated in item c.:
x² = (2k)²     (for some whole number k)

e.       Therefore         x² = 2² ∙ k² = 4k²

f.       also                  x² = 2(y²)     (from b. above)

g.      Therefore         2(y²) = 4k²

h.      Reduce             2∕2 (y²) = 4∕2 k²

i.                                y² = 2k²

j.        This implies that is even and therefore y is even (using the same logic applied to in c. above).

k.      We have now shown that x and y are both even numbers.

l.        But x and y cannot share any common factor if they represent the numerator and denominator of a rational number fraction. Since they are both even numbers as we have shown them to be, they would have a common factor of 2.

m.    So our proof is in contradiction to the proper description of a rational number, therefore √2 cannot be rational. The only other option is that it's irrational.

And so it is proven what it is by proving what it is not.

by the way--->     √2  =  1.414213562... (on to infinity, no end, an irrational number...) GregO's Site