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 x²
would be an odd number because if you square an odd number, the product is odd.
But x²
= 2(y²) and y²
is any positive integer and is multiplied by 2
here, so that will always result in an even number. So x²
has to be an even number and therefore x
has to be even as well.
In summary, x² 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 y² is even and therefore y is even (using the same logic applied to x² 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...)
