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

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...)