Defintion
Rationalization is the expression of a number in as a ratio (or "fraction") of the form: a/b where a and b are integers and have no common factors greater than 1.
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color(black)(sqrt(2)/2) is an irrational number.
Proof:
Suppose that sqrt(2)/2 were a rational number.
Then, by definition of rational numbers, we could write
color(white)("XXX")sqrt(2)/2=a/b
where a and b were integer values with no common factors.
This would mean
color(white)("XXX")sqrt(2)b=2a
Squaring both sides:
color(white)("XXX")2b^2=4a^2
color(white)("XXX")b^2=2a^2
color(white)("XXX")rarr b^2 must be even
and since squared odd numbers are odd
color(white)("XXX")rarr b must be even.
So we could replace b with 2c for some integer c
color(white)("XXX")sqrt(2)/2=a/(2c)
color(white)("XXX")cancel2sqrt(2)c=cancel2a
color(white)("XXX")cancel2c^2=cancel4^2a^2
color(white)("XXX")rarr a^2 is even
color(white)("XXX")rarr a is even.
Therefore both a and b would need to be even;
that is both a and b would have a common factor of 2
...but this is contrary to the initial declaration that a and b have no common factors.
