**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.