Question on proof of no rational number whose square is 2 in the proof that there is no rational number whose square is 2:

Suppose x is a rational number whose square is 2.

Then x can be written in lowest terms as $\frac{a}{b}$ , where a is an integer and b is a positive integer.

Since $x^2 = 2$, $\left (\frac{a}{b}  \right )^2$, so $\frac{a^2}{b^2}$
But then a is even, so $a = 2n$ for some integer n.

Then $(2n)^2$ = $2b^2$, so $4n^2 = 2b^2$.

Then $2n^2 = b^2$ , so $b^2$ is even, and thus b is even.

Then a and b both have 2 as a common factor, so $\frac{a}{b}$ cannot be in lowest terms, a contradiction.
Thus x cannot be rational.

The part I am struggling with is why we can assume that x can be written in lowest terms. Or in other proofs, they say that the gcd(a,b) must be 1, but I do not understand why this is important to the proof. Any help is appreciated
You can assume that $\dfrac{a}{b}$ is in lowest terms because any fraction can be written in lowest terms, so we just take $\dfrac{a}{b}$ to be that lowest terms representation to make the proof easier. This is exactly the same as assuming $\gcd(a,b)=1$. This assumption is crucial, as it allows us to go through the proof, and eventually conclude that $2\mid a$ and $2\mid b$, which gives a contradiction because of our construction that $\gcd(a,b)=1$. Since $\gcd(a,b)=1$ there cannot be any common factors of $a$ and $b$ other than $\pm 1$. In other words, if $\dfrac{a}{b}$ is in lowest terms, the numerator and denominator cannot both be even, because then they would both be divisible by two, and thus the fraction is not in lowest terms.
You might think of the problem in another way. It just makes the proof finite.

Say you don't consider a &amp; b to be the lowest terms. Follow it through...
You will find something along the lines of:
a = 2c
c = 2d
d = 2e
e = 2f
f = 2g
...  

This is analogous to numerical methods whereby each iteration increases accuracy but you never get to 100% accurate - wherever you stop you only have an approximation.

This is the nature of irrational numbers compared to rational numbers - they cannot be defined precisely with integers. Resting upon the lowest factors argument is shorthand for saying "this iteration shouldn't occur infinitely for rational numbers".