# Abstract Algebra Square Roots Are Irrational

For part (a), I begin by trying to prove $$S$$ is empty implies the square root of $$D$$ is irrational. If we take the contrapositive of this implication, this is equivalent to proving that if the square root of $$D$$ is rational then $$S$$ is not empty. Let $$D$$ be a positive integer and suppose $$D$$ is rational. Then it follows that $$D^2$$ is a perfect square. Moreover, S = n*D^1/2 and since n and D are positive integers, then by definition, S is non-empty. Conversely, we wish to prove that the square root of D is not rational implies S is empty. But this trivially follows from the product of a rational and irrational number. I'm not sure if my reasoning is sound, but constructive criticism would be appreciated.

For part (b), I'm not sure how exactly to use Well-Ordering to prove the inequality. I suppose my initial thought was to square both sides of the inequality and then rearrange some terms. Because the square root of D is not rational, it follows D^2 is not a perfect square. Then we can bound it appropriately?

For part (c), I think what we should begin with is to consider the number m*(D^1/2 - a). From there, I do not know how to proceed. Any help would greatly be appreciated.

• Can you please type your question out instead of pasting an image? Also, can you use MathJax to typeset your formulae? – Morgan Rodgers Jan 29 '19 at 7:08
• Sorry I just need help – Sanjoy Kundu Jan 29 '19 at 19:06

In part (a) you are right about the contrapositive of the implication but you proved it wrong because you said that "and since $$n$$ and $$D$$ are positive integers, then by definition, S is non-empty" but I don't see that this is true because $$\sqrt D \in \mathbb Q$$ doesn't imply that $$D\in \mathbb Z$$
So instead you can say that if $$\sqrt D \in \mathbb Q \implies \sqrt D= \frac{a}{b}$$ where $$a,b\in \mathbb Z$$
(Note that $$a$$ and $$b$$ are positive here since a square root is always positive so $$a,b\in \mathbb N$$)
Which gives that $$b\sqrt D=a \in \mathbb Z$$ so S is not empty since $$b\in S$$.
Now for the sufficient condition, yes it is trivial since if $$\sqrt D \not \in \mathbb Q \implies n\sqrt D \not \in \mathbb Q, \forall n\in \mathbb N \implies S=\phi$$