# Thread: Rationals and irrationals - properties

1. ## Rationals and irrationals - properties

Hey guys!

So I have two questions which are similar, but not the same. The first asks me to prove that between any two distinct rational numbers there exists an irrational number - I haven't managed to do this. The question after however, which asks me to show that between any two real numbers there exists an irrational number, I've had an attempt at but I'm not sure whether the proof is adequate enough - it is written below (please bear in mind that I have already proven that between any two real numbers lies a rational number, so the first statement of my proof follows from that theorem, and in addition that $\sqrt2 \notin \mathbb{Q}$):

Consider $\dfrac{a}{\sqrt2} < \dfrac{p}{q} < \dfrac{b}{\sqrt2}$ where $p,q \in \mathbb{Z}$ with $q\not= 0$ and $a,b \in \mathbb{R}$. That is, by definition, $\dfrac{p}{q} \in \mathbb{Q}$. Multiplying this inequality by $\sqrt2 > 0$ means that the inequality still holds, hence $a < \dfrac{p\sqrt2}{q} < b$ and therefore it follows that between any two real numbers there lies an irrational number as $\dfrac{p\sqrt2}{q} \notin \mathbb{Q}$.

2. ## Re: Rationals and irrationals - properties

some things you need to add, for clarity:

since a,b are assumed distinct, without loss of generality you may assume a < b (or else switch them, a standard tactic).

you should show why p√2/q is not rational (it's not hard, and only takes a line or two).

p needs to be non-zero, or else your argument fails. for example, what if a = -1/n, and b = 1/n, where n is a VERY large positive integer (like 3 billion)? this is a rather serious defect.

3. ## Re: Rationals and irrationals - properties

Originally Posted by Deveno
some things you need to add, for clarity:

since a,b are assumed distinct, without loss of generality you may assume a < b (or else switch them, a standard tactic).

you should show why p√2/q is not rational (it's not hard, and only takes a line or two).

p needs to be non-zero, or else your argument fails. for example, what if a = -1/n, and b = 1/n, where n is a VERY large positive integer (like 3 billion)? this is a rather serious defect.

Hmm, first of all thanks very much for your input; I really appreciate it.

I understand your points apart from the last one - sorry I don't follow. I sort of understand that p must be non-zero but how does the following argument where a = -1/n and b = 1/n relate?

Also, I've just proved that between two distinct real numbers there lies an irrational number. But how do I prove that between two distinct rational numbers there lies an irrational number? I thought that surely if all rational numbers are real then haven't I kind of just proved that already?

4. ## Re: Rationals and irrationals - properties

Originally Posted by Femto
Also, I've just proved that between two distinct real numbers there lies an irrational number. But how do I prove that between two distinct rational numbers there lies an irrational number? I thought that surely if all rational numbers are real then haven't I kind of just proved that already?
You have done that.
Between any two numbers there is a rational number.