# Thread: Finding rational and irrational between two numbers

1. ## Finding rational and irrational between two numbers

My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of $\frac{1}{2}n$ and write the irrationals in the form of $\frac{1}{\sqrt{2}}n$, where n is rational.

Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?

2. Originally Posted by Jameson
My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of $\frac{1}{2}n$ and write the irrationals in the form of $\frac{1}{\sqrt{2}}n$, where n is rational.

Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?
Scrolling down this link might help get the ball rolling ...?

3. Here is a proof that I like very much. I first saw in it in Martin Davis’ Applied Nonstandard Analysis.
LEMMA: $x - y > 1 \Rightarrow \quad \left( {\exists n \in Z} \right)\left[ {y < n < x} \right]$.
The proof of the is straight forward using properties of the floor function.
$\left\lfloor y \right\rfloor \le y < \left\lfloor y \right\rfloor + 1 \Rightarrow \quad y < \left\lfloor y \right\rfloor + 1 \le y + 1 < x$. Note that $\left\lfloor y \right\rfloor + 1$ is an integer.

Suppose that ${a < b}$ then $\left( {\exists J \in Z} \right)\left[ {\frac{1}{{b - a}} < J} \right].$ Because $Jb - Ja > 1 \Rightarrow \quad \left( {\exists K \in Z} \right)\left[ {Ja < K < Jb} \right]$.
But this means $a < \frac{K}{J} < b$ or there is a rational between a & b.

If $c < d \Rightarrow \quad c\sqrt 2 < d\sqrt 2$, then $\left( {\exists r \in Q\backslash \{ 0\} } \right)\left[ {c\sqrt 2 < r < d\sqrt 2 } \right]$.
This means that ${c < \frac{r}{{\sqrt 2 }} < d}$.