Show a set of numbers are irrational

Dec 2006
170
3
Show that
\(\displaystyle (p,q \in\mathbb{N}^+) \wedge \left((\sqrt{p+q} - \sqrt{q})^2 < 1/e \right)\Rightarrow \left(\ln\left(1+\frac{p}{q}\right) \in \mathbb{R \setminus Q}\right)\)
 
Nov 2010
288
1
hello there
i think the key to the proof lies in the fact that:
\(\displaystyle ln(1+p/q)=ln(q+p/q)\)
now this number meaning
\(\displaystyle (q+p)/q\)
is a rational number no matter what p and q are
(as long as they are natural) but it has been proved
that ln(rational number) is always irrational
except for the case where the rational number is 1
but in ur case 1+p/q is always bigger than 1
so u just need to search the proof that ln(rational number)
is always irrational :mrgreen: