> I'm sure the following is true but is it a known theorem.

> If so what is it called.

> Given a pair of rational (i.e. they dont need to be integer)

> factors of an integer N; p/q and Nq/p then the rational

> (p+Nq)/(q+p) lies between p/q and Nq/p This is effectively

> saying (p+Nq)/(q+p) is a better approximation to

> SQRT(N) than p/q

> A proof would be nice but if someone could just point me at

> the appropriate theorem I would appreciate it.

Let a < b and c > 1. Then a < (a+c*b)/(1+c) < b

Proof. (a+c*b)/(1+c) = a + c*(b-a)/(1+c) > a,

and (a+c*b)/(1+c) = b - (b-a)/(1+c) < b. End proof.

Your result:

Note (p+Nq)/(q+p) = (p/q + N)/(1 + p/q) = (Nq/p + 1)/(1 + q/p)

follows from:

if a=p/q < b=Nq/p, then let c=p/q.

if a=Nq/p < b=p/q, then let c=q/p.