TITLE: Is $22/7$ the closest to $\pi$, among fractions of denominator at most $50$?
QUESTION [6 upvotes]: Is $22/7$ the closest to $\pi$, among fractions of denominator at most $50$?
I am currently studying continued fractions, while I know that for all denominators at most $Q_n$, $\frac{P_n}{Q_n}$ is the closed approximation. But what about the denominators between $Q_n$ and $Q_{n+1}$?

REPLY [2 votes]: (1). If $a,b,c,d\in \Bbb N$ with $|ad-bc|=1$ then $(ma+nc)/(mb+nd)$ is in lowest terms whenever $m,n\in \Bbb N$ with $\gcd(m,n)=1,$ and every rational between $a/b$ and $c/d$ is equal to $(ma+nc)/(mb+nd)$ for some co-prime $m,n \in \Bbb N.$
(2). Let $\delta=3+1/7 -\pi.$ We have $3+1/8 <\pi-\delta<\pi<
\pi+\delta=3+1/7.$
If $q\in \Bbb Q$ and $|\pi-q|<\delta$ then $1/8<q-3<1/7$ so by (1), for some $m,n \in \Bbb N$ with $\gcd(m,n)=1$ we have $1/7 -2\delta <q-3=(m+n)/(8m+7n).$
This implies $0<1/7 -(m+n)/(8m+7n)<2\delta$ and hence $7n>m(-8+1/14\delta).$ Since $1/14\delta>56,$ this implies $7n>48m\ge 48,$ so $n\ge 7.$
So by (1) the lowest-terms denominator for $q,$ which is  $8m+7n,$ is at least $8(1)+7(7)=57.$
BTW. $\pi-\delta<3+8/57<\pi.$