Mediant

of two fractions $a/b$ and $c/d$ with positive denominators

The fraction $(a+c)/(b+d)$. The mediant of two fractions is positioned between them, i.e. if $(a/b)\leq(c/d)$, $b,d>0$, then

$$\frac ab\leq\frac{a+c}{b+d}\leq\frac cd.$$

A finite sequence of fractions in which each intermediary term is the mediant of its two adjacent fractions is called a Farey series. The mediant of two adjacent convergent fractions of the continued-fraction expansion of a real number $\alpha$ is positioned between $\alpha$ and the convergent fraction of lower order (cf. also Continued fraction). Thus, if $P_n/Q_n$ and $P_{n+1}/Q_{n+1}$ are convergent fractions of orders $n$ and $n+1$ in the continued-fraction expansion of $\alpha$, then

$$\left|\alpha-\frac{P_n}{Q_n}\right|>\left|\frac{P_n+P_{n+1}}{Q_n+Q_{n+1}}-\frac{P_n}{Q_n}\right|=\frac{1}{Q_n(Q_n+Q_{n+1})}.$$

References

[Kh]	A.Ya. Khinchin, "Continued fractions", Univ. Chicago Press (1964) (Translated from Russian) MR0161833 Zbl 0117.28601

References

[HaWr]

G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001