View source for Mediant

''of two fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633302.png" /> with positive denominators''

The fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633303.png" />. The mediant of two fractions is positioned between them, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633305.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633306.png" /></td> </tr></table>

A finite sequence of fractions in which each intermediary term is the mediant of its two adjacent fractions is called a [[Farey series|Farey series]]. The mediant of two adjacent convergent fractions of the continued-fraction expansion of a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633307.png" /> is positioned between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633308.png" /> and the convergent fraction of lower order (cf. also [[Continued fraction|Continued fraction]]). Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m0633309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m06333010.png" /> are convergent fractions of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m06333011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m06333012.png" /> in the continued-fraction expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m06333013.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063330/m06333014.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,   "Continued fractions" , Univ. Chicago Press  (1964)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)</TD></TR></table>