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''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''
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''of two fractions $a/b$ and $c/d$ 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
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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
  
<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>
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$$\frac ab\leq\frac{a+c}{b+d}\leq\frac cd.$$
  
A finite sequence of fractions in which each intermediary term is the mediant of its two adjacent fractions is called a [[Farey series|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
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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 $\alpha$ is positioned between $\alpha$ and the convergent fraction of lower order (cf. also [[Continued fraction|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
  
<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>
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$$\left|\alpha-\frac{P_n}{Q_n}\right|>\left|\frac{P_n+P_{n+1}}{Q_n+Q_{n+1}}-\frac{P_n}{Q_n}\right|=\frac{1}{Q_n(Q_n+Q_{n+1})}.$$
  
 
====References====
 
====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>
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|valign="top"|{{Ref|Kh}}||valign="top"|A.Ya. Khinchin, "Continued fractions", Univ. Chicago Press (1964) (Translated from Russian) {{MR|0161833}} {{Zbl|0117.28601}}
  
  
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====References====
 
====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>
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|valign="top"|{{Ref|HaWr}}||valign="top"|G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}}
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Revision as of 12:59, 1 September 2014

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 Template:Zbl


Comments

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
How to Cite This Entry:
Mediant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mediant&oldid=33226
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article