Difference between revisions of "Mediant"
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− | ''of two fractions | + | {{TEX|done}} |
+ | ''of two fractions $a/b$ and $c/d$ with positive denominators'' | ||
− | The fraction | + | 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|Farey series]]. The mediant of two adjacent convergent fractions of the continued-fraction expansion of a real number | + | 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 |
− | + | $$\left|\alpha-\frac{P_n}{Q_n}\right|>\left|\frac{P_n+P_{n+1}}{Q_n+Q_{n+1}}-\frac{P_n}{Q_n}\right|=\frac{1}{Q_n(Q_n+Q_{n+1})}.$$ | |
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |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==== | ||
− | + | {| | |
+ | |- | ||
+ | |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}} | ||
+ | |- | ||
+ | |} |
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
CommentsReferences
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Mediant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mediant&oldid=16953