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of two fractions and with positive denominators

The fraction . The mediant of two fractions is positioned between them, i.e. if , , then

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 is positioned between and the convergent fraction of lower order (cf. also Continued fraction). Thus, if and are convergent fractions of orders and in the continued-fraction expansion of , then

References

[1] A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) (Translated from Russian)


Comments

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)
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