The Ultimate Mediant (mathematics) - American History Information Guide and Reference

For mediant in music, see mediant.

In mathematics, the mediant of two fractions

$\frac {a} {c}$ and $\frac {b} {d}$

(where c > 0, d > 0) is

$\frac {a + b} {c +d}$

that is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively.

Properties of the mediant:

An important property (also explaining its naming) of the mediant is that it lies strictly between the two fractions of which it is the mediant: If $a / c < b / d$ , then

$\frac a c < \frac{a+b}{c+d} < \frac b d \ .$

This property follows from the two relations

$\frac{a+b}{c+d}-\frac a c={{bc-ad}\over{c(c+d)}} ={d\over{c+d}}\left( \frac{b}{d}-\frac a c \right)$

and

$\frac b d-\frac{a+b}{c+d}={{bc-ad}\over{d(c+d)}} ={c\over{c+d}}\left( \frac{b}{d}-\frac a c \right) \ .$

Assume that the pair of fractions a/c and b/d satisfies the determinant relation $b c - a d = 1$ . Then the mediant has the property that it is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. More precisely, if the fraction $a' / c'$ with positive denominator c' lies (strictly) between a/c and b/d, then its numerator resp. denominator can be written as $\,a'=\lambda_1 a+\lambda_2 b$ and $\,c'=\lambda_1 c+\lambda_2 d$ with two positive real (in fact rational) numbers $\lambda_1,\,\lambda_2$ . To see why the $λ i$ must be positive note that

$\frac{\lambda_1 a+\lambda_2 b}{\lambda_1 c+\lambda_2 d }-\frac a c=\lambda_2 {{bc-ad}\over{c(\lambda_1 c+\lambda_2 d)}}$

and

$\frac b d-\frac{\lambda_1 a+\lambda_2 b}{\lambda_1 c+\lambda_2 d }=\lambda_1 {{bc-ad}\over{d(\lambda_1 c+\lambda_2 d )}}$

must be positive. The determinant relation

b c - a d = 1

then implies that both $\lambda_1,\,\lambda_2$ must be integers, solving the system of linear equations

$\, a'=\lambda_1 a+\lambda_2 b$

$\, c'=\lambda_1 c+\lambda_2 d$

for

λ 1,λ 2

. Therefore $c'\ge c+d$ .

Mediants commonly occur in the study of continued fractions and in particular, Farey fractions . The nth Farey sequence F_n is defined as the (ordered with respect to magnitude) sequence of reduced fractions a/b (with coprime a, b) such that b ≤ n. If two fractions a/c < b/d are adjacent (neighbouring) fractions in a segment of F_n then the determinant relation $b c - a d = 1$ mentioned above is generally valid and therefore the mediant is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. Thus the mediant will then (first) appear in the (c + d)th Farey sequence and is the "next" fraction which is inserted in any Farey sequence between a/c and b/d. This gives the rule how the Farey sequences F_n are successively built up with increasing n.

External links

Mediant Fractions

Categories: Fractions | Elementary arithmetic

Last updated: 05-27-2005 05:33:29

Your American History Reference Guide! - Mediant (mathematics)

Mediant (mathematics)

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Your American History Reference Guide!
- Mediant (mathematics)