Mediant (mathematics)

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For mediant in music, see mediant. "Mediant" should not be confused with median.

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.

The Stern-Brocot tree provides an enumeration of all positive rational numbers, in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.

[edit] Properties

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 $c'\ge c+d$ .

The converse is also true: Assume that the pair of reduced fractions a/c < b/d has the property that the reduced fraction with smallest denominator lying in the interval (a/c, b/d) is equal to the mediant of the two fractions. Then the determinant relation bc − ad = 1 holds. This fact may be deduced e.g. with the help of Pick's theorem which expresses the area of a plane triangle whose vertices have integer coordinates in terms of the number v_interior of lattice points (strictly) inside the triangle and the number v_boundary of lattice points on the boundary of the triangle. Consider the triangle $Δ(v 1, v 2, v 3)$ with the three vertices v₁ = (0, 0), v₂ = (a, c), v₃ = (b, d). Its area is equal to

$\mbox{area}(\Delta)={{bc-ad}\over 2} \ .$

A point

p = (p 1, p 2)

inside the triangle can be parametrized as

$p_1=\lambda_1 a+\lambda_2 b,\; p_2=\lambda_1 c+\lambda_2 d,$

where

$\lambda_1\ge 0,\,\lambda_2 \ge 0, \,\lambda_1+\lambda_2 \le 1 \ .$

The Pick formula

$\mbox{area}(\Delta)=v_\mathrm{interior}+{v_\mathrm{boundary}\over 2}-1$

now implies that there must be a lattice point q=(q₁,q₂) lying inside the triangle different from the three vertices if bc-ad >1 (then the area of the triangle is $\ge 1$ ). The corresponding fraction q₁/q₂ lies (strictly) between the given (by assumption reduced) fractions and has denominator

$q_2=\lambda_1c+\lambda_2d \le \max(c,d)<c+d$

$\lambda_1+\lambda_2 \le 1 \ .$

Relatedly, if p/q and r/s are reduced fractions on the unit interval such that |ps − rq| = 1 (so that they are adjacent elements of a row of the Farey sequence) then

$?\left(\frac{p+r}{q+s}\right) = \frac12 \left(?\bigg(\frac pq\bigg) + {}?\bigg(\frac rs\bigg)\right)$

where ? is Minkowski's question mark function.

In fact, 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.