mediant (mathematics)

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

and

\frac b d-\frac{a+b}{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 $bc-ad=1$ . 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 $a'/c'$ 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 numbers $\lambda_1,\,\lambda_2$ . The relation $bc-ad=1$ then implies that both $\lambda_1,\,\lambda_2$ must be integers. 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

bc-ad=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

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