Mediant (mathematics)

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

In mathematics, the mediant of two fractions

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

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. It is sometimes called the freshman sum, as it is a common mistake in the usual addition of fractions.

In general, this is an operation on fractions rather than on rational numbers. That is to say, for two rational numbers q1, q2, the value of the mediant depends on how the rational numbers are expressed using integer pairs. For example, the mediant of 1/1 and 1/2 is 2/3, but the mediant of 2/2 and 1/2 is 3/4.

A way around this, where required, is to specify that both rationals are to be represented as fractions in their lowest terms (with c > 0, d > 0). With such a restriction, mediant becomes a well-defined binary operation on rationals.

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.

Properties

{\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).
{\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
bc-ad=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 \lambda _{1},\lambda _{2}. Therefore c'\geq c+d.
{\text{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}\geq 0,\,\lambda _{2}\geq 0,\,\lambda _{1}+\lambda _{2}\leq 1.\,
The Pick formula
{\text{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 \geq 1). The corresponding fraction q1/q2 lies (strictly) between the given (by assumption reduced) fractions and has denominator
q_{2}=\lambda _{1}c+\lambda _{2}d\leq \max(c,d)<c+d
as
\lambda _{1}+\lambda _{2}\leq 1.\,
?\left({\frac {p+r}{q+s}}\right)={\frac {1}{2}}\left(?{\bigg (}{\frac {p}{q}}{\bigg )}+{}?{\bigg (}{\frac {r}{s}}{\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 Fn 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 Fn 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 Fn are successively built up with increasing n.

Generalization

The notion of mediant can be generalized to n fractions, and a generalized mediant inequality holds,[1] a fact that seems to have been first noticed by Cauchy. More precisely, the weighted mediant m_{w} of n fractions a_{1}/b_{2},\ldots ,a_{n}/b_{n} is defined by {\frac {\sum _{i}w_{i}a_{i}}{\sum _{i}w_{i}b_{i}}} (with w_{i}>0). It can be shown that m_{w} lies somewhere between the smallest and the largest fraction among the a_{i}/b_{i}.

References

  1. Bensimhoun, Michael (2013). "A note on the mediant inequality" (PDF).

External links

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