Mediant (mathematics)

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 very 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[edit]

  • The mediant inequality: An important property (also explaining its name) of the mediant is that it lies strictly between the two fractions of which it is the mediant: If a/c < b/d and a,b,c,d\geq 0, 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 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' 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  \lambda_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
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'\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/cb/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 vinterior of lattice points (strictly) inside the triangle and the number vboundary of lattice points on the boundary of the triangle. Consider the triangle  \Delta(v_1,v_2,v_3) with the three vertices v1 = (0, 0), v2 = (ac), v3 = (bd). Its area is equal to
 \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\ge 0,\,\lambda_2 \ge 0, \,\lambda_1+\lambda_2 \le 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 = (q1q2) lying inside the triangle different from the three vertices if bc −&nsbp;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
as
 \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 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/cb/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[edit]

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[edit]

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

External links[edit]