In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the mediant (sometimes called freshman sum) of two fractions


is


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

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.

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

Stern-Brocot tree

In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond precisely to the positive rational numbers, whose values are ordered from left to right as in a search 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

• 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 , then

This property follows from the two relations

and

• Assume that the pair of fractions a/c and b/d satisfies the determinant relation . 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 with positive denominator c' lies (strictly) between a/c and b/d, then its numerator resp. denominator can be written as and with two positive real (in fact rational) numbers . To see why the must be positive note that

must be positive. The determinant relation

then implies that both must be integers, solving the system of linear equations

for . Therefore

• 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
  Pick's theorem
  Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
  
   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 with the three vertices v₁ = (0, 0), v₂ = (a, c), v₃ = (b, d). Its area is equal to

A point inside the triangle can be parametrized as

where

The Pick formula

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

as

• 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
  Farey sequence
  In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....
  
  ) then

where ? is Minkowski's question mark function

Minkowski's question mark function

In mathematics, the Minkowski question mark function, sometimes called the slippery devil's staircase and denoted by ?, is a function possessing various unusual fractal properties, defined by Hermann Minkowski in 1904...

.

In fact, mediants commonly occur in the study of continued fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

s and in particular, Farey fractions. The nth Farey sequence

Farey sequence

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....

 F_n is defined as the (ordered with respect to magnitude) sequence of reduced fractions a/b (with coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 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 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 at cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  
• MATHPATH: Cube roots via Mediants
• MATHPAGES, Kevin Brown: Generalized Mediant

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.