Mediant (mathematics)

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In mathematics, the mediant of two fractions, generally made up of four positive integers

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and is defined as

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 early stages of learning about addition of fractions.

Technically, this is a binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational numbers as equivalence classes of fractions. For example, the mediant of the fractions 1/1 and 1/2 is 2/3. However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique representatives from the respective equivalence classes.

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

Properties

  • Componendo: [1]
  • Dividendo: [1]

Graphical determination of mediants

Determining the mediant of two rational numbers graphically. The slopes of the blue and red segments are two rational numbers; the slope of the green segment is their mediant. Mediant.png
Determining the mediant of two rational numbers graphically. The slopes of the blue and red segments are two rational numbers; the slope of the green segment is their mediant.

A positive rational number is one in the form where are positive natural numbers; i.e.. The set of positive rational numbers is, therefore, the Cartesian product of by itself; i.e.. A point with coordinates represents the rational number , and the slope of a segment connecting the origin of coordinates to this point is . Since are not required to be coprime, point represents one and only one rational number, but a rational number is represented by more than one point; e.g. are all representations of the rational number . This is a slight modification of the formal definition of rational numbers, restricting them to positive values, and flipping the order of the terms in the ordered pair so that the slope of the segment becomes equal to the rational number.

Two points where are two representations of (possibly equivalent) rational numbers and . The line segments connecting the origin of coordinates to and form two adjacent sides in a parallelogram. The vertex of the parallelogram opposite to the origin of coordinates is the point , which is the mediant of and .

The area of the parallelogram is , which is also the magnitude of the cross product of vectors and . It follows from the formal definition of rational number equivalence that the area is zero if and are equivalent. In this case, one segment coincides with the other, since their slopes are equal. The area of the parallelogram formed by two consecutive rational numbers in the Stern–Brocot tree is always 1. [2]

Generalization

The notion of mediant can be generalized to n fractions, and a generalized mediant inequality holds, [3] a fact that seems to have been first noticed by Cauchy. More precisely, the weighted mediant of n fractions is defined by (with ). It can be shown that lies somewhere between the smallest and the largest fraction among the .

See also

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References

  1. 1 2 3 Milburn, R. M. (1880). Mathematical Formulae: For the Use of Candidates Preparing for the Army, Civil Service, University, and Other Examinations. Longmans, Green & Company. pp. 18–19.
  2. Austin, David. Trees, Teeth, and Time: The mathematics of clock making, Feature Column from the AMS
  3. Bensimhoun, Michael (2013). "A note on the mediant inequality" (PDF). Retrieved 2023-12-25.