词条 | Mediant (mathematics) |
释义 |
For mediant in music, see mediant. "Mediant" should not be confused with median. In mathematics, the mediant of two fractions, generally made up of four positive integers 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
This property follows from the two relations and
and must be positive. The determinant relation then implies that both must be integers, solving the system of linear equations for . Therefore
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 − 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
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 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. Graphical determination of mediantsA 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.[1] GeneralizationThe notion of mediant can be generalized to n fractions, and a generalized mediant inequality holds,[2] 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 . References1. ^Austin, David. Trees, Teeth, and Time: The mathematics of clock making, Feature Column from the AMS 2. ^{{cite journal| first=Michael|last=Bensimhoun| url=https://commons.wikimedia.org/wiki/File:Extension_of_the_mediant_inequality.pdf|format=PDF| title = A note on the mediant inequality|year=2013}} External links
3 : Fractions (mathematics)|Elementary arithmetic|Binary operations |
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