Mediant (mathematics) - Properties

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
\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
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 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 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
p_1=\lambda_1 a+\lambda_2 b,\; p_2=\lambda_1 c+\lambda_2 d,
where
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 −&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
q_2=\lambda_1c+\lambda_2d \le \max(c,d)<c+d
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) then
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.

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