In
mathematics, the mediant of two
fractions, generally made up of four positive integers
:
and
is defined as
That is to say, the
numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
and
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
(nonzero denominator), considered as
ordered pairs of appropriate integers, a priori disregarding the perspective on
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
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
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). I ...
, 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
* 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
and
, 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 and denominator can be written as
and
with two ''positive'' real (in fact rational) numbers
. To see why the
must be positive note that
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 holds. This fact may be deduced e.g. with the help of
Pick's theorem
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1 ...
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 ''v''
1 = (0, 0), ''v''
2 = (''a'', ''c''), ''v''
3 = (''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 lying inside the triangle different from the three vertices if (then the area of the triangle is
). The corresponding fraction ''q''
1/''q''
2 lies (strictly) between the given (by assumption reduced) fractions and has denominator