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 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