halbsumme

Halbsumme is the value obtained by averaging two numbers a and b, defined as (a+b)/2. In German mathematical usage, Halbsumme is synonymous with the arithmetic mean of two numbers and represents the midpoint of the closed interval [a,b] on the real number line. It is symmetric in a and b and always lies between them; if a ≤ b, then a ≤ (a+b)/2 ≤ b.

In numerical methods, the halbsumme is used as the equal-weight representative of two values, for example as the midpoint in the bisection method for root finding, where the interval is halved at each step. In geometry, it is the midpoint of the segment connecting the points a and b on the real line.

In trigonometry the term halbsumme appears in half-sum formulas, where expressions such as sin x + sin y and cos x + cos y are rewritten using the half-sum (x+y)/2 and the half-difference (x−y)/2. Thus (x+y)/2 is called the half-sum of the angles x and y.

For more than two numbers, the analogous concept is the arithmetic mean or average: the mean of numbers x1,...,xn is (x1+...+xn)/n, which generalizes the halbsumme when n=2.

Example: the halbsumme of 4 and 8 is (4+8)/2 = 6; the halbsumme of −3 and 7 is 2.

See also: arithmetic mean, midpoint, bisection method, sum-to-product formulas.