Gegebenes

Gegebenes is a German term used in mathematical problem statements to denote the initial data or givens. It typically appears at the outset of a problem, either within a sentence that begins with “Gegeben ist” or as a labeled section that lists the objects, quantities, and conditions assumed to be true. The function of Gegebenes is to specify the information from which a solution proceeds and to distinguish what is already known from what must be determined.

Linguistically, Gegebenes is formed from the verb geben (to give) and is used as a nominalized form meaning “the given” or “the givens.” In standard German, one can also encounter the noun das Gegebene, referring to the same idea; in practice, many texts present the data under the heading Gegebenes. The format helps readers quickly identify the problem’s constraints and the data upon which the reasoning will rest. It is common to contrast Gegebenes with Gesucht (or Ziel), which specifies the quantity, object, or condition to be found or proven.

In usage, Gegebenes may list geometric objects (a triangle, a circle), numerical values (lengths, angles, coordinates), and relations (parallelism, equalities, containment). An example might read: Gegebenes: a triangle ABC with side lengths a, b, c; angle C = 60 degrees; point P on AB with AP = 2 and PB = 3. The concept is widespread in German-language textbooks, exercise sheets, and proofs, serving as a concise preface to the problem’s statement.