gegeben

Gegeben is the past participle of the German verb geben (to give). In standard usage it appears in perfect tenses with the auxiliary haben, for example: Ich habe dir das Buch gegeben. It is also used in other compound tenses with habe or hat, such as Ich hatte ihm das Geld gegeben, to describe actions completed in the past.

As an adjective, gegeben means “given” or “provided” and can modify a noun. It follows normal German declension patterns, for instance: die gegebene Zeit, die gegebenen Daten, unter den gegebenen Umständen. In this adjectival use, it indicates something that is specified, assumed, or already stated in a context.

In mathematical and scientific writing, geben is often used in a fixed phrase Gegeben sei, which translates roughly to “let be given” or “let us suppose.” This introductory formulation is common when establishing assumptions or initial data, as in Gegeben seien die Werte x und y. In this context Gegeben is typically treated as a participle used in a formulaic clause rather than as a standalone adjective.

Etymology and grammar: geben is a strong verb from Germanic origins; the prefix ge- marks the past participle in German, producing gegeben. The word does not form a passive voice with a separate participle; instead, given is expressed through the participle in perfect constructions or as an attributive adjective.

In English, gegeben translates as “given.” In German texts, it serves both as the past participle of geben and as an adjective meaning specified or assumed, with a specialized usage in mathematical discourse.