natürlichen

natürlichen is the plural, definite-determiner inflected form of the adjective natürlich in German. It is not a standalone term, but an attributive form used before plural nouns when a definite article or a determiner that requires weak inflection accompanies the noun. In this use, the ending -en is standard across cases in the plural: die natürlichen Zahlen (nominative or accusative), der natürlichen Zahlen (genitive), den natürlichen Zahlen (dative). The same ending appears with other plural nouns in phrases like die natürlichen Ressourcen or den natürlichen Lebensräumen.

In mathematics, the phrase natürliche Zahlen designates the natural numbers, the basic counting numbers. The exact inclusion of zero varies by tradition: some texts define the natural numbers as {0, 1, 2, …}, others as {1, 2, 3, …}. When discussing this set, the word is often capitalized as Natürliche Zahlen or Naturzahlen in German mathematical literature, reflecting its status as a named mathematical object.

Beyond mathematics, natuerlichen appears in general language to describe things related to nature or occurring in nature, such as natürliche Umgebung (natural environment), natürliche Ressourcen (natural resources), or natürliche Ursachen (natural causes). In these phrases, the adjective takes the -en ending because of the definite article or other determiner in plural usage.

Summary: natuerlichen is a grammatical form used for plural adjectives with definite reference; it also commonly occurs in the mathematical term for the natural numbers and in everyday phrases describing natural phenomena.