ungeraden

Ungeraden is the German adjective used to describe numbers that are not divisible by two, i.e., odd numbers. In mathematics and German-language texts the phrase die ungeraden Zahlen (the odd numbers) is common. The set of odd integers can be expressed as {2k+1 | k ∈ Z}, and all odd numbers have parity 1 modulo 2. The form ungeraden is the declined form used with definite articles and plural nouns, as in die ungeraden Zahlen.

Characteristics of odd numbers include their simple characterization by the modulo operation: an integer n is odd if n mod 2 = 1. In decimal notation, a nonnegative odd number ends with 1, 3, 5, 7, or 9; negative odd numbers share the same last-digit pattern ignoring the sign. Arithmetic relations involving odd numbers follow familiar parity rules: the sum or difference of two odd numbers is even; the sum of an odd and an even number is odd; the product of two odd numbers is odd; a product that includes any even factor is even. Every odd number can be written in the form 2k+1 with k ∈ Z.

Uses and context include foundational role in number theory and parity problems. In German-language pedagogy, ungerade is contrasted with gerade (even). In computer science and programming, parity checks often determine oddness by evaluating n mod 2. As a grammatical form, ungeraden appears with definite articles in plural phrases, while the underlying mathematical concept remains the same across contexts.