summen

Summen is a German term with more than one primary meaning. In mathematics, Summen refers to the result of adding together numbers, and it can appear as the plural form of Summe. The standard word for a single total is Summe, while Summen can be used when discussing several sums or when the plural is specifically intended. In mathematical writing, phrases like endliche Summen (finite sums) and unendliche Summen (infinite sums) are common, and the overall operation is often denoted by the sigma symbol ∑ as in 1 + 2 + 3 = 6 or the infinite series 1/2 + 1/4 + 1/8 + … which converges to 1. The concept underpins summation in calculus and analysis, where properties such as linearity, commutativity, and associativity of addition play a role. Another related term is Teil-Summen or Partialsummen, meaning partial sums of a sequence.

In another sense, Summen is the infinitive form of the German verb sum­men, meaning to hum or to buzz with a low, continuous sound. It is used for bees, machines, or voices producing a background drone. Conjugation follows regular weak-verb patterns: ich summe, du summst, er summt, wir summen, ihr summt, sie summen. The verb is common in descriptive language about sound in nature or machinery, e.g., “Die Bienen summen im Garten.”

Usage notes: while Summen as a plural noun exists, in many mathematical contexts writers prefer Summe (singular) and Gesamtsumme for the total. The term Summen is still correct when referring to multiple sums or in set phrases. The two senses—mathematical and acoustic—are distinct yet share the same word form.