betrag

Betrag is a German noun with several related meanings. In everyday language, Betrag means amount, quantity, total, or price. In a mathematical context, the term is used for the absolute value or magnitude of a quantity. In German mathematics, der Betrag von x is written as |x| and is read as the absolute value of x.

Etymology: The word derives from be- and tragen, meaning to bear or carry; literally, what is borne.

In real numbers: the absolute value |x| is defined as x if x is nonnegative and as -x if x is negative. It is always nonnegative and satisfies the multiplicative property |xy| = |x||y| and the triangle inequality |x+y| ≤ |x| + |y|.

Complex numbers and vectors: for a complex number z = a + bi, the magnitude or modulus is |z| = sqrt(a^2 + b^2), representing the distance from the origin in the complex plane. In vector spaces, the magnitude generalizes to the norm, which shares key properties such as non-negativity, definiteness, and the triangle inequality.

Applications: the concept of magnitude is central in measuring size, length, and distance, in normalizing vectors, and in various quantitative contexts where only the size of a quantity matters. In German usage, Betrag can also refer to the total sum or price in finance and accounting, beyond its mathematical sense.