Absolutbeträge

Absolutbeträge, also known as the absolute value, is a mathematical concept representing the distance of a number from zero on the number line. It is always a non-negative value. The absolute value of a number 'x' is denoted by |x|. For example, the absolute value of 5 is |5| = 5, and the absolute value of -5 is |-5| = 5. This means that both 5 and -5 are the same distance away from zero.

Mathematically, the absolute value is defined piecewise: if x is greater than or equal to zero, then |x| = x. If x is less than zero, then |x| = -x. This definition ensures that the result is always positive or zero. The absolute value function has several important properties. It is always non-negative, |x| >= 0. It satisfies the triangle inequality, |x + y| <= |x| + |y|. Also, the absolute value of a product is the product of the absolute values, |xy| = |x||y|.

The concept of absolute value is fundamental in various areas of mathematics, including algebra, calculus, and geometry. It is used in defining distances between points, solving equations and inequalities, and in many statistical and physics applications where magnitude rather than direction is important. For instance, when calculating the error between an estimated value and an actual value, the absolute difference is often used to express the magnitude of the error.