itseisarvoon

Itseisarvo, often denoted by vertical bars, is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. For any real number x, its absolute value, written as |x|, is always non-negative. If x is a positive number or zero, its absolute value is simply x itself. If x is a negative number, its absolute value is its opposite, which is a positive number. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

The definition can be formally expressed as a piecewise function:

|x| = x if x ≥ 0

|x| = -x if x < 0

This concept extends beyond real numbers to complex numbers as well. For a complex number z = a + bi, where a and b are real numbers and i is the imaginary unit, the absolute value (or modulus) |z| is calculated as the square root of the sum of the squares of its real and imaginary parts: |z| = √(a² + b²). Geometrically, this represents the distance of the complex number from the origin in the complex plane.

Absolute value has several important properties. It is always non-negative, as mentioned, and the absolute value of zero is zero. For any two real numbers x and y, the absolute value of their product is the product of their absolute values: |xy| = |x||y|. Similarly, the absolute value of their quotient (if y is not zero) is the quotient of their absolute values: |x/y| = |x|/|y|. The triangle inequality is another key property, stating that for any two real numbers x and y, |x + y| ≤ |x| + |y|. This property is particularly important in fields like analysis and geometry.