# Absolute value

*the initial statement of logic is unclear*. (Discuss)

**Absolute value** is a function measuring a number's distance from zero. The absolute value of is written .

- If is a positive number, then .
- If is a negative number, then .
- If then .

For example, , and . Notice that is never negative.

Absolute value has several useful properties. One is the *multiplicative* property. If and are two numbers, then . Another is the *triangle inequality*, which is the fact that . For example, if and , then , while . In this case, the triangle inequality is the fact that 2 is not more than 8.

If is a real number, then . The absolute value is necessary because the principal square root is, by definition, nonnegative.

Complex numbers also have an absolute value (sometimes called the modulus). If is a complex number with real part and imaginary part , then . If we represent as a point in the complex plane with coordinates , then is the distance from this point to the origin. The absolute value of complex numbers also has the multiplicative property and satisfies the triangle inequality.