# Absolute value

The **absolute value** of a real number , denoted , is the unsigned portion of . Geometrically, is the distance between and zero on the real number line.

The absolute value function exists among other contexts as well, including complex numbers.

## Real numbers

When is real, is defined as For all real numbers and , we have the following properties:

- (Alternative definition)
- (Non-negativity)
- (Positive-definiteness)
- (Multiplicativeness)
- (Triangle Inequality)
- (Symmetry)

Note that

and

## Complex numbers

For complex numbers , the absolute value is defined as , where and are the real and imaginary parts of , respectively. It is equivalent to the distance between and the origin, and is usually called the complex modulus.

Note that , where is the complex conjugate of .

## Examples

- If , for some real number , then or .
- If , for some real numbers , , then or , and therefore or .

## Problems

- Find all real values of if .
- Find all real values of if .
- (AMC 12 2000) If , where , then find .