Difference between revisions of "Absolute value"

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.