Absolute value

Absolute value is a function measuring a number's distance from zero. The absolute value of $\text{[math]}$ is written $\text{[math]}$ .

If $\text{[math]}$ is a positive number, then $\text{[math]}$ .
If $\text{[math]}$ is a negative number, then $\text{[math]}$ .
If $\text{[math]}$ then $\text{[math]}$ .

For example, $\text{[math]}$ , and $\text{[math]}$ . Notice that $\text{[math]}$ is never negative.

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

If $\text{[math]}$ is a real number, then $\text{[math]}$ . 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 $\text{[math]}$ is a complex number with real part $\text{[math]}$ and imaginary part $\text{[math]}$ , then $\text{[math]}$ . If we represent $\text{[math]}$ as a point in the complex plane with coordinates $\text{[math]}$ , then $\text{[math]}$ 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.

Absolute value

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