Absolute value

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

• If $x$ is a positive number, then $|x|=x$.
• If $x$ is a negative number, then $|x|=-x$.
• If $x=0$ then $|x|=0$.

For example, $|-3|=3$, and $|5|=5$. Notice that $|x|$ is never negative.

Absolute value has several useful properties. One is the multiplicative property. If $x$ and $y$ are two numbers, then $|xy|=|x|\times |y|$. Another is the triangle inequality, which is the fact that $|x+y|\leq |x|+|y|$. For example, if $x=3$ and $y=-5$, then $|x+y|=|3+(-5)|=|3-5|=|-2|=2$, while $|x|+|y|=|-5|+|3|=5+3=8$. In this case, the triangle inequality is the fact that 2 is not more than 8.

If $x$ is a real number, then ${\sqrt {x^{2}}}=|x|$. 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 $z=x+iy$ is a complex number with real part $x$ and imaginary part $y$, then $|z|={\sqrt {x^{2}+y^{2}}}$. If we represent $z$ as a point in the complex plane with coordinates $(x,y)$, then $|z|$ 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.