Absolute value

Notation

Vertical bars mean absolute value:

$$|5| = 5 \qquad |-5| = 5 \qquad |0| = 0$$

Read $|x|$ as “the absolute value of $x$”, or “the distance from $x$ to zero”.

Why it is never negative

Distance measures how far apart two points are, and there is no such thing as a negative distance. This single fact explains two things that otherwise look arbitrary:

• $|x| = 5$ has two answers, because two points sit five units from zero.
• $|x| = -3$ has no answer, because no point sits a negative distance away.

A common misreading

$|-x|$ is not always $x$. If $x$ is negative then $-x$ is positive, and the bars leave it alone. Substituting a number settles it faster than reasoning about symbols: with $x = -4$, $|-x| = |4| = 4$.

Distance between two numbers

$|a - b|$ gives the distance between $a$ and $b$, in either order. That is why tolerance problems are written this way — $|L - 50| \le 0.4$ says ”$L$ is within $0.4$ of $50$”. See absolute value equations.