Absolute Value: Distance from Zero

Distance from zero

The absolute value of a number is how far it is from zero, written with two vertical bars.

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

Both are five steps from zero. The direction differs and the distance does not, and absolute value records only the distance.

Why it is never negative

A distance has no direction built into it. Walking five steps left and five steps right are both walks of five steps.

$$|{-12}| = 12 \qquad |0| = 0 \qquad |9| = 9$$

So an absolute value is always positive or zero. An answer of $-4$ for an absolute value is always a mistake, in the same way a length of $-4$ metres would be.

Zero is the only number whose absolute value is zero, because it is the only number sitting no distance from zero.

The two bars are grouping symbols

Work out what is inside the bars first, then take the absolute value:

$$|3 - 8| = |-5| = 5$$

Doing it the other way round — taking absolute values first — would give $|3| - |8| = -5$, which is a different answer and the wrong one.

The bars behave like brackets in the order of operations: they group what is inside them.

Absolute value is not the same as order

This is the distinction the standards single out, and it is worth being exact about.

$$-7 < -3 \qquad \text{but} \qquad |-7| > |-3|$$

Both statements are true, and they answer different questions:

Question	Comparison	Answer
Which is greater?	$-7$ against $-3$	$-3$
Which is further from zero?	$	-7

A temperature of $-7$ is lower than $-3$, and it is also further from freezing. Neither statement contradicts the other.

The same split appears with money. A debt of $200$ is a smaller balance than a debt of $50$, and a larger debt:

$$-200 < -50 \qquad \text{and} \qquad |-200| > |-50|$$

Order asks which side of the other a number sits on. Absolute value asks how far it sits from zero. Most real questions about debts, depths and temperatures are really absolute-value questions dressed up in the language of size.

Distance between two numbers

Subtract them and take the absolute value:

$$\text{distance} = |a - b|$$

The distance between $-3$ and $5$:

$$|-3 - 5| = |-8| = 8$$

Subtracting the other way gives the same answer:

$$|5 - (-3)| = |8| = 8$$

That is the point of the absolute value here — distance does not depend on which end you start from, and the bars remove the sign that the order of subtraction introduces.

Worked examples

Common mistakes

Practice problems