Two tools built straight from the number line: flipping sides, and measuring distance.
Point 5 of 5 in this lesson: 5.2.5 Using absolute value for distance
In Lesson 5.1 the number line grew a left half, and every number found a home on it. Look closely at that line and it quietly hands you two brand-new tools, no extra machinery required. The first is the opposite: stand on any number, walk straight across 0 to the mirror spot on the other side, and you have landed on its opposite. The second is absolute value: forget which way you walked, and just ask how far you are from 0. One tool is about direction; the other throws direction away and keeps only distance.
By the end you will be able to write the opposite of any number as −a, untangle a stack of minus signs like −(−(−5)), read |a| aloud as "the distance from zero," and measure the gap between two points as |a − b|. We keep one steady color habit: positive numbers are teal, negative numbers are red, zero is slate, and any distance or absolute value is amber.
Pick a number — say 3. It lives three steps to the right of 0. Now walk straight across zero to the matching spot three steps to the left: you land on −3. These two numbers are the same distance from zero but on opposite sides, and that makes them opposites. So 3 and −3 are opposites; so are −7 and 7.
There is a tidy way to write "the opposite of." The opposite of a number a is written −a. That little dash out front does not mean "negative" so much as it means "flip me to the other side of zero." If a is already on the left, flipping sends it back to the right — which is why −a can actually be a positive number when a itself is negative.
What about 0 itself? Zero sits exactly on the fold. Flipping it across zero leaves it right where it was, so the opposite of 0 is 0 — zero is its own opposite, the one number that does not move.
The opposite of a is its mirror image across 0, written −a. Opposites are the same distance from zero on opposite sides. The only number that is its own opposite is 0.
Step the teal point a left and right. Watch its opposite −a reflect across 0 in real time — same distance out, other side.
Because each leading − means "flip to the other side," you can stack them — and stacking flips is easy to follow. Start at −3. Put one more minus in front, −(−3), and you flip −3 across zero to its opposite, +3. Flip twice and you are right back home. So −(−3) = +3.
So you never have to think hard — just count the minus signs:
| Number of leading − signs | Result |
|---|---|
| even (0, 2, 4, …) | stays positive |
| odd (1, 3, 5, …) | ends up negative |
Simplify −(−(−5)).
Count the minus signs in front: there are three of them. Three is odd, so the result stays negative: the answer is −5.
Now −(−(−(−5))): four minus signs, which is even, so it collapses to +5.
Only the minus signs out front count as flips here. The sign that belongs to the number itself — the − inside −5 — is part of the starting value. In −(−5) there is one flip applied to the starting number −5, giving +5.
Add or remove leading − signs in front of a number, and watch the whole expression collapse to a single signed value. Keep an eye on the flip count.
Here is the second free tool. Sometimes you do not care which side of zero a number is on — you only care how far out it sits. A thermometer reading of −3° and a reading of 3° are equally extreme, just in opposite directions; a debt of $8 and a credit of $8 are the same size. The tool that measures size while ignoring direction is the absolute value.
The absolute value of a, written |a| with a straight bar on each side, is its distance from 0 on the number line. Distance is never about direction — it is just how far you walked. So |−3| = 3 (three steps to get home from the left) and |3| = 3 (three steps to get home from the right). And |0| = 0, because you are already home — zero steps.
Never read |a| as "make it positive." Read it as "the distance from 0 to a." Distance is what is left over when you forget which way you went, and distance is the reason every answer comes out at zero or above.
Set a anywhere from −9 to 9. The amber arrow shows the walk back to 0, and |a| is exactly its length.
Once you hold on to the idea that |a| is a distance, three facts about it become almost obvious — because they are just facts about distances.
(1) |a| ≥ 0 always. A distance is never negative. You can be zero steps from home, or some steps from home, but you can never be a "negative number of steps" away. So an absolute value is always zero or positive — never below zero.
(2) |a| = |−a|. A number and its opposite are the same distance from 0 — that is what "opposite" meant in Section 5.2.1. So measuring either one gives the same answer: |−4| = 4 = |4|.
(3) |a| = 0 only when a = 0. The only point at distance zero from 0 is 0 itself. Every other number is at least one step out, so its absolute value is strictly more than zero.
Because |a| is a distance: it is never negative (|a| ≥ 0), it ignores the sign (|a| = |−a|), and it is zero only at 0. Re-use the "Walk it back to zero" tool above and step a past its opposite — the arrow length never changes.
Absolute value measures distance from 0. But the same idea measures the distance between any two points. The gap between two numbers a and b on the line is
distance from a to b = |a − b|
Subtracting lines the two points up against each other; the bars throw away the direction so the gap comes out as a clean, non-negative length. And because distance does not care which point you start from, |a − b| = |b − a| — measuring left-to-right or right-to-left gives the same number.
How far apart are −2 and 3?
|−2 − 3| = |−5| = 5.
Check it the other way: |3 − (−2)| = |3 + 2| = |5| = 5. Same gap, as it must be. On the line you can count it: 2 steps from −2 up to 0, then 3 more up to 3, for 5 in all.
Set the two points a and b. The bracket under the axis is the distance between them, and the readout shows |a − b| = |b − a|.
The number line hands you two tools. The opposite of a, written −a, is its mirror image across 0 — flip once to cross over, and stacked flips just count: even back to positive, odd ends negative, with 0 its own opposite. The absolute value |a| is distance from 0, so it is never negative, ignores the sign (|a| = |−a|), and equals 0 only at 0. Stretch that same distance idea between two points and you can measure any gap as |a − b| = |b − a|.
Opposites and absolute value are the groundwork. Next you will meet the whole rational number family — integers, fractions, and decimals together — and use the number line to compare any two of them, deciding which is larger even when both are negative.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson serves the U.S. Common Core grade-6 number-system standards and previews grade 7. It develops opposites and the fact that −(−a) = a (6.NS.C.6a), introduces absolute value as a number's distance from 0 on the number line (6.NS.C.7c), interprets and orders absolute value in real-world contexts such as temperature and debt (6.NS.C.7d), and uses |a − b| as the distance between two points — the seed of adding and subtracting signed numbers in 7.NS.A.1. The single most common misconception is treating |x| as a mechanical "make-it-positive button" rather than a distance; that mistake quietly powers the later error |a − b| = |a| − |b|. The antidote, used throughout this lesson and the quiz, is simple and relentless: always read |x| aloud as "the distance from 0," and draw the walk back to zero whenever there is doubt.