The number line gives you two ideas for free: −3 and 3 sit the same distance from 0 — they are opposites — and that shared distance, 3, is each one's absolute value.
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.
5.2.1 Opposite numbers
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.
A number a and its opposite −a are reflections across 0: equal arrows, opposite directions. Here a = 4, so −a = −4.
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.
Key idea
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.
🎮 Try itThe mirror across zero
Step the teal point a left and right. Watch its opposite −a reflect across 0 in real time — same distance out, other side.
a =3
5.2.2 Simplifying stacked signs
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.
Every − out front is one flip across 0. Two flips bring you home: −(−3) = 3. Three flips leave you flipped: −(−(−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
Worked example — counting the flips
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.
Watch out
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.
🎮 Try itStack and remove minus signs
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.
Starting number5
Leading − signs2
5.2.3 What absolute value means
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.
|−3| is the length of the amber walk from −3 back to 0: 3 units. Forget the direction; keep the distance.
Read it aloud, every time
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.
🎮 Try itWalk it back to zero
Set a anywhere from −9 to 9. The amber arrow shows the walk back to 0, and |a| is exactly its length.
a =-3
5.2.4 Properties of absolute value
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.
A symmetric pair: |−4| and |4| are equal-length arrows pointing back to 0 from opposite sides. Opposites are always equally far out.
The properties in one line
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.
5.2.5 Using absolute value for distance
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.
Worked example — from −2 to 3
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.
The amber bracket spans from −2 to 3. Its length, |−2 − 3| = 5, is the distance between the two points.
🎮 Try itMeasure the gap
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|.
a =-2
b =3
★ The big ideas, in one breath
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|.
Coming up next — 5.3
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.
✎ Exercises 5.2
Work each one out first, then open the answer to check your thinking.
Write the opposite of each number: 6, −9, 0.
Show answer
The opposite of 6 is −6; the opposite of −9 is 9; the opposite of 0 is 0 (zero is its own opposite). Each one is the same distance from zero, on the other side.
Simplify −(−7).
Show answer
+7. One leading minus flips −7 across zero to its opposite. Two minus signs in all — even — so the result is positive.
Simplify −(−(−8)).
Show answer
−8. Count the leading minus signs: three, which is odd, so the value stays negative. Three flips leave you flipped.
Evaluate |−6|, |10|, |0|.
Show answer
|−6| = 6, |10| = 10, |0| = 0. Each is the distance from 0: six steps, ten steps, and zero steps.
True or false: an absolute value can be negative. Explain.
Show answer
False. An absolute value is a distance from zero, and a distance is never negative — the smallest it can ever be is 0 (which happens only for 0 itself). So |a| ≥ 0 always.
Without computing each side fully, explain why |−12| and |12| must be equal.
Show answer
−12 and 12 are opposites — the same distance from 0 on opposite sides. Absolute value measures exactly that distance and ignores direction, so |−12| = |12| = 12. In general |a| = |−a|.
Find the distance between 2 and 9 on the number line.
Show answer
|2 − 9| = |−7| = 7. (Or |9 − 2| = |7| = 7 — same gap either way.)
Find the distance between −7 and 4.
Show answer
|−7 − 4| = |−11| = 11. On the line: 7 steps from −7 up to 0, then 4 more up to 4, for 11 in all.
Find the distance between −8 and −2 (both negative).
Show answer
|−8 − (−2)| = |−8 + 2| = |−6| = 6. The two points are both on the left of zero, six steps apart.
Context: At dawn the temperature was −5°, and by afternoon it had risen to 8°. How many degrees apart are those two temperatures?
Show answer
|−5 − 8| = |−13| = 13 degrees apart. (Counting on a thermometer: 5 degrees up to 0°, then 8 more up to 8°, totals 13.)
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.
eastmath.com · Stage 5 · 5.2 Opposites & Absolute Value · Intuition before notation
eastmath.com · 5.2 Opposites and Absolute Value · 5.2.2 Simplifying stacked signs