Negative numbers (integer arithmetic)
Negative numbers: order, add/subtract (and ×/÷ — DE / KS3) integers; absolute value.
Once numbers stretch past zero into the negatives, you can start doing arithmetic with them — adding, subtracting, and (a little later) multiplying and dividing whole numbers on both sides of zero.
What it means
The integers are the whole numbers together with their negatives and zero: ... −3, −2, −1, 0, 1, 2, 3 ..., written ℤ. Picture them as evenly spaced ticks on a line, with zero in the middle and a mirror running through it.
Ordering is just left-to-right position. Anything further left is smaller, so −5 < −1 < 0 < 2. This is the one idea that trips people up: −5 is less than −1, even though 5 is bigger than 1.
Adding means stepping right; subtracting means stepping left. Start on the first number and move. Adding a negative is the same as subtracting: 4 + (−6) moves 6 steps left from 4, landing on −2. Subtracting a negative reverses twice and so moves right: 3 − (−2) = 3 + 2 = 5. A short way to say it: two minus signs next to each other make a plus.
Multiplying and dividing follow a sign rule (introduced in Germany’s Klasse 5 and England’s KS3, held back to a later cycle in France):
- same signs → positive:
(−3) × (−4) = 12 - different signs → negative:
(−3) × 4 = −12
You multiply the sizes as usual, then attach the sign.
The absolute value of a number, written |n|, is its distance from zero, ignoring direction. So |−7| = 7 and |7| = 7. Distance is never negative, so an absolute value is always zero or positive.
Worked examples
Adding across zero, as jumps on the line:
−3 + 5: start at −3, jump 5 right → −2, −1, 0, 1, 2 → answer 2
2 − 5: start at 2, jump 5 left → 1, 0, −1, −2, −3 → answer −3Subtracting a negative (two minuses make a plus):
6 − (−4) = 6 + 4 = 10
−1 − (−3) = −1 + 3 = 2Sign rule for products:
| a | b | a × b |
|---|---|---|
| −3 | 4 | −12 |
| 3 | −4 | −12 |
| −3 | −4 | 12 |
Absolute value: |−9| = 9, and |4| − |−6| = 4 − 6 = −2.
The generative-art connection
The number line is the artwork here, and its defining feature is symmetry about zero. Plot a number n a fixed distance to the right and its opposite −n the same distance to the left, and the picture is a perfect mirror image — the origin is the axis of reflection. That is exactly what hue-pulse shows: a central point with a matched partner either side, so “positive and negative are mirror twins” becomes something you watch rather than memorise. The Math Learning Center Number Line turns every sum into a visible hop — a step right for adding, left for subtracting — so 4 + (−6) is literally an arrow that overshoots zero. Absolute value is the length of that arrow, drawn without its direction: colour a run of numbers by |n| and you get a symmetric V-shaped gradient, brightest at the edges and dark at zero.
Common misconceptions
- “Negatives with bigger digits are bigger.”
−8 < −2. Further left is always less; compare by position, not digit size. - “Minus a negative is still going down.”
5 − (−3)goes up to 8 — removing a debt is a gain. Two adjacent minus signs make a plus. - “Absolute value flips the sign.”
|n|does not just remove a minus; it returns distance from zero, so|5|is5, not−5. It only ever changes negatives.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
Put these five integers in order from smallest to largest: `3, −7, 0, −2, 6`.
Answer
`−7 < −2 < 0 < 3 < 6`. Further left on the number line is smaller, so the two negatives come first, with `−7` (further from zero) before `−2`; then `0`, then the positives.
Art hook Draw a horizontal number line from `−10` to `10` with a tick and dot at each of the five values. Colour each dot on a blue-to-red scale by its position (blue = far left/small, red = far right/large). The finished strip reads as a left-to-right cool-to-warm gradient, so 'ordering' becomes a colour sweep.
-
Work these out: `−4 + 9`, `3 − 8`, and `−6 + (−5)`.
Answer
`−4 + 9 = 5` (start at `−4`, hop 9 right). `3 − 8 = −5` (start at 3, hop 8 left, past zero). `−6 + (−5) = −11` (adding a negative is a left hop of 5 from `−6`).
Art hook Animate each sum as an arrow on a number line: green arrow rightward for adding, red arrow leftward for subtracting, starting at the first number. The arrow's tip lands on the answer. Chain the three sums so the arrows relay along one shared line.
-
Two minus signs sit next to each other — work these out: `7 − (−5)` and `−3 − (−8)`.
Answer
`7 − (−5) = 7 + 5 = 12`. `−3 − (−8) = −3 + 8 = 5`. Subtracting a negative reverses direction twice, so it becomes adding: a right hop.
Art hook Show a leftward 'subtract' arrow that, because the amount is negative, flips into a rightward arrow — an arrow that visibly u-turns. Draw the flip as a small arc, so 'two minuses make a plus' is a turnaround you can watch.
-
Use the sign rule to compute `(−6) × 4`, `(−7) × (−3)`, and `(−20) ÷ (−5)`.
Answer
`(−6) × 4 = −24` (different signs → negative). `(−7) × (−3) = 21` (same signs → positive). `(−20) ÷ (−5) = 4` (same signs → positive). Multiply/divide the sizes, then attach the sign.
Art hook Build a signed 'times-table square' grid: rows `−5…5`, columns `−5…5`, each cell coloured by the sign of the product (warm for positive, cool for negative). The result is a striking four-quadrant chequerboard, warm in the top-right and bottom-left, cool in the other two corners.
-
Evaluate `|−8|` and `|3| − |−10|`.
Answer
`|−8| = 8` (distance from zero). `|3| − |−10| = 3 − 10 = −7`. Take each absolute value first (distance, always `≥ 0`), then subtract.
Art hook Colour every integer from `−10` to `10` by its absolute value `|n|`: dark at zero, brightening outward both ways. You get a symmetric V-shaped brightness gradient, a perfect mirror image about the origin.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
A thermometer reads `−3 °C` in the morning. Is that warmer or colder than `−7 °C`? Which is the higher temperature?
Answer
`−3 °C` is warmer (higher). On the number line `−3` sits to the right of `−7`, and further right means greater, so `−3 > −7`.
Art hook Draw a vertical thermometer number line from `−10` to `10`. Fill it up to a value the reader picks and tint the bulb blue when the reading is below zero, red when above. Sliding the value animates the mercury crossing the zero mark.
-
Fill in `<`, `>`, or `=`: `−5 ▢ −9`, `0 ▢ −4`, `−6 ▢ 2`.
Answer
`−5 > −9` (right of `−9`), `0 > −4`, `−6 < 2`. Compare by position on the line, not by digit size.
Art hook Place two dots on a number line for each pair and draw a hungry-mouth `>`/`<` bracket that always opens toward the dot further right. Regenerating with random pairs makes a little comparison game.
-
Start at `2` on a number line. Hop 6 steps to the left. Where do you land? Write it as a subtraction.
Answer
You land on `−4`. As a subtraction: `2 − 6 = −4`.
Art hook Animate a dot hopping one tick at a time leftward from 2, dropping a fading footprint at each tick. When it crosses zero, the footprints switch colour, marking the moment the value turns negative.
-
Work these out: `−8 + 8`, `−2 + 5`, and `4 + (−9)`.
Answer
`−8 + 8 = 0` (a hop back to the start), `−2 + 5 = 3`, `4 + (−9) = −5` (adding `−9` is a 9-step left hop).
Art hook Show each sum as two arrows placed tip-to-tail from zero: a left arrow and a right arrow. Where the second arrow ends is the answer. When they exactly cancel (like `−8 + 8`) the arrows form a closed loop back at zero.
-
Compute `6 − (−2)` and `−4 − (−4)`. What is special about the second answer?
Answer
`6 − (−2) = 6 + 2 = 8`. `−4 − (−4) = −4 + 4 = 0`. The second is zero because subtracting a number's own value undoes it — every integer minus itself is `0`.
Art hook Pair each number with its opposite as two dots equidistant from zero and connect them with an arc that meets at the origin — a visual 'they cancel to 0'. Repeat for many pairs to draw a fan of arcs all pinned at zero.
-
Spot the mistake: a friend writes `−6 < −10 because 6 is smaller than 10`. Is the answer right? Explain.
Answer
The answer is wrong: `−6 > −10`. On the number line `−6` is to the right of `−10`, so it is the larger number. With negatives, the bigger the digit, the smaller (further left) the value.
Art hook Show both numbers as dots on a line with a faint mirror line through zero. Reflecting `6` and `10` across zero visibly swaps which is further right, illustrating why the digit-size rule flips for negatives.
-
Evaluate the absolute values `|−12|`, `|0|`, and `|7|`. Can an absolute value ever be negative?
Answer
`|−12| = 12`, `|0| = 0`, `|7| = 7`. No — absolute value is a distance from zero, and a distance is never negative.
Art hook For a chosen `n`, draw the segment from `0` to `n` and label its length `|n|` — same length whether `n` is left or right of zero. Two inputs that are opposites (like `−7` and `7`) draw mirror-image segments of equal length.
-
Use the sign rule: `5 × (−4)`, `(−9) × (−2)`, and `(−3) × (−3)`.
Answer
`5 × (−4) = −20` (different signs → negative). `(−9) × (−2) = 18` (same signs → positive). `(−3) × (−3) = 9` (same signs → positive).
Art hook Represent `(−3) × (−3)` as a 3×3 array of dots flagged 'negative × negative', tinted the positive colour to show the product came out positive. Toggling one factor's sign recolours the whole array to the negative tint.
-
Divide using the sign rule: `(−24) ÷ 6`, `18 ÷ (−3)`, and `(−15) ÷ (−5)`.
Answer
`(−24) ÷ 6 = −4` (different signs → negative). `18 ÷ (−3) = −6` (different signs → negative). `(−15) ÷ (−5) = 3` (same signs → positive).
Art hook Show division as sharing dots into equal groups on a signed grid; colour the quotient cell warm (positive) or cool (negative) by the sign rule, so a strip of divisions becomes a warm/cool bar code.
-
A diver is at `−15 m` (15 metres below the surface). She rises `9 m`, then descends `6 m`. What is her depth now? Write it as one integer calculation.
Answer
`−15 + 9 − 6 = −12`. She is at `−12 m`, i.e. 12 metres below the surface. Rising adds, descending subtracts.
Art hook Draw a vertical sea number line with the surface at 0. Animate the diver dot: up 9, down 6, leaving a trailing path. Tint the water darker with depth so the dot's colour tracks how far below zero it is.
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Reasoning challenge: I pick a negative integer `n`. Compare `n` with `|n|`. Which is always bigger, and why?
Answer
`|n|` is always bigger. If `n` is negative then `n < 0`, but `|n|` is its distance from zero, a positive number the same size — so `n` is negative while `|n|` is positive, giving `|n| > n` for every negative `n`.
Art hook For a draggable negative `n`, plot both `n` (left of zero) and `|n|` (its mirror, right of zero) and connect them with a symmetry arc across the origin. Dragging `n` shows the right-hand twin always sitting higher on the line.
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Open-ended: using the numbers `−4, −1, 3, 5`, put a `+` or a `−` in front of each (using each number once) to try to make the total exactly `0`. Can you do it with all four? Show what you find.
Answer
You cannot reach `0` using all four: the four sizes `4 + 1 + 3 + 5 = 13` is odd, and changing any sign only shifts the total by an even amount, so the result is always odd — never `0`. You CAN hit `0` if you are allowed to leave one out, e.g. `5 − 1 − 4 = 0`. Accept either: a valid explanation that all four can't make `0`, or any correct expression using three of them that evaluates to `0` (such as `5 − 1 − 4 = 0`).
Art hook Make draggable number cards on a line; as the child chains `+`/`−`, a running arrow walks left and right, and the line flashes when the arrow lands exactly on zero — a 'hit zero' target game.