Negative Numbers

Pick two numbers and an operation, then watch the moves happen on the line. Nothing here is a rule you have to take on trust.

Worked example — Negative Numbers

Flips about zero	0
Distance between them	8
Order, then negated	5 > -3 → -5 < 3
is the same move as	5 + 3

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Negative Numbers335 words

A negative number is the additive inverse of a positive one: −a is the unique number that adds to a to give 0. Everything else about negatives follows from that one sentence together with the ordinary rules of arithmetic, and this section shows the following.

What each symbol means

a: any number at all, positive or negative. The definition does not care which, which is why −(−3) = 3 needs no separate rule.
−a: the additive inverse of a — not "a with a minus painted on it", but the answer to "what do I add to a to get nothing?"
0: the additive identity, and the pivot of the whole argument. It is the only number that leaves every other unchanged under addition.

Where the formula comes from

Assume the distributive law, a(b + c) = ab + ac, holds for negatives as well as positives. This is the whole assumption, and it is a choice: we are insisting that extending the number line must not break the rules the positives already obey.
Apply it to (−1)(1 + (−1)). The bracket is 0, and anything times 0 is 0, so the whole expression is 0.
Now expand the same expression the other way: (−1)(1) + (−1)(−1), which is −1 + (−1)(−1).
Both readings must give the same answer, so −1 + (−1)(−1) = 0. There is exactly one number that adds to −1 to give 0, and it is +1. Nobody decided this; it is the only value that leaves the distributive law standing.

Assumes: That arithmetic on negatives should obey the same distributive law as arithmetic on positives. Give that up and you may define (−1)(−1) = −1 freely — but then (−3)((−3) + 1) is (−3)(−2) = −6 while (−3)(−3) + (−3)(1) is −9 + (−3) = −12, and the two ways of reading one expression no longer agree. That is the price, and it is why the sign rule is not a convention.

Subtraction was never a second operation 🖖

Press 5 − (−3). The tool rewrites it as 5 + (3) before it moves anything, because that is what subtraction is: a − b means a + (−b), for every a and every b. Nothing about negatives makes it a special case. What changes when b is negative is only which way the arrow points, and the arrow is the same arrow. So "two minuses make a plus" is what you get when you write the subtraction out and notice that the opposite of −3 is 3. Set the second number to anything, positive or negative, and the restated line above the number line never stops being true.

Multiplying by a negative flips the line 🖖

Press (−3) × (−4) and watch three separate moves: a stretch out to 12, a flip about zero to −12, and a second flip back to 12. Each negative factor contributes exactly one flip, which the readout counts for you. Two flips return every point on the line to where it began, and that is why the answer is positive. It also says what a negative number is: a direction. Multiplying by −1 does not shrink anything; it turns the line around. Now press ÷ with the same two numbers. The flip count does not move, because dividing by −4 is multiplying by −¼, and −¼ carries the same single negative. The sign rule taught twice is one rule.

Negation reverses order, and that is the whole story of the flipping inequality 🖖

Read the order line: −5 < −3, and negate both to get 5 > 3. The relation turned around. It has to, because negation is the flip from the block above, and flipping the line swaps left and right. Every pair of different numbers does it — try any two you like. This is the fact that makes an inequality change direction when both sides are multiplied by a negative, which most people meet as an unexplained exception years later. It is the same reflection, applied to a comparison instead of to a point.

Problems solved in full

The temperature rise of a freezer warming from −18 °C to −4 °C 7 steps
A freezer is at −18 °C. It warms to −4 °C. By how much did the temperature rise, and what are the two ways of getting it wrong?
1. Most people get 14 and are right, so the interesting question is what the wrong routes look like. Write the change as an ordinary subtraction: the new value minus the old one.
  
  $(- 4) - (- 18)$
2. Subtracting −18 means adding its opposite, so the answer is −4 + 18.
  
  $(- 4) + 18 = 14$
3. Fourteen degrees. The common wrong answer is 22, from adding the magnitudes as though both were distances from nothing: 18 + 4. The line makes that hard to do, because you can see that −18 and −4 are on the same side of zero.
  
  $18 + 4 = 22$
4. The other common wrong answer is −22, from treating both signs as instructions to go left. That is the sign of the number confused with the operation, and it is worth separating them explicitly.
  
  $- (18 + 4) = - 22$
5. Now the same question in the other direction. Cooling from −4 back to −18 is a change of −14, and the distance is still 14. Distance and change are different quantities, and only one of them carries a sign.
  
  $(- 18) - (- 4) = - 14, ∣ - 18 - (- 4) ∣ = 14$
6. Which matters the moment the numbers are multiplied. Doubling a temperature change of −14 gives −28, one flip, and the reading falls further. Halving it gives −7, and that is still one flip: the negative sits in the change, not in the multiplier.
  
  $2 \times (- 14) = - 28$
7. And here the tool stops being enough. −18 °C doubled is not −36 °C in any physical sense, because the Celsius zero is a convention rather than an absence of heat. Multiplying a temperature by a number is meaningless; multiplying a temperature change by a number is fine. The arithmetic cannot tell you which of the two you are holding.
  
  $2 \times (- 18^{\circ} C) \neq = - 36^{\circ} C$
Answer

A rise of 14 °C. Written out, −4 − (−18) = −4 + 18 = 14, and every step is the same move the tool draws. The two usual wrong answers, 22 and −22, both come from confusing the sign of a number with the direction of an operation.
From −3 to 5, and the two numbers that are 8 away 6 steps
A diver is 3 m below the surface, a gull 5 m above it. Work out the distance between them, then decide whether that distance alone would let you put the gull back.
1. The instinct to get rid of first is subtracting the sizes. Five and three, so two.
  
  $∣5∣ - ∣ - 3∣ = 5 - 3 = 2$
2. The subtraction itself. The change is −8, which is the result card, and the distance is its size, 8.
  
  $(- 3) - 5 = (- 3) + (- 5) = - 8, (- 3) - 5 = 8$
3. Eight because the walk crosses zero, so the two depths add instead of cancelling. Whenever the numbers sit on opposite sides of zero the distance is the sum of their sizes.
  
  $x < 0 < y ⟹ ∣ x - y ∣ = ∣ x ∣ + ∣ y ∣ = 3 + 5 = 8$
4. On the same side they subtract. A freezer warming from −18 to −4 moves 14, which is 18 − 4. One rule with two faces, and what decides which face you get is whether zero lies between the numbers.
  
  $x, y < 0 ⟹ ∣ x - y ∣ = ∣ x ∣ - ∣ y ∣ : - 18 - (- 4) = 18 - 4 = 14$
5. Now read the distance card backwards. It says 8, and 8 is all it says. Two numbers sit 8 away from −3, and the equation that finds them has two solutions because taking the size threw a sign away.
  
  $x - (- 3) = 8 ⟹ x + 3 = \pm 8 ⟹ x = 5, - 11$
6. The two answers always straddle the point they were measured from: their average is exactly it. Every pair of numbers has one distance, and every distance has two numbers behind it.
  
  $\frac{5 + ( - 11 )}{2} = - 3$
Answer

8 m. The diver and the gull are on opposite sides of zero, so their depths add rather than cancel. Read the same 8 backwards and it does not name the gull: −11 sits exactly as far from −3 as 5 does, and the distance card cannot tell them apart, because the sign went out of it when the size was taken.

$∣ x - a ∣ = d ⟹ x = a \pm d (d > 0)$

Learning path

Solving for x, from the minus sign upward

Leads to Linear equations

References (1)

A Fields Medallist on why the rule that looks like a convention is not one, and why it took so long to accept: D. Mumford, "What's so Baffling About Negative Numbers? — a Cross-Cultural Comparison." In Studies in the History of Indian Mathematics, Hindustan Book Agency, 2010, pp. 113–143.

By the 42Math team · Updated 2026-08-25

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

5 − (−3) - Subtracting −3 moves 3 to the right, ending at 8. That is not a special rule: subtracting anything means adding its opposite, and the opposite of −3 is 3.
(−3) × (−4) - Two negatives, two flips. Stretch to 12, flip to −12, flip back to 12. Flipping twice puts everything where it started, which is the whole of "two minuses make a plus".
Crossing zero - From −3 to 5 the distance is 8, not 2. Subtracting the magnitudes is the single most common mistake with signed numbers, and the line makes it hard to make.
Order reverses - −5 is less than −3, but 5 is greater than 3. Negation reverses order, which is exactly why an inequality flips when you multiply both sides by a negative.