---
title: "Why Does a Negative Times a Negative Make a Positive?"
url: https://www.learningloftinstitute.com/answers/why-does-a-negative-times-a-negative-make-a-positive
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Answers › Why Does a Negative Times a Negative Make a Positive?"
---
# Why Does a Negative Times a Negative Make a Positive?
> **Answer.** A negative multiplied by a negative gives a positive because arithmetic has to stay consistent with itself. Continue the pattern 3 × -2 = -6, 2 × -2 = -4, 1 × -2 = -2, 0 × -2 = 0 and the next line has to be -1 × -2 = 2. The distributive law forces the same conclusion independently.
## Continue the pattern past zero
Write out one column of a times table and let it run below the line where it usually stops. On the left, each number is one less than the one above it. On the right, watch what the answers do.
The answers climb by 2 every time: -6, -4, -2, 0. Nothing changes at zero - the left column keeps dropping by one, so the right column keeps rising by two - and the next three lines are forced. This is why the rule is not an extra piece of arithmetic bolted on for negative numbers. It is what the arithmetic already in front of you does when you keep going.
- 3 × -2 = -6
- 2 × -2 = -4
- 1 × -2 = -2
- 0 × -2 = 0
- -1 × -2 = 2
- -2 × -2 = 4
- -3 × -2 = 6
## The argument that proves it rather than suggests it
A pattern is evidence, not proof, and a sceptical student is entitled to ask who says it carries on. So here is the same conclusion reached without any pattern at all, using two facts nobody disputes: anything multiplied by zero is zero, and multiplication distributes over addition.
Start with -2 × (3 + -3). The bracket is zero, so the whole thing is zero. Now expand the bracket instead, which is allowed because that is what distributing means, and you get (-2 × 3) + (-2 × -3), which must therefore also be zero. The first part is -6. So -6 plus the unknown product equals zero, and the only number that does that is 6.
That is the whole proof, and it is worth noticing what it rules out. If -2 × -3 were -6, the line would read -6 + -6 = 0, which says -12 = 0. Keeping the negative answer costs you the distributive law, and every technique in algebra - expanding brackets, factorising, collecting terms - is the distributive law wearing different clothes.
- -2 × (3 + -3) = -2 × 0 = 0
- -2 × (3 + -3) = (-2 × 3) + (-2 × -3)
- so -6 + (-2 × -3) = 0
- so -2 × -3 = 6
## Why two negatives make a positive is a dangerous thing to remember
The slogan is a summary of the rule with the conditions removed, and the conditions were the useful part. Applied where it does not belong it produces confident wrong answers, which are harder to fix than uncertain ones.
It is false for addition: -3 + -4 = -7, and two negatives there have made a bigger negative. It is false for subtraction of a positive: -3 - 4 = -7 as well. Students who have learned only the slogan reach for it in both cases, because both cases contain two minus signs.
The rule properly stated has a scope. Two negatives multiplied give a positive; two negatives divided give a positive; and subtracting a negative is the same as adding, so 5 - (-3) = 8. Outside those three situations the slogan should be left alone.
- -2 × -3 = 6 and -12 ÷ -3 = 4 - the rule applies
- 5 - (-3) = 5 + 3 = 8 - the rule applies
- -3 + -4 = -7 - the rule does not apply
- -3 - 4 = -7 - the rule does not apply
## The version to use under pressure
For a chain of multiplications, count the negative factors rather than tracking signs pair by pair. An even number of negatives gives a positive result, an odd number gives a negative one. So -2 × -3 × -4 = -24, because there are three of them, and it takes a second to check rather than three separate sign decisions.
The same count explains powers. (-2)³ = -8 and (-2)⁴ = 16: odd power negative, even power positive. Which brings up the error that costs more marks than any other in this topic - the difference between -3² and (-3)². The first is -9, because the power applies to the 3 and the minus sign is applied afterwards. The second is 9, because the bracket puts the minus inside.
On a calculator these are two different keys and two different meanings. The subtract key and the negative key look similar and do not do the same job, and a calculator entering -3² will return -9 whatever you intended. Where a negative number is being squared, put the bracket in.
## Questions
### Does a negative divided by a negative also give a positive?
Yes, for the same reason. Division is multiplication by the reciprocal, so -12 ÷ -3 asks what number multiplied by -3 gives -12, and the answer is 4. Any sign rule that holds for multiplication holds for division unchanged, because the two operations are the same operation wearing different notation.
### Why is -3² equal to -9 but (-3)² equal to 9?
Order of operations. In -3² the power is applied first, giving 9, and then the negative is applied, giving -9. In (-3)² the bracket makes -3 a single quantity, and -3 × -3 = 9. If you mean to square a negative number, the bracket is not optional.
### Do two negatives make a positive when you add them?
No. -3 + -4 = -7. Adding a negative moves you further down the number line, so two negatives added give a larger negative. The rule about two negatives belongs to multiplication, to division, and to subtracting a negative - nowhere else.
### Is -2 × -3 = 6 just a convention someone decided on?
No, it is forced. You could invent an operation that returns -6 instead, but it would not distribute over addition, so it would not be multiplication and none of the algebra built on multiplication would work with it. The rule is a consequence of wanting arithmetic to keep behaving, not a choice made for tidiness.
### How do you work out the sign of something like -2 × 3 × -4 × -1?
Count the negative factors: there are three, which is odd, so the answer is negative. Then multiply the sizes: 2 × 3 × 4 × 1 = 24. The answer is -24. Counting first is quicker and less error-prone than carrying the sign through each step.
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