Negative numbers
We work on the number line: adding a positive number moves to the right, subtracting moves to the left. Taking away a negative number is like adding: 5 − (−3) = 8. In multiplication and division, the same signs give plus and different signs give minus.
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Plus and minus on the number line
… −5 −4 −3 −2 −1 0 1 2 3 4 5 …
Negative numbers are to the left of zero. The further to the left, the smaller the number.
−8 < −3
We start at the first number. Adding a positive number moves to the right, subtracting moves to the left.
- −3 + 5: five steps to the right from −3 gives 2.
- −3 − 4: four steps to the left from −3 gives −7.
- 2 − 9: two steps to zero, then seven more steps gives −7.
Adding a negative number is like taking on a debt.
6 + (−4) = 6 − 4 = 2
Why minus minus makes plus
We want to understand why 5 − (−2) is 7.
| Calculation | Answer |
|---|---|
| 5 − 2 | 3 |
| 5 − 1 | 4 |
| 5 − 0 | 5 |
| 5 − (−1) | 6 |
| 5 − (−2) | 7 |
Each row takes away a smaller number, so the answer gets bigger. Subtracting −2 gives the same answer as adding 2.
From 6 degrees down to −4 degrees is ten degrees:
6 − (−4) = 6 + 4 = 10
Multiplication and division
(−3) · 4 is four debts of 3 kronor.
(−3) · 4 = (−3) + (−3) + (−3) + (−3) = −12
When the second number goes down by one, the answer goes up by three: (−3) · 2 = −6, (−3) · 1 = −3, (−3) · 0 = 0. The next step is (−3) · (−1) = 3: minus times minus makes plus.
| Signs | Answer | Example |
|---|---|---|
| plus times plus | plus | 3 · 4 = 12 |
| plus times minus | minus | 3 · (−4) = −12 |
| minus times plus | minus | (−3) · 4 = −12 |
| minus times minus | plus | (−3) · (−4) = 12 |
Division follows the same rules: (−12) ÷ (−4) = 3, because 3 · (−4) = −12.
Example
Example 1: −7 − (−3)
We want to take −3 away from −7.
Subtracting a negative number is like adding the positive number.
−7 − (−3) = −7 + 3
From −7 we go three steps to the right: −6, −5, −4.
−7 + 3 = −4
−4 + (−3) = −7. We got back to −7, so the answer is correct.
Example 2: (−2) · (−3) · (−4)
We want to multiply three negative numbers, two at a time.
Minus times minus makes plus.
(−2) · (−3) = 6
Plus times minus makes minus.
6 · (−4) = −24
Three minus signs: two cancel each other out and one is left. 2 · 3 · 4 = 24, so −24 is correct.
Pitfalls
The right digits but the wrong sign: 3 − 8 = 5.
From 3 we go eight steps to the left: three to zero and five more. The answer is −5. Check: −5 + 8 = 3.
Using "minus minus makes plus" on −4 − 3 and getting −1.
The rule is for two minus signs in a row, as in 5 − (−3). In −4 − 3 we go three steps to the left: −7.
Thinking that −5 + (−2) is 7.
The sign rule is for multiplication and division. A debt of 5 plus a debt of 2 is a debt of 7: −5 + (−2) = −7.
Practise on your own
0 of 4 correct1Work out −4 + 9.
2Work out 3 − 10.
3Work out −2 − (−8).
4Work out (−4) · (−6) ÷ (−3).
More walkthroughs with Fox
Common questions
Why does minus times minus make plus?
The pattern shows it: (−3) · 1 = −3 and (−3) · 0 = 0. The answer goes up by three with each step, so (−3) · (−1) = 3.
Is −10 bigger or smaller than −2?
−10 is smaller. It is further to the left on the number line, and −10 degrees is colder than −2 degrees.
Why do you put brackets around negative numbers?
The brackets separate the operation sign from the sign of the number. 5 − (−3) shows that we are taking away the number −3.
What does plus minus make?
Adding a negative number is like subtracting: 5 + (−3) = 5 − 3 = 2.
Read also
- Years 4–9Order of operations (BIDMAS)What is the order of operations?
- Years 7–9 and upper secondaryPowers and the laws of exponentsHow do you calculate with powers?
- Years 6–9Solving equations step by stepHow do you solve an equation?
- Years 7–9 and upper secondarySimplifying expressionsHow do you simplify an expression?
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