Both positive and negative numbers can be multiplied and divided using two rules. [1] Two signs the same gives a positive answer. [2] Two different signs give a negative answer.
Contents
Core Rules
- Same signs make a positive: A negative number multiplied or divided by a negative number results in a positive number ((-2 \times -3 = +6)).
- Different signs make a negative: A negative number multiplied or divided by a positive number (or vice versa) results in a negative number ((-2 \times 3 = -6)).
Key Points
- Multiplication
- Multiplication is commutative. This means that the order in which you multiply a pair of numbers does not make a difference, e.g., 3 x 5 = 5 x 3
- Factors: The numbers being multiplied together.
- The product is the answer when two or more factors are multiplied together.
- Division
- Division is NOT commutative. This means that the order in which you divide a pair of numbers does make a difference, e.g., 5 ÷ 3 ≠ 3 ÷ 5.
- Dividend: The total amount or number you start with (the number being divided).
- Divisor: The number you are dividing by (how many groups you are making, or the size of each group).
- The quotient is the answer or result of the division of the dividend by the divisor.
- Remainder (if applicable): Any amount left over that cannot be divided equally into whole numbers.
- Calculations can be written with brackets around negative numbers because this can make a calculation easier to read, e.g., -3 × -1 is the same calculation as (-3) × (-1).
- Learning about positive and negative numbers will help when multiplying and dividing negative numbers.
Multiplying Positive and Negative Numbers
When multiplying negative numbers it is often useful to complete the calculation using positive numbers initially. Remember that:
- Multiplying two numbers together with the same sign gives a positive answer.
- Multiplying two numbers together with different signs gives a negative answer. Multiplying two numbers with different signs gives a negative answer because multiplication is repeated addition of a negative amount or repeated subtraction of a positive amount.
Two numbers with the same sign. (Also see the Commutative Property of Multiplication.)
3 × 5 = 5 × 3 = 15
(-4) × (-6) = (-6) × (-4) = 24
Two numbers with different signs. (Also see the Commutative Property of Multiplication.)
3 × (-5) = (-5) × 3 = -15
(-4) × 6 = 6 × (-4) = -24
Dividing Positive and Negative Numbers
When dividing negative numbers it is often useful to complete the calculation using positive numbers initially. Remember that:
- Dividing two numbers with the same sign gives a positive answer.
- Dividing two numbers with different signs gives a negative answer. Dividing two numbers with different signs gives a negative answer because division is the exact opposite of multiplication.
Two numbers with the same sign.
8 ÷ 4 = 2
(-9) ÷ (-3) = 3
Two numbers with different signs.
8 ÷ (-4) = -2
(-9) ÷ (3) = -3
Notes
Negative Numbers (Google Slides)
Please wait for the Google Slides to load.
Download the PDF version of the slides.
Negative Numbers Using a Number Line
Multiplication
Multiplication on a number line represents repeated addition, visualized by skip counting. Multiplying positive numbers involves moving right from zero, while positive and negative combinations move left. Multiplying two negative numbers results in a positive value, mirroring the logic of positive multiplication. Furthermore, number lines effectively illustrate fraction multiplication by subdividing intervals into equal segments. These methods provide a clear, geometric way to solve diverse multiplication problems across various number types.
See Multiplication on a Number Line.
Division
Division on a number line represents division as repeated subtraction. For positive integers, the divisor is subtracted from the dividend until zero is reached, with the number of steps indicating the quotient. Negative division follows similar rules regarding signs, while non-divisible numbers result in remainders. Fractions are solved by reversing the divisor and treating the operation as multiplication, using a partitioned number line to visualize the final result.
See Division on a Number Line.
Multiplying Two Numbers With Different Signs
The “Repeated Groups” Logic
Think of multiplication as adding up groups of the same size.
If you have 3 × (-4), this means you have 3 groups of (-4).When you add them together:
(-4) + (-4) + (-4) = -12
No matter how many positive groups of a negative number you add, the total will always keep moving deeper into the negative numbers.
REMEMBER: Multiplication is commutative, therefore, (-4) × 3 = 3 × (-4).
The Pattern Logic (Maintaining the Math)
Math must follow consistent patterns. If you follow a multiplication table downward, you can see that the answer must become negative to keep the pattern unbroken:
- 3 × 2 = 6
- 3 × 1 = 3 (down by 3)
- 3 × 0 = 0 (down by 3)
- 3 × -1 = -3 (down by 3)
- 3 × -2 = -6 (down by 3)
If 3 × (-1) equaled a positive 3, it would completely break the constant pattern of subtracting 3 at each step.
Dividing Two Numbers With Different Signs
Dividing two numbers with different signs gives a negative answer because division is the exact opposite of multiplication. If multiplication moves you in one direction, division must undo that movement by tracking how you got there.
The “Missing Factor” Logic
Division is just a multiplication problem with a missing piece. It asks: “What number do I multiply the divisor by to get the dividend?”
Let’s look at the problem:
-12 ÷ 3 = x
This is the exact same as asking:
3 × x = -12
We already know from multiplication rules that to get a negative answer (-12), a positive number 3 must be multiplied by a negative number. Therefore, the missing number has to be (-4).
The Real-World “Debt” Logic
Think of a negative number as money that you owe (debt), and division as splitting that debt equally among a group of people.
Imagine a group of 3 friends collectively owes a store (-15) dollars. They decide to split the bill evenly.
(-15) ÷ 3 = (-5)
Each individual person now owes (-5) dollars. Because you are splitting a negative total among positive people, each person’s individual share must also be a negative amount.
Skip Counting
Skip counting means counting forward or backward by a fixed number other than one, rather than counting every single number.
How It Works
- Equal steps: You add or subtract the same amount every time.
- Skipped numbers: The numbers in between are passed over and not said aloud.Starting points: You can start at zero, or at any other number.
Common Examples
- By 2s: 2, 4, 6, 8, 10
- By 5s: 5, 10, 15, 20, 25
- By 10s: 10, 20, 30, 40, 50
- Backward by 5s: 70, 65, 60, 55, 50
Videos
References
“How to Multiply and Divide Positive and Negative Numbers – KS3 Maths – BBC Bitesize.” BBC Bitesize, November 23, 2021. https://www.bbc.co.uk/bitesize/articles/z8x44xs.
“Multiply and Divide Positive and Negative Numbers.” BBC Bitesize, August 5, 2020. https://www.bbc.co.uk/bitesize/guides/zkqf6g8/revision/1.
“Grid Method for Multiplying Numbers.” BBC Bitesize, August 5, 2020. https://www.bbc.co.uk/bitesize/guides/zkqf6g8/revision/2.
“Column Method for Multiplying Numbers.” BBC Bitesize, August 5, 2020. https://www.bbc.co.uk/bitesize/guides/zkqf6g8/revision/3.
“Division on a Number Line – Examples and Diagrams.” Math Monks, August 3, 2023. https://mathmonks.com/number-line/division-on-a-number-line.
“Number Line Multiplication – Examples and Diagrams.” Math Monks, August 3, 2023. https://mathmonks.com/number-line/multiplication-on-a-number-line.
“Why do we get a positive number when we multiply two negative numbers?” Mathematical Mysteries, December 23, 2023. https://mathematicalmysteries.org/why-do-we-get-a-positive-number-when-we-multiply-two-negative-numbers/.