Properties of Integers & Inverses

The commutative, associative and distributive properties applied to integers, plus additive and multiplicative inverses — what they are, how exams ask for them, with worked examples, a worksheet with memo and a quiz.

The properties you met with whole numbers keep working when negatives join in. Exams ask about them in two ways: name the property shown, and use a property to make a calculation easier.

1Commutative and associative

Additive inverses sit the same distance either side of zero, so they add to 0.

Commutative: a + b = b + a and a × b = b × a
Associative: (a + b) + c = a + (b + c)

Worked ExampleName the property: (−4) + 7 = 7 + (−4)

1
The same two numbers appear on both sides, just swapped around.
Order changed, grouping did not.
2
This is the commutative property of addition.
Commutative = order can change.

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Subtraction and division are not commutative: 5 − 3 is not the same as 3 − 5, and (−8) ÷ 2 is not 2 ÷ (−8). Only + and × may be rearranged.

2Distributive

Distributive: a × (b + c) = a × b + a × c

Worked ExampleUse the distributive property: (−3) × (10 + 4)

1
Distribute the −3 into each part: (−3) × 10 + (−3) × 4.
Multiply the outside number by everything inside.
2
= −30 + (−12).
Different signs each time → negative.
3
= −42.
Check: (−3) × 14 = −42 too.

3The additive inverse

The additive inverse of a number is what you add to it to get 0.
a + (−a) = 0

Worked ExampleWrite down the additive inverse of 7 and of −5

1
For 7: we need 7 + ? = 0, so the inverse is −7.
Just flip the sign.
2
For −5: we need −5 + ? = 0, so the inverse is 5.
The inverse of a negative is positive.

4The multiplicative inverse

The multiplicative inverse (reciprocal) of a number is what you multiply it by to get 1.
a × 1a = 1

Worked ExampleWrite down the multiplicative inverse of 4 and of −3

1
For 4: 4 × ? = 1, so the inverse is 14.
Turn the number upside down.
2
For −3: (−3) × ? = 1, so the inverse is −13.
Keep the negative sign — a negative times a positive could never give +1.

⚠️

Don't confuse the two inverses. The additive inverse of 4 is −4 (they add to 0). The multiplicative inverse of 4 is ¼ (they multiply to 1). Read the question carefully.

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0 has no multiplicative inverse — there is no number you can multiply 0 by to get 1. That is another way of saying you can never divide by 0.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Name the property: (−6) × 3 = 3 × (−6)
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Commutative property of multiplication
2
Name the property: (2 + (−5)) + 4 = 2 + ((−5) + 4)
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Associative property of addition
3
Name the property: (−2) × (7 + 3) = (−2)×7 + (−2)×3
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Distributive property
4
Use the distributive property to calculate (−5) × (20 + 3).
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(−5)(20) + (−5)(3) = −100 + (−15) = −115
5
Write down the additive inverse of −11.
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11 (because −11 + 11 = 0)
6
Write down the additive inverse of 8.
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−8
7
Write down the multiplicative inverse of −6.
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−16 (because −6 × −16 = 1)
8
Is subtraction commutative? Give an example to justify your answer.
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No — e.g. 5 − 3 = 2 but 3 − 5 = −2, so the order matters.