Mathematics KnowledgeLesson 3 of 21
Properties of Arithmetic
The commutative, associative, distributive and identity properties, and recognising which one an example shows.
Table of ContentsShow
- The four properties
- Telling commutative from associative
- Subtraction and division have neither
- The distributive property
- Distributing backward is factoring
- Mental arithmetic
- Identity and inverse
- Naming a property from an example
- What you can skip
- Where people lose points
- Work one in under a minute
- Where this leads
This topic is asked in two ways. Either you are shown an equation and asked to name the property, or you are asked which property justifies a step. Both need the same thing: knowing what each one physically does to the expression.
The four properties
| Property | What moves | Addition | Multiplication |
|---|---|---|---|
| Commutative | the order of the terms | a + b = b + a | ab = ba |
| Associative | the parentheses | (a + b) + c = a + (b + c) | (ab)c = a(bc) |
| Identity | nothing; a value is added or multiplied | a + 0 = a | a x 1 = a |
| Inverse | a value that returns the identity | a + (-a) = 0 | a x (1/a) = 1 |
And the one that stands alone:
Distributive: a(b + c) = ab + ac.
Telling commutative from associative
This is what the question actually tests, and the distinction is mechanical.
Look at the order the letters appear in.
- 3 + 7 = 7 + 3 - the numbers swapped places, so commutative.
- (3 + 7) + 2 = 3 + (7 + 2) - the numbers are in the same order and only the parentheses moved, so associative.
If the order changed, it is commutative. If the order held and the grouping changed, it is associative. An example that does both is usually written to test whether you noticed the reordering, and reordering is the stronger claim.
Subtraction and division have neither
Worth stating because it is a question in its own right.
- 10 minus 4 is 6; 4 minus 10 is -6. Subtraction is not commutative.
- 12 divided by 3 is 4; 3 divided by 12 is 0.25. Division is not commutative.
- (20 minus 8) minus 3 is 9; 20 minus (8 minus 3) is 15. Subtraction is not associative.
- (24 divided by 4) divided by 2 is 3; 24 divided by (4 divided by 2) is 12. Division is not associative.
This is why rewriting subtraction as adding a negative is such a useful habit: addition is commutative and associative, so once everything is an addition you can reorder and regroup freely.
The distributive property
The only one that matters for the rest of algebra.
a(b + c) = ab + ac. The factor outside multiplies every term inside.
6(x + 4) = 6x + 24
The sign travels too. This is where the points go.
-3(x - 5) = -3x + 15
The -3 multiplies the x to give -3x, and multiplies the -5 to give +15, because a negative times a negative is positive. Writing -3x minus 15 is the standard error and it is always a choice.
Distributing backward is factoring
Read the property right to left and it pulls a common factor out.
12x + 18 = 6(2x + 3)
Same property, opposite direction, and it is the basis of the factoring lesson.
Mental arithmetic
Distribution is also how you multiply awkward numbers without a calculator.
7 x 98 = 7 x (100 - 2) = 700 - 14 = 686
15 x 24 = 15 x (20 + 4) = 300 + 60 = 360
Worth practicing, because on a no-calculator test this is faster and more reliable than stacking the multiplication.
Identity and inverse
Identity leaves a number alone. Adding 0 changes nothing; multiplying by 1 changes nothing.
Inverse takes a number back to the identity.
- The additive inverse of 8 is -8, because they sum to 0. It is the opposite.
- The multiplicative inverse of 8 is 1/8, because they multiply to 1. It is the reciprocal.
The reciprocal of a fraction is that fraction flipped: the multiplicative inverse of 3/5 is 5/3. Zero has no multiplicative inverse, because nothing multiplied by zero gives one.
The reason dividing by a fraction means multiplying by its reciprocal is exactly this property. Dividing by 3/5 is multiplying by the multiplicative inverse of 3/5, which is 5/3. It is not a trick anyone invented for fractions; it is what division is.
Naming a property from an example
The question usually looks like this:
Which property is shown by 4 x (6 + 2) = 4 x 6 + 4 x 2?
Scan for the signature.
| What you see | Property |
|---|---|
| a factor outside a parenthesis appears on every term inside | distributive |
| the same terms in a different order | commutative |
| the same terms in the same order, different parentheses | associative |
| a 0 added or a 1 multiplied, nothing else changes | identity |
| two things combining to give 0 or 1 | inverse |
The example above has a 4 outside a parenthesis reappearing on both inner terms, so it is distributive.
What you can skip
Across all 1,150 Mathematics Knowledge questions in our bank, trigonometry, logarithms, matrices, complex numbers, sequences and standard deviation never appear. If a study guide spends pages on them, that time is better spent here.
- The property names as vocabulary. Commutative, associative and distributive appear by name only rarely. What is asked is using them to rearrange a calculation, which is what this lesson practices.
Where people lose points
Calling a regrouping commutative. Check whether the order changed.
Losing the sign when distributing a negative. The most expensive error in this topic because it propagates into equation solving.
Distributing across a multiplication. 3(4x) is 12x, not 12 times 3x. The distributive property is about multiplication over addition, not multiplication over multiplication.
Assuming subtraction or division commute because addition and multiplication do.
Confusing the two inverses. The opposite is additive, the reciprocal is multiplicative.
Work one in under a minute
Which property justifies rewriting 5 + (3 + 9) as (5 + 3) + 9?
The terms are in the same order - 5, then 3, then 9 - and only the parentheses moved. That is the associative property of addition.
Had it been rewritten as 5 + (9 + 3), the 9 and 3 would have swapped and it would be commutative.
Where this leads
The distributive property is the engine of expanding and factoring, and the associative and commutative properties are what license combining like terms.
Related lessonsReference
- Algebraic Expressions - distributing and combining like terms in practice
- Factoring and Quadratics - the distributive property read backward
- Order of Operations - why grouping has to be resolved before anything else
- Signed Numbers and Absolute Value - the sign rule that distribution depends on