Commutative, associative and distributive properties

Why this exists

Algebra is mostly rewriting: a sum put in a handier order, a bracket multiplied out, a common factor pulled to the front. Each rewrite promises that the new expression is worth exactly what the old one was, for every number the letters could stand for.

Trying numbers cannot keep that promise, since a thousand trials leave infinitely many real numbers untested. A short list of laws true of all real numbers at once does keep it, and three of them do most of the work: the commutative, associative and distributive properties. With whole numbers they are counting facts, met in the properties of addition. Here they are stated with letters for every real number, so that every step of a simplification can name the law that allows it.

Definition

The letters a, b and c stand for any real numbers: whole numbers, negatives, fractions, decimals, and irrational numbers such as \displaystyle \sqrt{2}. Each law says two expressions are equal for every choice of them.

The commutative properties: order does not matter. Two numbers may be added, or multiplied, in either order.

The associative properties: grouping does not matter. Adding or multiplying three numbers, it does not matter which two are combined first. The numbers keep their order; only the brackets move.

The distributive property: multiplication spreads over addition. A number times a sum equals the sum of the separate products.

It is the only one of the three that joins two operations: the factor a is handed out, or distributed, to each term of the sum. Commutative is about order, as in commuting back and forth; associative is about which numbers are associated in a bracket.

Intuition

Think of a positive number as a length. Adding two lengths lays them end to end, and in either order the rods in Figure 1 reach the same point: the commutative property of addition. With a third rod, (a + b) + c joins the first two and then the third, while a + (b + c) joins the last two first; it is the same line of rods either way.

Multiplying two lengths gives the area of a rectangle. Turning it on its side changes which side is called the width, not its area. For three numbers, picture a box a long, b wide and c tall: its volume is the base times the height, (ab)c, or the length times the end, a(bc).

Figure 3 is a tall and b + c wide, so its area is a(b + c). The dashed line cuts it into an a by b piece and an a by c piece, so its area is also ab + ac. It is the area picture behind the distributive property of arithmetic, now with letters.

The pictures show positive lengths only. The laws hold for every real number, negatives and zero included.

Results

Worked examples

Boundaries

Only addition and multiplication. Subtraction and division are neither commutative nor associative (Example 2). To move terms in a difference, first write subtraction as adding the opposite, a - b = a + (-b); then each term moves with its sign attached:

Multiplication distributes over addition, not the other way round. With a = 1, b = 2, c = 3:

Multiplication does not distribute over multiplication. 2(3 \cdot 5) = 30, while (2 \cdot 3)(2 \cdot 5) = 60.

Division distributes from one side only. For d \neq 0, (b + c) \div d = b \div d + c \div d, because dividing by d is multiplying by \tfrac{1}{d}. A sum underneath does not split: 12 \div (2 + 4) = 2, while 12 \div 2 + 12 \div 4 = 9.

Powers have none of these properties. 2^5 = 32 but 5^2 = 25; (3^2)^3 = 729 but 3^{(2^3)} = 6561; (3 + 4)^2 = 49 but 3^2 + 4^2 = 25.

These are laws about real numbers. Later operations need not obey them: multiplying matrices or composing functions in the other order usually gives a different answer.

Where this goes next

These three laws are half of the rules the real numbers obey. The rest, about 0, 1, opposites and reciprocals, are in identity and inverse elements, with closure alongside; together they are the properties of real numbers.

The distributive property carries most of the everyday work: forwards in distributing over a sum, backwards in factoring out a common factor and combining like terms, and twice over in multiplying two binomials. Naming a law for each step is also the first form of proof in algebra; solving equations adds the properties of equality to the list of reasons.

Exercises

Naming the property

Name the property that makes each statement true for all real numbers.

1. p + (-9) = (-9) + p
2. (2.5 \cdot r) \cdot 8 = 2.5 \cdot (r \cdot 8)
3. \tfrac{1}{2}(6c + 10) = \tfrac{1}{2}(6c) + \tfrac{1}{2}(10)
4. (d + 4)(d - 1) = (d - 1)(d + 4)

True for every real number?

Is each statement true for every real number? If so, name the properties that make it true. If not, give a counterexample.

5. x - 8 = 8 - x
6. (y \div 4) \div 2 = y \div (4 \div 2)
7. 3(5w) = (3 \cdot 5)(3 \cdot w)
8. (k + 2) \cdot 7 = 7k + 14

Making the arithmetic easy

Work out each value without a calculator, naming each property you use.

9. 25 \cdot 19 \cdot 4
10. \tfrac{2}{9} + 4.3 + \tfrac{7}{9} + 5.7
11. 34 \cdot 96 + 34 \cdot 4

Expanding and factoring

Expand each product.

12. -4(3t - 2.5)
13. (6 - b) \cdot 7

Factor out the largest common factor.

14. 18f + 24
15. g \cdot h - 5g

A reason for every step

16. Simplify 3(2 + 5s) + 4s, giving the property or arithmetic behind each step.
17. Someone wrote 40 \div (8 \div 2) = (40 \div 8) \div 2 = 5 \div 2 = 2.5. Which step is not allowed, and what is the correct value?

Reasoning

18. Division is not commutative, but sometimes a \div b = b \div a. Find every b that makes 2 \div b = b \div 2, and then say for which nonzero a and b it happens in general.
19. Define a \star b = \dfrac{a + b}{2}, the number halfway between a and b. Is \star commutative? Is it associative? Explain.
20. Show that a - (b + c) = (a - b) - c for all real numbers, naming each property. Explain why this does not make subtraction associative.