Commutative and Associative Properties

Multiplication shares the commutative property with addition. You can multiply two numbers in either order and the product stays the same. Think about arranging tiles on a floor. If you place $4$ rows with $6$ tiles in each row, you cover $24$ tiles total. If you rotate the arrangement so that there are $6$ rows with $4$ tiles each, the floor is the same size: still $24$ tiles. You have simply changed the perspective.

In symbols, for any numbers $a$ and $b$:

$a \times b = b \times a$

Knowing this can cut your memorization work in half. If you already know that $6 \times 7 = 42$, then you automatically know $7 \times 6 = 42$ as well. Every fact you learn in a multiplication table comes with a free partner.

Not every operation behaves the same way. Subtraction and division are not commutative, which means the order of the numbers absolutely matters. If you have $10 - 3$, you get $7$. But if you reverse the numbers and calculate $3 - 10$, you get $-7$, which is a completely different answer. Changing the order in subtraction changes the sign of the result.

Division has the same issue. $12 \div 4 = 3$, but $4 \div 12$ is not a whole number at all.

A good way to remember this is to ask yourself: "Does switching the numbers change the real-world situation?" With addition, putting 3 apples in a basket and then 5 more gives the same total as putting 5 in first. But taking 3 dollars away from 10 is very different from taking 10 dollars away from 3. Whenever the physical situation changes, the operation is not commutative.

Once three or more numbers are involved, a new question arises: does it matter which two numbers you add first? The associative property of addition says the answer is no. You can group any two adjacent numbers together first without changing the final sum. The word "associative" comes from "associate," meaning to pair or group together.

Imagine filling a cart with three boxes weighing $4$, $6$, and $3$ pounds. You might add the first two weights first: $(4 + 6) + 3 = 10 + 3 = 13$. Or you might add the last two first: $4 + (6 + 3) = 4 + 9 = 13$. Both routes arrive at the same total. Formally:

$(a + b) + c = a + (b + c)$

This property is especially useful when some pairs of numbers add up to a nice round number. If you need to add $7 + 3 + 18$, you might spot that $7 + 3 = 10$ and then easily add $10 + 18 = 28$. The associative property gives you permission to pair the numbers however you like.

Multiplication is also associative. When you multiply three or more numbers together, you can choose which pair to multiply first and the final product will not change. This flexibility is very powerful for mental arithmetic.

Consider finding $5 \times 7 \times 2$. If you go left to right, you get $(5 \times 7) \times 2 = 35 \times 2 = 70$. But if you spot that $5 \times 2 = 10$ first, you can instead compute $5 \times (7 \times 2) = 5 \times 14$... or, even better, rearrange using the commutative property as well and compute $(5 \times 2) \times 7 = 10 \times 7 = 70$. Same answer, but the calculation is much quicker.

In general: $(a \times b) \times c = a \times (b \times c)$

Look for pairs that multiply to $10$, $100$, or other multiples of ten. The associative and commutative properties together give you the freedom to reorder and regroup any string of multiplication in whatever way makes the arithmetic simplest.

The commutative and associative properties work together as a pair. The commutative property lets you reorder numbers, and the associative property lets you regroup them. Used together, you have complete freedom to arrange a sum or product in any configuration that makes calculation easier.

For example, suppose you need to add $13 + 47 + 27 + 53$. If you add left to right, the numbers are awkward. But if you spot that $13 + 47 = 60$ and $27 + 53 = 80$, you can regroup: $(13 + 47) + (27 + 53) = 60 + 80 = 140$. That is a much smoother calculation, and both properties together justified every rearrangement you made.

In algebra, these properties are the reason you can simplify expressions like $x + 5 + x + 3$ by collecting the like terms in any order. The rules you learn here for numbers extend seamlessly into symbolic mathematics.