Commutative property of Addition and Multiplication

The commutative property of addition states that for any $a$ and $b$, $a + b = b + a$. Similarly, the commutative property of multiplication states that $a \cdot b = b \cdot a$.

The commutative property essentially means that the order in which we perform a single addition (or multiplication) does not matter. When combined with the associative property of addition and multiplication, it implies that the order of many additions (or multiplications) does not matter. For instance, the expressions

$$1 + 2 + 3 + 4 + 5 + 6$$

and

$$3 + 5 + 4 + 1 + 6 + 2$$

will necessarily have the same result.

At first glance, commutativity might look like a very obvious property: of course the order of addition doesn't matter! But in fact this is not immediately evident; while many operations (both in real life and in mathematics) are commutative, many are not.

For instance, these operations are both commutative:

• Putting on your left sock and putting on your right sock
• Addition (and multiplication)

But these two operations are not commutative: * Putting on your socks and putting on your shoes * Subtraction (and division)

Thus, commutativity is a rather strong condition! In fact, most operations are not commutative, and thus operations that have this property are rather special. Besides addition and multiplication, the union and intersection of sets are commutative, certain logical operators like AND and OR are commutative (which you would expect from real life -- saying "A and B" is the same thing as saying "B and A"!), and a handful of others. As a result, before rearranging things for convenience, it's important to check that the operation is commutative (lest you end up with socks over your shoes)!

Because the order of commutative operations does not matter, arithmetic can often be simplified by rearranging numbers. For instance, when adding many numbers, a common method is to "make tens" -- rearranging numbers so that some additions result in multiples of 10, which are easy to work with.

Determine the value of

$$3 + 8 + 6 + 2 + 17 + 3 + 5 + 1 + 2 + 4.$$

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The addition can certainly be carried out directly without too much trouble, but this process is relatively tedious and susceptible to error. A cleaner method is to rearrange the terms:

$$\begin{aligned} \color{#3D99F6}{3} + \color{#D61F06}{8}+\color{#20A900}{6}+\color{#D61F06}{2}+\color{#3D99F6}{17}+\color{skyblue}{3}+\color{skyblue}{5}+1+\color{skyblue}{2}+\color{#20A900}{4} &=\color{#3D99F6}{3}+\color{#3D99F6}{17}+\color{#D61F06}{8}+\color{#D61F06}{2}+\color{#20A900}{6}+\color{#20A900}{4}+\color{skyblue}{3}+\color{skyblue}{5}+\color{skyblue}{2}+1 \\ &=20 + 10 + 10 + 10 + 1 \\ &=51. \ _\square \end{aligned}$$

Utilizing the commutative property for multiplication is similar: "making-tens" is still the general strategy. This means being on the lookout for fives is particularly useful!

Compute the value of $$2 \cdot 3 \cdot 5 \cdot 3 \cdot 2 \cdot 3 \cdot 5.$$

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The multiplication can certainly be carried out directly without too much trouble, but this process is relatively tedious and susceptible to error. A cleaner method is to rearrange the terms: $$\begin{aligned} &\color{#D61F06}{2} \cdot \color{#20A900}{3} \cdot \color{#D61F06}{5} \cdot \color{#20A900}{3} \cdot \color{#3D99F6}{2} \cdot \color{#20A900}{3} \cdot \color{#3D99F6}{5} \\ &=\color{#D61F06}{2}\cdot \color{#D61F06}{5}\cdot \color{#3D99F6}{2}\cdot \color{#3D99F6}{5} \cdot \color{#20A900}{3} \cdot \color{#20A900}{3} \cdot \color{#20A900}{3} \\ &= 10 \cdot 10 \cdot 27 \\ &= 2700. \ _\square \end{aligned}$$

• Associative property of Addition and Multiplication