什么是交换律？

Commutative Properties

The four operations, addition, subtraction, multiplication, and division, are important skills in pre-algebra.

It is important to realize that oftentimes the order in which you write the numbers out matters, and sometimes it doesn't.

We're going to focus on addition and multiplication here, because for those operations you can flip the order in which you write the numbers.

In particular, if $a$ and $b$ are numbers, then $$a + b = b + a$$ $$a \times b = b \times a$$

As a result of this beautiful mirror-like symmetry we call addition and multiplication commutative operations.

Commutative Addition

While we can look at equations all day, I want to feature some mouth-watering examples that highlight these facts when talking about food.

We're going to talk about why adding is commutative by counting M&Ms. Let's say you have $7$ red M&Ms and $8$ orange M&Ms (your big brother ate all the other colors, sadly).

You're not sure if you want to eat the red M&Ms first, or the orange ones first, but one thing's for sure: you're eating ALL the M&Ms.

If you eat the red ones first, and then the orange ones, you eat $\color{Red}{7}$ M&Ms, followed by $\color{Orange}{8}$ M&Ms. Adding this together means you eat a total of $7 + 8 = 15$ M&Ms.

If on the other hand, if you eat the orange ones and THEN the red ones, you eat $\color{Orange}{8}$ M&Ms followed by $\color{Red}{7}$ M&Ms.

This still makes for a total of $8 + 7 = 15$ M&Ms. What we have shown is that $$7 + 8 = 8 + 7,$$ because both sides of the equation equal $15.$

While that's a tasty fact to keep in mind, what's even sweeter is that you can do this even if you have MORE than two different colors of M&Ms.

Let's say you now have $5$ green M&Ms in addition to the $7$ red M&Ms and $8$ orange M&Ms.

If you want to eat the M&Ms in the order green, red, orange then you eat $$5 + 7 + 8 = 12 + 8 = 20$$ M&Ms. On the other hand if you switch up the order and eat the M&Ms in the order orange, green, red, then you eat $$8 + 5 + 7 = 13 + 7 =20$$ M&Ms. This should make sense, because at the end of the day all of the M&Ms will be gone. Since both of these sums equal $20$ we have shown $$5 + 7 + 8 = 8 + 5 + 7.$$

Let us note this is very different from the associative property. The associative property introduces parentheses. It asks you to do the operations in a certain order. No parentheses here!

Commutative Multiplication

Now we're going to show why multiplication is commutative by figuring out how many ways to make a sandwich.

Let's say that a sandwich has three components: bread, cheese, and meat.

If you're building a sandwich you can choose the bread, cheese, and meat in any order and still have the same sandwich. It's the same ingredients all piled high on some tasty carbohydrates.

We'll use this fact to make it clear that multiplication is commutative.

You open up your pantry and see three different types of bread, five types of cheese, and two types of meat.

Let's say you want to choose one of each to build a sandwich for lunch.

In one case you choose the bread first, then the cheese, then the meat. The number of ways to do this is $$3 \times 5 \times 2 = 15 \times 2 = 30$$ because we multiply the number of choices for each step together. The key is that the order of the steps doesn't matter.

If instead we choose the meat first, then the bread, and then finally the cheese then our number of ways for building the sandwich is $$2 \times 5 \times 3 = 10 \times 3 = 30.$$

That is, no matter which component of the sandwich we choose first, we can build the same number of different sandwiches.

If you don't believe me, go buy a lot of bread, cheese, and meat and try this at home yourself. Happy eating!

Here are some examples:

Commutative Property of Addition

$$\begin{align} 3+1&=1+3\\ 21+19&=19+21\\ x+y&=y+x \end{align}$$

Commutative Property of Multiplication

$$\begin{align} 3\cdot 1&=1 \cdot 3\\ 21\cdot 19&=19\cdot 21\\ x\cdot y&=y\cdot x \end{align}$$

Bonus

Interested in mathematical properties? Check out some other cool properties, like the reflexive, symmetric, and transitive properties!

(Video) Commutative and Associative Properties of Addition

by mathman1024

This video by mathman1024 explains the commutative and associative properties.

$\large{\text{Summary}}$

The Commutative Property of Addition states: $$a+b=b+a$$ This means it does not matter what order you add numbers together in; it will always result in the same number. For example: $$\begin{align} 5+7&=7+5\\ 12&=12 \end{align}$$

$$\begin{align} (-4)+1&=1+(-4)\\ -3&=-3 \end{align}$$

The Associative Property of Addition states: $$(a+b)+c=a+(b+c)$$ This is similar to the commutative property in that it does not matter in what order you add things together. You can mix and match and put whatever terms first, they always add to the same value.

He illustrates this with the example: $7+(-12)+(-17)+10+(-33)$. We can rearrange the terms so that the positive values add together and the negative values add together. $$\begin{align} &7+(-12)+(-17)+10+(-33)\\ &={\color{Blue}{7+10}}+{\color{#8925AE}{(-12)+(-17)+(-33)}}\\ &={\color{Blue}{17}}+{\color{#8925AE}{(-62)}}\\ &=-45 \end{align}$$