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Section 3.4 Closing Ideas

In this section, we rearranged terms many different times and in many different ways. As long as we made sure that the negative signs moved with the appropriate terms, everything worked out just fine. But we didn't really discuss why things worked out fine. We just showed you how to do it, and then let you mimic that.

There are some fundamental properties of addition that you've probably seen at some point before.

Definition 3.4.1. The Commutative and Associative Properties of Addition.

Let \(a\text{,}\) \(b\text{,}\) and \(c\) be real numbers. Then the following properties hold:

  • The Commutative Property of Addition: \(a + b = b + a\)

  • The Associative Property of Addition: \((a + b) + c = a + (b + c)\)

Notice that this property does not apply to subtraction. You should be able to see that \(a - b\) and \(b - a\) are not the same value. If we were to switch their positions without changing the value, we would first have to rewrite it as addition, and then apply the commutative property of addition to swap their positions. If you've worked through the exercises, then this should feel very familiar.

\begin{equation*} \begin{aligned} a - b \amp = a + (-b) \amp \eqnspacer \amp \text{Subtraction is addition of the opposite} \\ \amp = (-b) + a \amp \amp \text{Commutative property of addition} \end{aligned} \end{equation*}

As it turns out, these two addition properties are what give us the ability to move things around the way we have this section. The reason that we need these properties is extremely subtle. Have you ever noticed that you can only add two numbers at a time? This doesn't mean that you haven't seen sums like \(3 + 5 + 5\) before, but when you actually go to calculate this, you don't actually work with all of the numbers all at once.

This means that when we do arithmetic, we are always implicitly putting parentheses all over the place. If you're just working from left to right, then you would see the calculation as \((3 + 5) + 5\text{.}\) That is, you add the and the first, and then you add to that result.

But maybe you've got a bit of intuition and recognize that \(5 + 5 = 10\text{,}\) which is potentially an easier or faster approach. In that case, you would see the calculation as \(3 + (5 + 5)\text{.}\) This is not a problem because the result is the same both ways!

The commutative property is also used when doing calculations. For most people, the calculation \(27 + 2\) is much easier to think about than \(2 + 27\text{.}\) And so when we see that calculation, we instinctively switch it around to the way that's easier for our brains to think about. And because of the commutative property, it doesn't change the result.