Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 7.4 Closing Ideas

In this section, we saw the last of the algebraic properties of arithmetic.

If this looks familiar, it's because we saw something very similar to it a few sections ago. It turns out that addition also has these two properties (Definition 3.4.1). And when we combine them together with the distributive property (Definition 3.1.4), we get a pretty thorough description of how arithmetic and algebra works.

It's important to notice that we've never proven any of these properties. We've relied on our experiences going all the way back to when we were first learning basic arithmetic as our foundation. And from those experiences, we've built out our understanding that this is how numbers work. The world would be a very different place if these things weren't true.

• (The commutative property of addition) If we start with an empty bag, then put one apple into bag followed by two apples, we get the same result as if we had put in two apples followed by one apple.

• (The associative property of addition) If we put one apple and two oranges in a bag, and then put three bananas in afterward, we get the same result as if we had put the two oranges and three bananas in the bag first, and then put the one apple in.

• (The commutative property of multiplication) Three bag of five apples have the same number as five bags of three apples.

• (The associative property of multiplication) Two boxes that each contain three bags with four apples has the same number of apples as (two times three) bags with four apples each.

So at a very basic level, we believe these statements are true because our experiences with reality tell us they should be true. These are not rules that mathematicians came up with and told everyone else they had to follow. Mathematicians simply created a language to describe these things.

When it comes to college level mathematics, it can be helpful to think about the work you're doing with that framework in mind. As much as you can, try to ground all the work that you do in terms of practical reality. You won't always be able to do it. Sometimes, the work that you're doing is an abstraction or generalization of a concept. If that's the case, try to bring yourself back to the last example that made sense, and see if you can use that knowledge to help you build the next piece.

This isn't always going to be easy. Some ideas will take more time to sink in than others. That's part of the nature of learning. But if you invest the time and the energy to slow down and learn to think clearly about what you're doing, you'll have a much better chance at being successful.