In many ways mathematics begins with addition and subtraction. After all, the uniqueness of the infinity of numbers depends upon the fact that some separation exists between them—and we can use addition and subtraction to cross the separations from one to the next.

Everybody seeing this no doubt already gets addition and subtraction at some level. But do you really feel them? Hopefully you developed an intuitive feel for their meaning way back when you first learned to count. But I’m a firm believer that it never hurts to reinforce our intuition with even more intuition to ensure that we really do feel things way, way deep down within our gut. So that’s our plan for the next few weeks.

With that in mind, the question for today is: How should we think about addition so that we develop a good deep-down-in-the-gut feel for it? Let’s find out.

Addition With Integers

To kick things off, let’s start with a simple question: What is addition? There are no doubt a bunch of ways you can think about it, but for our purpose I’m going to say that addition is the process of combining things or groups of things into a single bigger group of things. You can think of it as taking two or more piles of rocks, dollars, pumpkins, imaginary friends, or whatever else and putting them all together into a bigger pile, like this:

So addition is piling things up. Yes, I know this seems kind of obvious—it is—but that’s exactly what addition is! But does this way of thinking about it always make sense? Does it still work when we’re adding up stuff other than integers—stuff like fractions, irrational numbers, or maybe even variables?

Addition With Fractions

For the most part, yes … it’s the same. Although as soon as you expand your numerical horizons and start throwing these other items into the mix, things do get a bit more complicated. For example, let’s think about fractions. Of course, the actual meaning of addition doesn’t change when we move from integers to fractions—we’re still making a new pile from two or more initial piles—it’s just the mechanics of doing this piling up that changes. To see what I mean, take a look at this:

Suddenly, now that we’re dealing with fractions, it’s not so easy to add things up, is it? Sure, once you know how to do it, it’s not all that bad. But the picture is a bit different than when we were just dealing with positive integers making nice clean stacks of blocks. To see how to intuit the meaning of this slightly different form of pile-making, we can update our picture to look like this:

The trick is to find a way to rewrite 1/3 and 1/2 so that they both represent some number of the same type of somethings. In other words, we can’t straight-forwardly add 1/3 to 1/2 because they have different denominators—which means they represent different “types of somethings.” If we find a common denominator, we can express 1/3 and 1/2 in terms of some number of sixths. In particular, 1/3 = 2/6 and 1/2 = 3/6. Once they're drawn this way, hopefully it becomes clear that we’re still just piling things up to make a new pile.

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