 There are many ways to define mathematics, but here's one we'll find useful. Mathematics is the process of abstracting commonalities. For example, the pictures shown all have an abstract commonality, and we might say that they all have a fiveness about them. Now suppose we have a collection of objects, and we have another collection of objects. What happens when we put them together? Collection is the abstraction of this idea of putting things together. In this case, our separate collections could be associated with the concepts 4 and 3, and the combined collection could be associated with the concept 7, and so the abstraction of combining them can be expressed as 4 plus 3 equals 7. Now sometimes we can't do a lot. For example, if you have three apples and two oranges. If you put them together, you have three apples and two oranges. Since you can't convert an apple to an orange or vice versa, there's no alternate expression for what you have. But suppose you're combining amounts base n. Since you can bundle and trade, you can often simplify the final result. For example, let's say we want to add 1, 2, base 8, and 2, 4, base 8. So arithmetic is bookkeeping, and so what we have here is 1 medium and 2 small and 2 medium and 4 small. Let's go ahead and represent those on our placed value table. So when we put them together, arithmetic is bookkeeping. We have 3 medium and 6 small, so we can write our final answer as 3, 6, base 8. What happens if we add together larger amounts? The thing to remember here is that if we have more than n, we need to bundle and trade for a larger unit. Remember the largest amount you can recognize in base n is n. Anything more is lots. So let's add 3, 2, base 5, and 1, 4, 1, base 5. Arithmetic is bookkeeping, so this first add n tells us we have 3 medium, 2 small. Our second add n tells us that we have 1 large, 4 medium, 1 small. And if we combine them, we see that we have 1 large, many medium, and 3 small. Now we can't count the total number of medium because there's too many, but we can bundle and trade them. So we're working base 5, so it's bundle, sets of 5, and trade. So for example we can take this set of 5, and one advantage of using a physical placed value chart is we can see if we place things in the wrong space. So let's move this large unit to where it should be. And so bundling and trading gives us our final answer, 2, 2, 3, base 5. Believe it or not, this is everything you have to know about addition to be able to add any numbers in any base. And this is the basis for adding fractions and algebraic expressions. Anything else you do with addition is a way to make this process more efficient.