Binary Numbers

I feel presumptive writing this post ... and, I guess, I feel like I should apologize or something. This issue with binary numbers feels like an elephant in the room; I've never felt the matter was rightfully explained and I want to be clear about why I'm using them and how. So, spoiler alert, I'm going to write about very simple math ~ please feel free to skip this post. [There is a bit at the end that explains why I'm using binary numbers for my mapping].

In any case, the worst we're going to do is addition.

I'll begin with the obvious. We use a ten-based number system every day without much thought that the number of symbols, 0 through 9, are arbitrary. If we discarded the 9, there would be nine symbols remaining in what we would call a nine-based number system. The math we use everyday would be just the same ... but, admittedly, it would be weird.

Binary is a two-based number system, using only 0 and 1. Most talk about binary nowadays ends up being about computers, because a two-based system is perfect for a machine using electrical impulses ... but we don't have to talk about any of that. Suffice to say that the math in binary works like the math in the base-ten that we use.

When we see the number 123, we don't stop to think about it, but we're fairly clear that this is 100+20+3. It helps to think of this as three columns, as shown:

While some of you are having flashbacks to grade 9 math class, I'll take the next step and explain what happens when we add 123 to 869 ... the sort of thing that we do every day without a moment's hesitation. We begin by putting one number above the other, so the columns line up; then we start with the 1s column, adding 3 + 9 to get the product 12. We then divide the product by ten, putting the 1 in the 10's column and the 2 in the 1s column.

We don't think of it as dividing by ten. We do it so habitually that we just automatically divide the number 12 in half, putting the numbers where they belong, without a moment's thought. But we are dividing that product by ten.

The rest is clear. We add 2 + 6 + 1 (that we've added from the 1s column) which gives us 9. We write this in the 10s column. Then we add 1 + 8 to get 9, so that our final total is 992. And I'm sorry ~ I know this has to be old hat to most of us. But it's necessary we're all on the same page.

When we count in base 10, we count 1, 2, 3, 4 and so on. And this seems obvious too. But with binary numbers there is no 2 and no 3 or 4. So how do we count in binary? Well, how do we actually count in base-10?

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