Numbers
\[\renewcommand{bin}[1]{#1\text{b}}\]
Building Our Way Up
In the previous chapters, we discussed how to break down what we had into more fundamental parts. Such an approach has an obvious due-date attached to it; once you get to the fundamental building blocks, there’s nowhere left to go in that direction.
And so in this chapter, we turn our attention around, and look at how we can build more complicated things out of our simpler pieces.
Wires, as we’ve defined them, allow us to carry a single piece of information – whether they’re on or whether they’re off. While that’s great and all, lots of things we as humans care about are more complicated than yes-or-no. The good news, is that yes-and-no is all we need in order to build arbitrarily interesting things, if we’re clever in how we go about it.
But again, we must walk before we can run. And so we will turn our attention to construct the second-most-simple thing we can – numbers. We do this not only because it will be useful, but also as an exercise in deconstructing things into their constituent parts. Ready? Let’s go!
Frickin’ numbers. How do they work?
A (Very) Short History of Numbers
It might not seem like it, but numbers are actually a pretty radical invention. Not only in the sense of like “yeah, it’s kinda cool that we can count things,” which was an amazing technological advance for humanity when it happened. But there’s also a subtler marvel at work here, and that’s how we represent numbers.
Remember back in the late 1900s, when movie studies thought it was pretty cool to write the year in which films were copyrighted as Roman numerals? The credits would have things like ©MGM MCMXCIX at the end. I’d be willing to bet that you don’t know off the bat exactly what year MCMXCIX refers to. There’s actually a pretty complicated set of rules for deciphering Roman numerals into a real number that people without a Greco-Roman anthropology degree would understand.
What’s perhaps more interesting is that deciphering Roman numerals wasn’t even significantly easier for the Romans than it is for us. There’s a lot of arithmetic required to figure out what’s going on in that jumble of letters.
And then the Arabs came around and said “hey you guys, forget all that stuff. We have a much better system,” and the world agreed with them, and that’s why we still use the Arabic numeral system today.
It’s the Arabic numeral system that’s the marvelous technological innovation behind numbers. And it’s that invention that we’ll need to study in order to implement numbers in our wiring diagrams.
The Decimal System
We use the decimal system. What’s that? It means we have ten different symbols for numbers. You know ’em. 0 through 9. Somewhat remarkably, even though we have ten of them, we don’t have a symbol for the number ten. To represent ten, we use a combination of two symbols: 1 and 0, placed after one another.
In fact, the order in which we place these symbols turns out to matter. 12 is a very different number than 21, even though they have all the same symbols. The more-left a number is in a sequence, the more “number-ness” it represents.
I’m sorry, I know that you know this. But bear with me for a little while longer.
Let’s say we’re counting. We start with 0, and then 1, and so on, changing our number-symbol until we reach 9. Now we need to change it again to represent ten, but we don’t have any more symbols to use, since it’d be confusing if we reused one. So we cheat, and we go back to 0, and slyly add a new 1 to the left. We continue counting, and now we can change our right-most 0 again, changing it through all of our symbols again, until we hit 9.
Crap, we’ve run out of symbols again, so we pull another trick; we switch back to 0, and change that 1 we made up to a 2. We realize we can just cycle that symbol as well, albeit slower than our first one.
When we get all the way up to 99 – poop, no more ways we can change those symbols and not reuse a combination we used already. So we use the old tried and true method again, change everything back to a 0 and slap a new 1 at the front.
I just spent 300 words describing counting to you. It’s patronizing, I know. I’m sorry. The point I’m trying to illustrate is that this procedure we’ve been doing has really nothing to do with numbers whatsoever, we’re just shuffling symbols around.
The magic is in our strategy for shuffling these symbols. It turns out that because we have 10 symbols to switch between, then we can obviously represent 10 different things with them. If we use TWO of the symbols, and each has 10 different possibilities, then we can describe \(10*10=100\) things. Three symbols gives us \(10*10*10=1000\) and so on.
Your first thought might be “but the biggest number we can represent with three symbols is 999.” Which is true, but we also have the number zero, which conventional counting skips. So we really and truly have 1000 different numbers that we can represent with three symbols.
Because of this decimal system, we get an interesting mathematical fact, which is that any number can be decomposed into powers-of-ten. For example:
\[6934 = 6\times 10^3 + 9\times 10^2 + 3\times 10^1 + 4\times 10^0\]
If you don’t like math and the exponents make you unhappy, don’t fret, we’re not going to do much more with them. Intuitively, this says that the rightmost symbol describes how many ones your number has, well the next-to-rightmost describes how many tens your number has. And so on, until you run out of symbols.
The Binary System
I’ll bet you can see where we’re going with all of this. Since a wire can be either hot or cold, which can think of it as having two different symbols. If we stack a bunch of wires beside one another, we can decide on a convention for which one is the biggest, and then replace all of the “tens” in our argument above with “twos”.
The resulting system we get out of all of this work is known as the binary system. I’m sure you’ve heard of it.
To really whet your whistle for these things, let’s go through a few examples for deciphering a number represented in the binary system into one you’d be more familiar with, represented in the decimal system.
Remember that in the decimal system, even though we had ten symbols, we had no symbol for the number ten. Similarly, in the binary system, we have two symbols, but no symbol for two.
So as to not confuse anyone, we’ll end any binary numbers we’re working with with a final b. So the number \(\bin{10}\) should be considered a binary number, while the number \(10\) is the usual ten that we know and love.
Starting slowly, we’ll look at the number \(\bin{0}\), which is obviously the same as our regular decimal \(0\). Likewise, we still have a symbol to shuffle, so \(\bin{1}=1\).
If we increase one to both sides, our decimal side easily goes up to \(2\), but in binary, we’ve now run out of symbols, and so we need to steal a new one. Therefore \(\bin{10}=2\). In the same way that a decimal number full of 9s switches to a 1 followed by the same number of 0s when its increased, a binary number full of 1s switches too!
That decomposition of decimal numbers into their powers-of-ten trick that we saw earlier works equally well in binary land. For example:
\[\bin{101101} = 1\times 2^5+0\times 2^4 + 1\times 2^3 + 1\times 2^2 + 0\times 2^1 + 1\times 2^0 = 45\]
Your powers-of-two likely aren’t up to the same calibre as your powers-of-ten, but you have to admit, this method is much easier than trying to count all the way up from 0.
And so that’s it! We’ve come up with a scheme for representing numbers in terms of the building blocks we had, namely wires! This is the beginning of a trend. As we’ll see more and more, wires are, in fact, useful. We use them to describe things that are, while we use machines to describe how things change, by way of function tables. In this sense, our wires are “nouns”, and our machines are “verbs”.
In the next chapter we’ll look at putting these numbers directly into our diagrams, and reinterpret some of our existing machines in terms of how they act on numbers. Additionally, we’ll build some new machines for doing useful things with numbers.
Exercises
- Convert the number \(\bin{11111111}\) into its decimal representation. Recalling what you know about how numbers “roll-over”, is there an easier way of computing this?
- Change the decimal number \(17\) into its binary representation.
- Come up with a general strategy for changing decimal numbers into their binary equivalents. Hint: Start from the left, and try to find the biggest power-of-two that you can pull out of it.