manual computing devices
goodmath | October 12, 2006To do multiplication with your fingers in binary is very easy: it's just a mixture of addition and bit-shifting. The only real trick is memory: to multiply a×b, you need to remember the binary digits of both x and y, which can be a bit of a trick for 10 digit binary numbers.
The trick that I like is to use coins. Lay out a bunch of coins: one for each binary digit of a, and one for each binary digit of b. Put two lines on a piece of paper: one line will be the ones, the other will be the zeros. So, for example, to multiple 47 times 24, you'd start with the following:
The coins on the paper…
goodmath | October 11, 2006There is another way of doing math on your fingers, which gives you a much greater range of numbers, and which makes multiplication particularly easy. It's a bit more work to get used to than the finger abacus, but it has a lot less limitations. Someone in the comments of the finger-abacus post mentioned that they do something similar.
The methods for binary fingermath that I'll describe are my own creation; so if you think they're ridiculous, the blame is entirely mine. I know other people have come up with similar things, but this is my own personal variant.
The idea is to use your…
goodmath | October 9, 2006Suppose you want to do some math, but you don't have an abacus handy. Oh, the horror! What do you do?
No problem! Your hands make a *great* two-digit soroban-type abacus. The four beads on the lower deck are your four fingers; the bead on the upper deck is your thumb, as illustrated in this diagram (with apologies for my terrible artwork):
So the numbers from one to nine look like:
To get two digits, you use your right hand for the ones, and your left for the tens. So, for example, let's look at a simple addition:
Once you know the abacus, doing this with your hands is pretty simple. It'…
goodmath | October 2, 2006Doing square root on the abacus is a lot like doing it on paper. The big difference? It's actually *easier* on the abacus. What I find pretty cool is that I'm a rank beginner at the abacus. I never actually tried to use one before I started writing these posts. But I can do that root *faster* on the abacus than I can on paper.
The one difficult step in the paper square root is guessing the approximate digits; as you get beyond the third or fourth digit, the numbers start getting a bit large, and it can be hard to guess the correct estimate. On the abacus, you can very rapidly do repeated…
goodmath | September 28, 2006To do a square root on an abacus, you use partitions to do a paper algorithm for square root using the abacus. The catch is that most people don't even *remember* how to do square roots on paper, if they ever learned it at all. (In fact, in school, *I* didn't learn the classical paper algorithm; we never really did roots on paper; the closest we did was using square root as an example of Newton's method. Like so much of my basic math, I learned this from my father.)
So, for your entertainment and edification, today, I'll describe the classical algorithm for computing square roots on paper. I…
goodmath | September 25, 2006Now we're going to try something challenging on the abacus: *division*. Like multiplication, abacus division is close to the way you'd do it on paper. But just like doing paper division is trickier than paper multiplication, abacus division is tricker than abacus multiplication. But the technique that is used to do division on the abacus is an important fundamental one: it's what makes it possible to use the abacus for more advanced operations, like roots.
Before going into the algorithm, there's one important new technique that we need, called *partitioning* on the abacus. The idea is that…
goodmath | September 21, 2006Once you can add on an abacus, the next thing to learn is multiplication. Like addition, it follows pretty closely on the old pencil-and-paper method. But it's worth taking the time to look closely and see it step by step, because it's an important subroutine (to use a programming term) that will be useful in more complicated stuff.
Just for clarity, I'll write out the basic pencil and paper algorithm:
1. Write down a "0" for the initial value of the result.
1. For each digit si in the *second number* number, from right to left
1. For each digit dj in the first number, from right to left…
goodmath | September 19, 2006If you want to talk about mechanical computing tools, you can't ignore the abacus. It's the oldest computing tool in the world; and it's still very commonly used. It's also about as different from the slide rule as you could imagine. The abacus is really fundamentally an addition device; the slide-rule is fundamentally a multiplier. And the slide rule is very complicated - all those different scales, in logarithmic relationships; the abacus is thoroughly simple - just beads hanging on wires. But don't let that fool you: the abacus is is a remarkable device, which is capable of a really huge…
goodmath | September 13, 2006Slides rules are actually astonishingly powerful things. The simple slide rule does multiplication and division using the C and D scales; strictly speaking, you can have a basic rule with nothing but C and D. But you almost never see a rule that simple. (The only one I've ever seen with only the two scales was a circular rule used as a promotional giveaway.)
The other scales are where things get a bit complicated; but it's a lot of fun to figure out how they work, and to see how much you can actually do with, basically, two attached rulers with a bunch of different markings.
As an example,…
goodmath | September 12, 2006Several people in the geekout thread asked me to explain how a sliderule works, and I've been meaning to write a couple of article about manual computing devices. So I thought I'd do it. There's a nice slide-rule simulator at [Derek's Virtual Slide Rule Gallery][sr], which is what I used to generate the images in this article.
I know a lot of people think that the idea of learning to use something like a slide rule is insane in an age of computers and calculators, and that this is a silly thing to post about. But I really *love* slide rules, and not *just* because I'm a geek. Slide rules make…