Abacus and Slide Rule

Ye Olden Tools of Yore

I’ve been meaning to write an Abacus post for years. I used one in my first job, back in high school, and they’ve appealed to me ever since. Many years ago I learned there were people who had no idea how an abacus worked. Until then I hadn’t internalized that it wasn’t common knowledge (maybe a consequence of learning something at an early age).

Recently, browsing through old Scientific American issues before recycling them, I read about slide rules, another calculating tool I’ve used, although, in this case, mainly for fun. My dad gave me his old slide rule from when he considered, and briefly pursued, being an architect.

So killing two birds with one stone…

It was back in 1998 or 1999, I think. We threw a large Halloween party. (The whole nine yards with decorations: dry ice fog, orange and black colored food.)

But the party was too big. We’d invited people from three separate social groups, and they didn’t blend well. (Not foreseeing that possibility, we didn’t plan any mixer activities.)

Therefore the party fragmented — one might even say coagulated — and no one mingled. (Minnesotans can be shy.)

The little groups enjoyed themselves okay, but the party had no flow (lesson learned). A smaller clot was a couple, friends of my wife that I’d gotten to know.

I noticed them fiddling with my abacus. They seemed engrossed, but puzzled. It turned out they had no idea how it worked, so I gave them a demo.

Their reaction seemed on the, “Whoa! Cool!” side, so with that in mind, here’s how an abacus works:

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At heart, an abacus is a scratchpad (although “memory register” is a more exact analogy).

An abacus in the starting (aka reset, clear, “all zeros”) state.

Each column of an abacus represents a digit. The one shown above (and in all examples here) has 13 columns, so it can represent a 13-digit number (or multiple numbers with fewer digits).

Each column consists of a set of seven “stones” that slide up and down on a rail. The lower set of five stones each have a value of one. The upper set of two stones each have a value of five.

The condition shown above is the “all zeros” state. When the lower stones are down, they are not counted. When the upper stones are up, they are not counted.

What is counted is the stones touching the middle crosspiece.

The configuration here is designed to show off the full range of normal digits as well as some transitory states.

For example, notice how there are two ways to represent 5 and two ways to represent 10.

In both cases, the left-hand one is the “normal” way. When five stones accumulate in the lower part (as in the right-hand examples), you usually transition that to left-hand version.

As the left-most column shows, digits in an abacus can represent values up to 15. Values over 9 are usually carried to the column to the left.

Essentially, an abacus works almost exactly the same way as you do math by hand with paper and pen. Both deal with digits, add and subtract, carry and borrow. The abacus just acts as a memory register so you’re not scribbling all over the paper.

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Here’s a very simple step-by-step example that illustrates adding along with a little subtracting.

We’re going to start with 27 (two right columns), and we’re going to add 48 (two left columns).

Normally, we wouldn’t bother showing the 48, but doing so here provides the example of a little subtraction.

The first step is to add 8 to the 7 (note how we also subtract 8 from the left columns)…

In abacus terms, 8 is 5+3, so we’ll first move a 5-stone down, which results in a total of 12. That has to be carried to the next column to the left. (Note we also removed 5 from the left columns.)

We carry the 12 by resetting the two 5-stones and adding a 1-stone in the next column:

Now we add three 1-stones to complete the addition of 8 (and subtract three from the left side):

There are now five stones in the lower half, so we transition that by resetting those five and sliding down a 5-stone:

(As a self-check, 27+8=35, and 48-8=40, so we’re good so far.)

Now all we have to do is add the 4 to the 2 (which is now a 3 from the carry).

But there’s a problem: there are only two stones remaining, and we need four. So we’ll two-step by taking first taking two stones:

Which gives us five that we transition to the normal form:

Then we can do the other two:

And we’re done. (27+48=75)

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Larger numbers are just more of the same. A typical use involves summing a list of numbers and only the running total is maintained on the abacus.

Subtraction is a similar process, except sometimes we have to borrow from the left rather than carry.

It’s fairly easy to multiply any number by a small number, but multiplying two large numbers gets a bit tricky. Even multiplying by, say, 25 is better done as two operations of multiplying by 5.

I may post a Sidebar getting into more examples, including multiplication, especially if there’s an interest, but for now, that’s the basics of how an abacus works.

Note that the 5+2 design isn’t the only design. A fairly common alternate is 5+1, with only a single 5-stone.

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While an abacus is discrete (it uses digits), in contrast, a slide rule is analog — it uses a logarithmic scale.

Slide rules have multiple scales along their length. How many, and what type, varies. Mine (above) has the very standard C/D scales as well as the CI (Inverted) and CF/DF (Folded) scales.

These are all variations of the C/D scales. Note that the C and D scales are identical, and so are the CF and DF scales.

The C/D scale is a logarithmic scale that runs from 1.00 to 10.0. If you recall your high school math, logarithms transform multiplication into addition:

If we take two ordinary foot-long rulers, and place them end-to-end, we have a two-foot long ruler. We’ve added them.

If we place them next to each other, scales touching, and slide one 6 inches along the length, we are again adding. For example, the 2 of the ruler we slid now lies opposite the 8 of the other ruler: 6+2=8

However, when we do this with logarithmic scales, we end up multiplying.

Two logarithmic scales that run from 1.00 to 10.0, placed end-to-end, create a logarithmic scale that runs from 1.00 to 10.0 and 10.0 to 100, which comes from multiplying 10×10.

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One thing about slide rules: They have an accuracy of about three digits, at best (two digits in some cases).

In the days before calculators they provided a “good enough” handy multiplication device (“handy” in all senses of the word).

And, the truth is, for a lot of work, two or three digits of precision are quite good enough, and there are ways to refine an answer if needed.

Another property of slide rules is that, when set for a specific answer, they actually provide a continuum of answers along their length.

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Here’s an example:

The slide rule above is set to 1/12 (on the lower scales labeled C and D at the far left).

The 1.00 of the C scale is directly above the 1.20 of the D scale, but we can call it 12.0, if we like. This brings up another aspect of slide rules: we have to keep track of our decimal points.

The reticle (sliding piece with vertical hairline) is set to 1.50 over 1.80, but since we’re moving the decimal point (1.20 = 12.0), we see it as being 1.50 over 18.0 (which is the same as 1/12).

Over on the far right, you can see 2.00 over 24.0, which is also 1/12. That fraction is expressed along the entire length all the way to the end:

On the left of the image above, 5.00 over 60.0 (keeping track of our decimal point), is the same as 1/12, and so is 6.00 over 72.0 (or 7.50 over 90.0).

Everywhere you look, 1/12.

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The CI scale inverts the C scale, and provides a quick inverse for any number on the C scale.

For example, in the image above, on the left side, the red 2 is directly above the 5. The inverse of 5.00 is 0.20, and the inverse of 2.00 is 0.50.

The CF/DF scales just fold the C/D scales, which make it easier to deal with numbers that are on the edges of the C/D scale. Rather than trying to read a value of  the ends of the ruler, you can be comfortably in the middle.

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An abacus might seem like a cumbersome tool, but it’s surprising how a bit of practice makes it fairly easy and fast.

That first high school job was a retail position, and at the end of the night we (two of us) had to “count our banks” (add up the cash in our cash register drawers and reconcile it with our sales).

The boss, a Chinese man, had a mechanical desk calculator — the kind with 10 buttons for each digit and a level you pulled to cycle the mechanism. (This was 1971 or so.) Our boss also had an abacus.

So I’d take the abacus, and my co-worker would take the adding machine, and we’d race to see who finished first.

Believe it or not, sometimes I won. When I lost, it wasn’t by much.

Stay calculating, my friends!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

5 responses to “Abacus and Slide Rule

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