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:

**§ §**

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

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.

**§**

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)

**§**

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.

**§ §**

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**.

**§**

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.

**§**

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**.

**§**

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.

**§ §**

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!*

∇

December 20th, 2019 at 1:41 pm

892 posts, and counting (7 to go). Also, tomorrow is the Winter Solstice, so yay!

December 20th, 2019 at 2:05 pm

A slide rule was required for my high school physics. The good old days. 🙂

December 20th, 2019 at 2:24 pm

The

olddays, anyway… 😉December 21st, 2019 at 10:47 am

By my high school days (early 80s), the calculator was firmly established, although my chemistry teacher impressed upon us how easy we had it compared to his own slide rule days.

December 21st, 2019 at 11:34 am

Yeah, it’s one place that kicked off the debate about how important it is (or isn’t) to teach fundamentals and learning to think about numbers. (Another was the debate about cash registers telling cashiers how much change to give rather than, as I learned it, how to figure and count out change.)

I tend to be strongly in favor of fundamentals, but I also have the ethic of not bothering to learn or memorize things I can easily look up.

But I do think learning how a slide rule (or abacus) works can be helpful. But using one regularly? No thanks! Pass me that calculator, please!

November 24th, 2020 at 8:18 pm

Seeing increased activity on this post lately. I guess people really do find it interesting.