How to Add and Subtract with a Counting Board

The mathematical notation and the techniques we use to compute with it are so concise and removed from the physical world that it is easy for mathematics students to lose awareness of its meaning when using it, so I think it is a useful thing for helping students understand the meaning of arithmetic to learn some more primitive methods for doing it. One such method is to use the counting board. (See my articles on Egyptian arithmetic, Egyptian Addition , Egyptian Subtraction , Egyptian Multiplication , and Egyptian Division , for another interesting method.) Writing was not such an easy thing to do in early times, various physical devices were used to make calculating easier. The counting board is a nice simple one that is interesting to learn about. Until comparatively recent times, most merchants in Europe used it, and even when the methods we commonly teach for arithmetic were introduced many teachers of arithmetic thought it was useful to first learn it to help with understanding.

The picture above shows a competition between the counting board and arithmetic done by paper and quill arithmetic. (I didn't say "paper and pencil" because I don't think they had pencils in those days.) It is from G Reisch, Margarita Philosophica (1508). The counting board is quite a simple device. It is a way of extending the method of adding by piles of pebbles to larger numbers without having to use too many pebbles. The idea is that pebbles on one line stand for just what they are, one pebble, but pebbles on other lines stand for different numbers of pebbles, 10, 100, 1000 pebbles. You can make a counting board for yourself by drawing lines on an ordinary piece of paper like this.

Alternatively if you want you can print the above image. To do this either download it and print it from any application you have that reads gif files or open it in your browser and print it. Whichever way you do it you must choose landscape orientation in your printing setup.  For counters you can use any small item that you have at least 40 of, pebbles, coins, poker chips, paper clips, etc.. I find the little glass stones that are used for flower and floating candle arrangement kind of nice. In the middle ages and early renaissance counting boards like this were widely used by merchants. There were made out of various things. Some were made of stone with the lines carved in them. Others out of wood with the lines drawn in chalk for temporary use or painted on for more permanent use. Portable counting boards were made by embroidering the lines in cloth. The traveling merchant would then use the cloth as a table cloth when conducting business.

Counters placed on the bottom line would represent one, those on the next line ten, the next one hundred, and the next one thousand, and they probably didn't very often need to add numbers larger than a thousand. I have written the values of the lines in Roman numerals, because most of the people who used these counting boards would not have known about the Hindu-Arabic number system. Also it is really easier to use the counting board with Roman Numerals than with our present system. To get a better feel for what this was like, it is helpful to learn something about the Roman Numerals. Here is a chart with the values of the Roman Numerals.

              I - 1
              V - 5
              X - 10
              L - 50
              C - 100
              D - 500
              M - 1000

To write a number in Roman Numerals, mostly you just use however many of each of these letters it takes to make the number using as few as possible letters and writing the letters in decreasing order of size from left to right, so for example thirty two would be written XXXII and two thousand six hundred eighty seven would be MMDCLXXXVII, sort of like counting money. It is made slightly more complicated than this, though, because in order to use fewer symbols they introduced some shortcuts. They introduced the convention that when you put a letter representing a smaller number to the left of one representing a larger number it subtracts instead of adding, but you are only allowed to do this with one smaller valued letter. This way you never have to use more than three of the same letters in a row, so for example instead of write IIII for four you write it as IV. There are some subtleties to the system with regard to this convention and they weren't entirely consistent about their rules. For a more thorough discussion of Roman Numerals see Paul Lewis' page  Roman Numerals and Dates. There are a lot of other pages as well on the web about Roman Numerals, so if you are interested in further information you should be able to find a lot by doing a search. For our purposes here with the counting board I'm going to use a simplified version of the Roman Numerals without the subtracting aspect, so it will be okay to use four of the same letters in a row, so four will be written IIII and nine will be written VIIII. But if you are used to normal Roman Numerals and want to write four as IV and nine as IX that's okay too. I will also write the Hindu-Arabic numeral in parentheses, because I know that in our modern world most of us a lot more familiar with them than the medieval merchants were.

The simplest way to represent a number on the counting board would be to simply place one counter on the ones line for each one, one on the tens' line for each ten, one on the hundreds' line for each hundred, and one on the thousands' line for each thousand, so MMMDCCCLXXIIII (3874) would be represented like this.

But if they once did it this way someone at some time got tired of using so many counters and came up with a way to use less of them by using the spaces between the lines to represent 5 times the value of the line below, so a counter placed between the I and the X line would represent V (5), a counter between the X and the C line would represent L (50), a counter between the C and the M line would represent D (500), and a counter above the M line MMMMM (5000). This also corresponds better with Roman Numerals, because the old way would be like write MMMDCCCLXXIIII (3874) as MMMCCCCCCCCXXXXXXXIIII, more like the Egyptian system. (See Egyptian Addition.) So using this system the same number MMMDCCCLXXIIII (3874) would be represented instead like this. The counter between the M line and the C line would correspond to the D and the counter between the C and the X line would correspond to the L and if there were any V's in the number, counters for them would go between the X line and the I line. You can kind of think of this kind of representation as like counting coins except that instead of having different coins for the different values, the values of the counters change when they are on different lines, so it's like pennies become dimes when they are on the X line and then become dollars when they are on then C line and 10 dollar bills when they are on the M line. You could do this whole thing with different coins for the different values, but it is less different things to carry around to use the same counter but just change its location to give it a different value.

So now that we know how to represent numbers on the counter board we can start talking about adding. Now let's suppose you want to add MMDCCXXXVII (2737) to MMMDCCCLXXIIII (3874), the number represented above. To do this you would represent the other number MMDCCXXXVII (2737) on the other side of the counting board like this.

Then you push the counters from the left side over to the right side so that they are all together to represent the answer to the addition problem and the board will look like this. The next thing to do is neaten it up so that it will be represented by the least number of counters as possible. To do this you use the rules that there can't be more than 4 on the lines or more than one in the spaces. I suggest that you try this on your on counting board and follow along with my steps and then check your answer with mine.

1. Working from the bottom up first you notice that there are more than four counters on the I line and five of them is the same as one on the space above, so you grab a hold of five of them, (One fun way to do this is to grab one with each finger.) and put aside all but one of them and shove the one up to the space above the I line.
2. Now you have two in the space, which is too many. Two fives make a ten which is the value of the next line, so put aside one of them and promote the other one.
3. Now there should be six counters on the X line, and just like in step one, five on the X line is the same as one on the space above, so you can again grab a hold of five of them and put aside four and promote the remaining one.
4. Now like in step 2 there are too many in the space between the X and the C, so since 2 fifties, make one hundred, you can put aside one and promote the other.
5. Now just like in steps 1 and 3 there are too many on the hundreds line, so you need to take a hold of five of them and promote it to a five hundred by putting away four and pushing up the remaining one.
6. This now makes three five hundreds. Two of them make a thousand, so put away one of them and promote the other to the thousands line.
7. Now you have six on the thousands line. Now I'm not so sure whether a medieval merchant would really do anything about this, since there is no separate Roman Numeral symbol for five thousand, so you can decide for yourself whether you want to do anything about it. To be consistent with what we have done before, though, you would again take five of them off of that line and put four away.

So now for the final answer. It should look like this if you didn't clean up the thousands. And this if you did. In either case, you should get MMMMMMDCXI (6611) for your final answer.

Now let's suppose you want to subtract MMDCXXXVIII (2638) from MMMMCCCCXXI (4321). To do this you would represent  the two numbers on the two sides of the counting board like this.

Then the idea is match up the counters on the two sides and remove them together, but you can't quite do that in this example, so you have to replace some of the counters on higher value positions with the equivalent number of ones on lower value positions. So to have enough in the bottom line you need to break up one from the space above it, but there is nothing there, so you have to go to the next line and take one of the counters from the X line that represents ten and represent it as instead one five and five ones. But then you also don't have enough in the X line, so you have to look to the line above it, which is also empty, so you have to take one of the counters from the C line and break it into one fifty on space above the X line and ten tens for the X line. Now the C line is all right, but there aren't enough in the space above it, the five hundreds space, so you have to change one of the counters on the M line, representing one thousand to two counters on the line above the C line, representing two five hundreds. After you get done with this regrouping the counting board should look like this. Now it is ready for you to do the subtraction by removing counters in pairs, so you just have to remove the counters in pairs from the two sides until there is nothing left on the right side. Then what is left on the left side will be your answer. A nice way to keep track of this is to remove the counters with your two hands, working together, with your right hand removing the counter on the right and your left hand removing the one on the left. When you get done the counting board should look like this representing MDCCLXXXIII (1783) for your answer.