In this blog, I’ve surveyed a number of different representation systems for quantities, including the decimal number system, the number line, tally marks, and prime bags, and I’ve tried to see and say something interesting and fresh about each. When I think of the suitability of different representations for quantities, I tend to think of their suitability for operations like counting, adding, multiplying, comparing. That is, I think of representations in relationship to common or important actions. I want to see if in a given system of representation it is easy to see which quantity is larger, and I want to see if in a given system of representation it is easy to multiply.
Yet there is one important action that I had completely overlooked until I started to look into tally marks more deeply. And that is the action of copying. For sure, it is not common to think of copying as having anything to do with arithmetic or mathematics or with representation of quantity. And indeed, for many representation systems, copying of quantities is largely transparent. In the decimal number system, you need a lot of digits to make copying a number something you’d even be concerned about. But even in the decimal number system, we resort to various tricks when the numbers get larger. If I asked you to write down a phone number, 3105551212, so you could call it, it suddenly wouldn’t look so easy. We’re used to seeing a phone number split in smaller chunks, 310-555-1212, and we do a similar thing with social security numbers. You can copy down the un-split large number, but you might find yourself doing it by tracing the number with your finger. Using your finger to mark progress is not something to be ashamed about – I see middle schoolers use their fingers routinely when trying to find the median of a set of numbers. It is simple, and it works.
Each system of representation has its own issues with copying, some trivial, some not so much. The act of copying a decimal number is quite straightforward: you can copy one digit at a time, while tracing the digits carefully. As I mentioned, it is rarely something we pause about and mention its difficulty. It also helps that most numbers we encounter don’t have that many digits. It is rare to be asked to write down 30 digits of or of . And when you encounter really big numbers, such as the speed of light in miles per second, or Avogadro’s number in chemistry, you use an approximation with few “real” digits. For the speed of light, you use 187,000 miles per second, or you use 300,000 kilometers per second. The more precise 299,792.458 kilometers per second is not something you’d likely be asked to quote (i.e. copy).
If you represent quantity by a distance on the number line, copying a number comes down to approximating the distance on a different line. The number 1.5 is easy to copy precisely, the number 1.5178 not so easy.
With tally marks, copying isn’t particularly hard, but it can be quite cumbersome. A hundred tally marks, done as 20 groups of five, can be copied one group at a time, while tracing your progress with your finger or by other means. When larger groupings are allowed, as in our fanciful proposal here, it may be easier to keep track, but it still requires a considerable amount of writing to copy the number from one place to another. Of course, the niche in which tally marks are used is almost exclusively for counting. You rarely copy a number in tally mark representation, you convert it to a number in decimal representation. The cumbersome nature of copying tally marks is surely one of the reasons its use is restricted to such a narrow niche.