Tally Marks and Roman Numerals

In earlier posts on tally marks, I mentioned that one of the drawbacks of tally marks as a representation system for numbers is that it isn’t easy to copy numbers from one place to another.  In our normal use of tally marks, that isn’t a problem, because we tend to only use tally marks in the process of counting, and then we report the count as a decimal number.  So if we were counting the cars that passed by, using tally marks, and we ended up with this:

tally marks

we wouldn’t report on it by giving an exact copy of this, we’d report it as 28. The awkwardness of copying is one of the reasons why tally marks, if viewed as a representation system for quantity, is only used in a relatively narrow niche.

In one of the posts about tally marks, I was rather dismissive of Roman numerals, saying that whatever niche they still occupied was diminishing rather fast.

I’ve since come across a hypothesis about the origin of Roman numerals that gave me a renewed respect for them.  Though I have no way of assessing the historical accuracy of this hypothesis, it does allow us to see Roman numerals as a solution to a problem.  The hypothesis is that Roman numerals started out as a tally mark system, and then was abbreviated in a way to make copying numbers more practical.

roman-ordinals

The top row of the diagram shows a system of tally marks, marking a count of sixty.  After four vertical marks, the fifth mark is distinct, represented by a V.  Each second V is changed to an X.  Each fifth X is marked as an L instead.  As a way to tally, this system has both advantages and disadvantages to the schemes I’ve posted about before, but it fits squarely as a tally system.  If this is the only system you’ve got -your only way to represent quantity – then if you want to tell somebody else “how many” you would need to copy the entire top row.

The hypothesis that I came across suggests that Roman numerals act as an abbreviation of this tally system.  The next three rows in the diagram show how this works for the quantity eleven.  First, eleven tally marks are shown, highlighted in yellow.  In a pure tally system, this is how you mark eleven.  In the row after, I’ve highlighted just the eleventh mark.  If we had a way to show its location in a unique way, then we wouldn’t have to write all the preceding marks.  In the next row we highlight the preceding X in teal.  The location of the yellow-highlighted vertical mark smack to the right of the teal-highlighted X tells us what we need to know to appreciate its position in the stream of tally marks.  Reporting “XI” is sufficient; I don’t need to report the entire stream “IIIIVIIIIXI”.

In the next set of rows, I look at the quantity nine.  First, in true tally form, the representation for nine would be “IIIIVIIII”.  Then, the ninth mark alone is highlighted, in yellow.  In the row after, we show its distinctive neighborhood, which allows us to give an abbreviated form that is yet unique: “IX”.

The next examples show that this idea of neighborhood is powerful, but shouldn’t be taken too literally.  For the quantity sixteen, I first show the position of the sixteenth mark, and then the (proximate) neighborhood that makes it unique.  If you imagine squinting a bit, and ‘dropping’ some intermediate vertical marks where they aren’t really needed, you could justify reporting the sixteenth marker as “XVI”.  If you don’t like the idea of squinting, you could say that the I is positioned relative to the immediately preceding V, and that that V is positioned in turn by the preceding X.

For the quantity thirty nine, the vertical mark “I” is positioned relative to the “X” mark immediately following, and that “X” in turn is characterized by its position to the left of the “L”.  This would justify reporting thirty nine as “IXL”.  Now, in traditional Roman numeral notation, thirty nine would be rendered “XXXIX”, which you can justify just as well.

Where are we left with this?  I really like the kind of thinking that came up with this hypothesis.  Completely independent of whether the historical development was indeed as the hypothesis suggests – and we may or may not ever find out – the hypothesis looks for a problem for which the Roman numerals were a solution.  That strikes me as very powerful.

In math education, so often we present stuff, whether methods, techniques, or notations, as if they just fell out of the sky.  Many kids learn better and retain better if they have a sense of what the problem environment was to which the technique or the notation was a response.  We can appreciate it as a rich heritage, a step in the history of thinking.

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One Response to Tally Marks and Roman Numerals

  1. Robert says:

    The way you put it, it makes Roman numerals look like a cool and clever idea. Which they are, compared to tallying.
    I believe that the “difficult” part of Roman numerals comes from the “subtraction” principle: 4 is IV, 6 is VI, maybe tricky to keep straight. If 4 were always IIII, and 6 always VI, then maybe they would be easier to keep straight. Also, addition would be almost ridiculously easy: to add two numbers, just mash them together. I randomly chose 13+19 for an example. Mash XIII and XVIIII together to get XXVIIIIIII. Clean this up by rubbing out VIIIII and writing another X. (Easy with chalk.) You then have XXXII (32) before you. I’m sure that medieval peasants were very happy with the simplicity of this system, as from what I’ve read, this is not too far from what they used in real life.
    You might also want to read about Suzhou numerals, which look like a *very* abbreviated form of Roman numerals, but can also handle decimals.

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