In previous posts, I looked at tally marks as a representation of quantity. Well-suited to keeping track of counts, it will yet never compete seriously with the decimal number system. The system of tally marks is simple enough and familiar enough that we can play with extension and variations of it, so we can discover freshly what is important in a representation system for quantity, so we can discover what makes a number system a good number system.
In this post, we will imagine a small and remote country where tally marks are the de facto number system. Queen Tully spearheads an effort to improve on the standard tally mark system so that keeping track of quantities and doing arithmetic operations can be made more effective. They recently saw the need for organizing the five-groups so that larger quantities could be recognized at a glance, and came up with the five-five-group as discussed in a previous post.
Jon, a mathematician in Queen Tully’s academy, suggests that a group of 5 five-groups could be marked in a way similar to the way a single five-group is marked, by a line slashed through them, connecting them.
The advantage, Jon suggests, is that we now have a general principle by which to mark larger groups: take five groups of the same type, and slash a line through them, and that way we’ve made a bigger group. Just like we organize large amounts of tally marks by grouping them by fives through a slashed line, he says, you can mark 5 five-groups with a slashed line making a five-five-group, and if you had a really BIG number, you could take 5 five-five-groups and make them into a single five-five-five-group! Moreover, if you had a REALLY BIG number, you could take 5 five-five-five-groups and slash them into a single five-five-five-five-group! Really, Jon exclaims, there is no end to how big a number you could write and still recognize at a single glance. You would never have to look at anything that has more than five pieces in it (though each of the pieces could have smaller groups inside of it), and we’re very good already in working with quantities from none to five.
Jane, another mathematician in the academy, likes the idea that Jon put forward, but doesn’t quite buy Jon’s claim that the grouping principle Jon suggests is the same all the way through. She draws four tally marks and draws a slash through them. “When I put the slash through them,” she notes, “the quantity changed from four to five. When I have five five-groups, and I put a slash through them, does the quantity also go up by one?” “No,” Jon admits. “In my scheme, the slash through the five five-groups is purely to bind together the smaller groups. It isn’t supposed to add anything to the quantity.” “Wouldn’t it be more consistent,” Jane offers, “if we stopped after 4 five-groups, and when we needed to add one more, we would then add a slash through the 4 five-groups, just like we do after 4 single tally marks?” Jon is puzzled for a moment, then speaks: “I see your point, but your solution doesn’t quite work either. When you count the slash through the 4 five-groups as a tally mark itself – adding one to the quantity – then you don’t end up with the same quantity as when you have 5 five-groups. You would just have 4 five-groups and 1.” Jane agrees with Jon, but doesn’t see why this would be a problem. “We would just need names,” she says, “for the larger groupings. Even when you talked about five-five-five-groups, the names get confusing quickly. We could call the groups whatever we want, like ‘white’ for a five-group, ‘red’ for the four five-groups-and-one,and green and lavender for the next sizes up.”
Jean, who has been listening quietly, now pipes up. “Couldn’t we adopt Jon’s idea of using the slash purely as a grouping marker, identifying five-groups and five-five-groups, and make it consistent in a different way? What if we didn’t slash after 4 single tally marks, but slashed them after five? That way, the slash would never indicate a quantity itself, and represent group-ness only.” Jon accepts the consistency of the idea, but – like Jane – is concerned that the slash mark from four to five is too ingrained, too familiar, and used in too many textbooks and old documents to be changed easily. Jean thinks about this for a while, and agrees. “OK,” she says, “we could leave the traditional way of marking fours and fives alone. After all, Jon’s idea of marking groups of five doesn’t require us to use a slash. We could draw a circle around groups of five, or rectangles. Or use a color highlighter, like Jane suggested.” Jon isn’t quite convinced that a slash couldn’t work, but has no argument to offer against using circles or rectangles or highlighting in a particular color as a way to mark groups of five smaller groups. They decide to offer a proposal to Queen Tully.
Jean presents the proposal. On the left, she shows the traditional marks, up to five. She shows the box with 5 single marks in it, and proposes that this will henceforth be an alternative way to indicate groups of five single marks. A box with five single marks in it is to be called a white group. A box with 5 five-groups in it is called a red box. A green box has 5 red boxes in it. The principle is that you can always make a larger box by surrounding 5 smaller boxes of equal size. Larger sizes will have their own names, after green comes lavender, after lavender comes yellow. But, Jean insists, these are meant as names. They don’t require actual colors, you could still write them or print them in black and white. “We just think ‘red’ is an easier name than ‘five-five-group’ and green a better name than ‘five-five-five-group.’ If you please, we’ll call this system a system of tully marks.”
Queen Tully is pleased with the proposal, but tasks the academy with working out how counting, addition and subtraction would be done in this representational system. “We should know how those work before pushing this system for wide adoption,” she judges. “And how could you do multiplication in this system?”