Notes on Representation – Tally Marks

I’m assuming you are quite familiar with tally marks:

As a system of representation of quantity, tally marks play a very interesting role.  You can view it as a number system, yet as a number system it does not compete well with the decimal number system that has spread all over the world.  Yet unlike Roman numerals, tally marks don’t appear to be on their way out.  (Roman numerals, for a long time, survived in traditional settings like cornerstones and clocks, and they are quickly losing ground on clocks.)

Tally marks have found a niche for themselves for counting.  You know the kind of situation where tally marks are very helpful, because that’s where you use them too.  You use them when items or events come in at a quick pace, and you want to keep track of the number of them.  The process of tallying with tally marks is simple: you add a tally mark for every item to be counter, and every fifth tally mark is rendered horizontally or diagonally, marking a group of five by claiming the previous four.  The process is additive: you never need to correct or strike out a previous marking.  This is in contrast with the markings that would result from counting in the usual decimal number system:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17…

No matter how well-suited (and fast) the tally system is for counting, what really allows it to exist in its niche and coexist with the decimal numbers system is the ease of translating a quantity represented in tally marks to a decimal number representation.  The tally marks up above represent 28, as you’ve no doubt already figured out.  Conversion from a decimal number to a tally mark representation is just as simple, but I doubt you have ever encountered a situation where you needed to.

This highlights two key aspects of any representational system: for it to survive, it needs to be good at supporting some important operation.  It makes no sense to look at a representational system separate from the operations it supports.  Tally marks live and die with counting.  If nobody ever counted on paper, tally marks would die out fast.
A representation only needs to be good in supporting one important operation, and can be poor in supporting other operations provided it is easy and efficient to convert in and out of the widely used representational system.  Since tally marks are easy to convert into decimal numbers, it is fine that tally marks are good for counting and perhaps not so good for multiplication.  Conversely, no matter how good a representational system is at supporting certain operations, if it cannot easily transpose back and forth into our decimal number system, it will not find much currency.

Though tally marks have little chance of displacing the decimal number system, it is nevertheless instructive to see how tally marks, as a representation, support other operations.  Can you add with tally marks?  Can you multiply with tally marks?  Just because we never do doesn’t mean it cannot be done.  As it turns out, adding with tally marks is quite convenient and fast.  Multiplication is not.

To add two quantities represented by tally marks, you simply shove them together and consider it one big collection of tally marks.  The only “gotcha” is that each of the piles of tally marks, separately, may have a number of marks that didn’t quite make an entire group of five, and yet together, there may be enough loose tally marks to now make a group of five.  If such is the case, we can slash a group of four tally marks, to make a group of five, and scratch out one of the single (vertical) tally marks to compensate for the slash tally mark we just added.  You could argue that this process is quite a bit simpler than adding of decimal numbers.  A similar observation can be made for subtraction.  One area where tally marks lose out is in comparing large numbers.  Though 2008 and 2009 look very different when expressed in tally marks, 2009 and 2004 differ only in the number of completed 5-groups, and there are some 400 of those – hard to distinguish 400 five-groups from 401 five-groups.  The simplicity of tally marks as a representation of quantity works well for small numbers but becomes a liability when it comes to large numbers.  Tally marks have their niche: I venture to guess they will continue to do well in that niche, but are unlikely to ever break out of it.