## 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.

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### 10 Responses to Notes on Representation – Tally Marks

1. Mike says:

Just passing by.Btw, your website have great content!

2. I hope you will have a look at Funforms, a tally mark, place order, binary number system. You can see and listen to a narrated power point presentation at:
http://www.authorstream.com/Presentation/joxl-1251628-funforms/

• Bert Speelpenning says:

Joel,
There is a long history in mathematics for people in different professions making strong contributions. One of the most famous is Pierre de Fermat, who was a busy attorney, for whom mathematics was merely a hobby.

From looking at your presentation, I see essentially two things there:
1. a very thorough treatment of arithmetic in the binary number system, covering addition, subtraction, multiplication and division – both for whole numbers and for what you call fractions but which I’ll call binary terminating fractions.
2. a pictorial representation of the binary numbers, involving flags on a staff. The characteristics of the staff representation are that positions are equally spaced and that each position has either zero or one flag on the right, though on a temporary basis, staff positions can have multiple flags on the right or they can have flags on the left indicating negative values. One of the positions on the staff is associated with the unity position and all other positions have double the value of the position above and half the value of the position below. In this pictorial representation, “F” represents the value 3, assuming the unity value is at the top, and assuming that the lowest horizontal bar of the “F” is exactly one position below the top one. “L” would represent the value 4, and “E” would represent the value 7, making these same assumptions.

What the pictorial representation allows you to do is to imagine arithmetic operations as a series of simple actions on the flags. No addition tables or multiplication tables are needed, just some simple recipes as to what to do with a temporary situation of two flags in a single flag position.

What I’m curious about is how you would mark the unity value in your system, and how you would deal with very large numbers. A pictorial representation of 2^10+1 and 2^11+1 would look almost exactly the same. In the much maligned system of Roman numerals, one hundred and one (CI) looks very different from one thousand and one (MI). In our normal decimal system, 10001 looks somewhat different from 100001, though 1000000001 doesn’t stand out as different from 10000000001. At the very least, though, in our decimal system the numbers that we are likely to encounter in our daily lives all look quite distinct. For numbers outside that range, we use something called scientific notation, which on a calculator may show up as 3.4E8. This is intended to mean that the number isn’t really 3.4, but what you get if you move the decimal point to the right 8 places, hence 340000000.
In a binary system you can do the same thing with a special notation that indicates that the unity position is some number of places above or below the unit position shown.

In the context of computer science, people often represent large binary numbers by breaking the binary number up in groups of 4. Instead of writing 100100111000, you would break it up s 1001 0011 1000 and write each of these as a “hexadecimal digit”, here 934. The translation between binary notation and hexadecimal notation is so straightforward that you can think of the hexadecimal notation as a shorthand for the binary notation rather than a separate system with radix 16.

Since there are 16 hexadecimal digits, and normal digits only carry us from 0 to 9, we need six more, and traditionally the letters a,b,c,d,e, and f are used for that. So fa8 is shorthand for 1111 1010 1000 or 111110101000. As an alternative, I’ve seen a system of flags before, but different from your scheme in that ones are marked with a flag on the right and zeroes are marked with a flag on the left. What people quote as an advantage of that scheme is that you can be sloppy in writing it, not needing lined paper to pull it off. Instead, you are using the flags on the left to mean negative values.

Your use of the flags on the left for negative values allows for number representations other than the ones you describe. Instead of writing 15 as four flags on the right, you could represent it as 16-1, with one flag on the right (16) and one flag on the left (-1). Implicit in your representation is that flags are either all on the left or all on the right, and any other situation is what you call ‘temporary’, just like having two flags in one position is something you only allow during operations.

Thank you for an interesting take on the binary number system.

• WOW! Thank you so much for looking and for commenting. You are correct about using FF as an octal or hexadecimal [or 32 or 64 bit] unit. We always figured we could demonstrate that by some sort of link across the top of two or three or more staffs. For even larger numbers, there is no reason not use a FF exponent. So that way E (7) to the E power would be 7 to the seventh power. If that does not get you to large enough numbers, you could have exponents of exponents or at least mutiples of exponents.

We were not so interested in very large numbers, but in helping people see operations from this very simple, totally mechanical viewpoint. This gives the learner a second language to look back at what s/he had to learn by rote before the possibility of critical thinking was available. [I have learned alot aabout what I already “knew”.]

Also we don’t use zero as a number. Zero is an empty staff or an empty position, a much more representational feature, I believe.

I want to introduce FF into schools [and study what effects, if any,]. I think there are people with dyscalcuia who might be able to understand FF and then, maybe, move forward into our more complicated decimal numeral system. One of my kids at his age 15 had a Eureka moment with fractions and said to me, “Oh! That’s why half of 1/4 is 1/8.”
I said, “You didn’t know that???!!!”
He said, “Oh yes, I did know that, I just didn’t know WHY.”
So he got an insight that had previously been unavailable to him. {And he is a bright fellow.]

I look forward to further correspondence with you and some good ideas about how to move forward.

js

• another important advantage to use of FUNFORMS is the ease and transparency with which operations take place and how seeing that happen reflects on the understanding of the same operations in the decimal system. I called that the second language effect.

• Bert Speelpenning says:

When it comes to how to get ideas integrated into a school system, that’s the 64million dollar question. I don’t have any particular track record in this.
Depending on one’s preference, one can start with interesting an individual teacher, or an individual textbook publisher, or somebody who puts on professional development for teachers.
For any of these to get very far, you are going to have to get specific. In which grade would this be introduced, and what is the sequence of instruction, exercises, etc. that are designed to get to the understanding and the skill. Addition, subtraction, multiplication and division are not usually taught all at the same time, so how would the Funforms stuff be interwoven with all the stuff that’s currently in the various curricula – specifically, how would you interweave the binary stuff with decimal stuff?
Even though any specific proposal you make will still have to be adjusted and completely reworked once somebody expresses a cautious interest, without such a specific proposal, you may find that people will have trouble in making the translation from what you have sketched to some set of actions for them to take in their classrooms.
I mentioned professional development for teachers because it occurs to me that the first people who would have to really understand the binary system are the teachers themselves, and finding a way to have them learn and understand the intricacies of a number system is not a trivial thing.