Keep posting.

Danial from TI 89 Titanium

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

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.