For setting up a local development environment, see: Setup Page
More details about the architecture and design can be found here: Architecture
This document refers to number lines that can be found here: FlexibleTenNumberLines
The calculator itself is a browser app currently hosted here: http://www.webaddict.games/flexibleten/index.html
The Flexible Ten Calculator is a proof of concept calculator designed to demonstrate that the apparent special behavior of ten (10) in mathematics is not a difference rooted in mathematical logic, but is instead a notational artifact generated by where we arbitrarily decide to begin repeating digits on a number line. The calculator allows you to designate multiple number lines with differing numbers of repetitive digits and does quantitatively accurate math within the defined number line. It demonstrates that any quantity defined notationally as ten "10" will behave with "powers of ten" and retain quantitative accuracy, only the notation will be different.
Flexible Ten Mathematics is an extension of the fact that the logic of mathematical operations is not determined by quantities, and in fact, it is the other way around: quantities are determined by mathematical logic. This fact is, of course, elementary to higher math, take for example the following expressions:
((axa)^a)/a
((bxb)^b)/b
((cxc)^c)/c
Any of these terms can be defined as any quantity and their logical behavior remains identical, only the quantity resultant from the operations changes. This is, of course, why quantities can be designated as letters in math without their behavior in the math changing. For example:
a = 10, b = 5, c = 2 vs. a = 5, b = 2, c = 10
Changing the quantitative value of the variables has no effect on their logical behavior in the math; the expressions stay the same. "a" defined as 10 behaves identically in logic to "a" defined as 5. Again, this is an elementary fact of mathematics, but it logically follows that any apparent difference in logical behavior between two different quantities through the same function is, in fact, not logical but instead notational. Flexible Ten Mathematics points out that the fact that any variable can be given any quantity value applies to the number line as well. We can define any number line we wish of arbitrary length with digits defined as any quantity of equal magnitude increase and mathematical logic will not change when applying that number line to calculations; quantifications will remain accurate so long as we stick to the defined number line when doing our calculations. Further, what ever point we designate as the repetition point on that number line will behave notationally as 10.
For example, take two number systems: a system based on ten numerals and a system based on four.
System one: 0,1,2,3,4,5,6,7,8,9.
System two: 0,1,2,3.
In the first system the quantity normally defined as ten is depicted with the symbol 10 because that is where the number line begins repeating; in the second system the quantity commonly defined as four is depicted with the symbol 10 because, again, that is where the number line starts repeating.
Now lets do some basic math starting with addition.
System one: 10 + 10 = 20 ===> 1 1 1 1 1 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1 1 = 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
System two: 10 + 10 = 20 ===> 1 1 1 1 + 1 1 1 1 = 20
1 2 3 10 11 12 13 20
In system two you can see that choosing to repeat the number line every fourth position has caused the quantity typically denoted by the symbol "4" to be denoted by the symbol "10" instead, and this has caused it to behave identically in notation to the positional value typically defined as "10". Further, this behavior is perfectly accurate quantitatively so long as we stick to our new number line when doing our math.
Another example.
Multiplication:
System one: 10 x 10 = 100 ===> We all know this; I won’t bother showing it.
System two: 10 x 2 = 20 ===> 1 1 1 1 | 1 1 1 1 = 20
1 2 3 10 11 12 13 20
10 x 3 = 30 ===> 1 1 1 1 | 1 1 1 1 | 1 1 1 1 = 30
1 2 3 10 11 12 13 20 21 22 23 30
10 x 10 = 100 ==> 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 = 100
1 2 3 10 11 12 13 20 21 22 23 30 31 32 33 100
We can see in the above examples with system two that they are quantitatively identical to 4x2, 4x3, and 4x4 in system one, but now rather than being notationally depicted as 8, 12, and 16 they are depicted as 20, 30 and 100. Despite the notational difference, the quantitative results of the math are the same. In other words, identical quantities put through identical mathematical logic have resulted in identical quantities, despite the notational differences. It is also worth noting that the quantity defined as 10 in system one is defined as 22 in system two; if you multiply 22x22 in system two, the result on the four digit repeating number line is the position denoted as 1210, the position that corresponds exactly in magnitude to the position denoted as 100 on the ten digit repeating number line. These things are true because, once again, mathematical logic isn't dictated by quantity; it is the other way around. Stated differently (again) mathematical logic behaves the same for all quantities; any apparent difference in logical behavior between one quantity and another through the same operation is a notational difference rooted only in the symbol that we have chosen to represent that quantity. That is why any variable in any mathematical expression can be defined as any quantitative value and the expression remains logically valid. Flexible Ten Mathematics simply points out that this fact applies to the number line as well: mathematical logic remains valid irrespective of how we depict values on our number line. The special behavior of 10, relatedly, in our number system is a notational difference caused by where we have arbitrarily decided to repeat our digit system on our number line. We can set that repetition point anywhere we want and the quantitative value of that point will begin behaving notationally as 10 without causing any mathematical logic to become invalid. Quantifications will be accurate as long as we stick to the defined number line.
Accepting all of that, a logical consequence of defining a different number system, such as our four digit system, is that sub-increment (“decimal”) values between whole numbers now have different proportional magnitudes compared to our standard ten digit system. Said differently, when we, for example, divide 1 by 10 using system two above (our four digit system) it results in the answer 0.1, but 0.1 in system two is now one quarter of the way between zero and 1:
Four digit system: .1 + .1 + .1 + .1 = 1
Similarly:
Four digit system: 1.1 + 1.1 + 1.1 + 1.1 = 2
Effectively, on a four digit number line dividing 1 by 10 (1/10) is quantitatively equivalent to dividing 1/4 on a ten digit number line. Though the operational logic and quantitative value are no different, the notation is different:
Standard system: one quarter = 0.25
Four digit system: one quarter = .1
An immediate consequence of this notational difference is that ratios that are not expressible as perfect decimal values when using our standard ten digit number line are expressible as perfect values on a Flexible Ten number line. For example, take a three digit system: 0,1,2:
Standard system: one third = 1/3 = 0.33333333333333333 – forever
Three digit system: one third = 1/10 = .1
I have not done enough math to prove that there are other ratios that are not expressible in a standard ten system which would be in a flexible ten system, but I suspect there are many. I would argue that there is no logical nor quantitative reason to insist on using only a number line with a number of repeating units equal to the number of digits on a human hand; I would further argue that there may be some instances when there is a really good reason not to. For example, when trying to put an exact value on a logical ratio that is not expressible due to notational artifacts caused by the arbitrary insistence that all number lines must be graduated into the standardly defined 10 units. Insisting that all math be done using a number line with a number of repeating digits equal to the number of digits on two human hands may even be logically similar to insisting that the planets circle the earth.
It is easy to speculate how this thinking may make certain computations possible that would otherwise not be. A notation similar to the following might be useful in some instances:
10 = 16 (an example short hand for designating a 16 digit repeating number line)
(some math to demonstrate that two hypothetical values are) = (or some perfect relationship to each other expressible in abstract terms)
10 = 10 (short hand to designate moving back to the standard system)
(some more math that wouldn't be possible without knowing that the two values manipulated above have a perfect relationship to one another)
Again, this is only speculative, but it does seem reasonable to suspect that something like the above example might allow for solutions that would not be possible otherwise.