So far, every numbering system I have found seems to require a ‘base’ number. More precisely, if that system were used outside of its standard arena of use, one would need to demarcate it with a “base subscript”. The intent of this is so that someone who uses a different base system would realize that it is a different base, what base it is, and be capable of transcribing it into their base numbering system. While working on my Proof Invalidating Fermat’s Theorem in conjunction with seeking answers to correcting our Mathematical Base System, I decided to try and determine if it was possible to create a mathematical system that would not require a “base subscript”. The following article details what I came up with.
I know, I know, everybody will look at that and go, “but Mr. Heemeyer, that’s not a real number…”. Actually, not only is this a number, it is a single digit representing the decimal number of: 825,768. How about another example? Ok…
This one’s a bit crowded, here… let me zoom in a bit…
There we go, so how about that one? Guess what… this is also a single digit representing the decimal number of: 825,768. Ok, now I am certain many of you are quite confused. How can both of these be single digits? How can both of these be from a single numerical system? And, how can both of these equal the same number?
Well to understand the answers to these questions one must first understand how this numbering system works, and how each digit is determined. First, let’s look at the digit itself. A single digit can consist of any number of various Locai. Each Locai represents a specific numerical value as shown below:
As is shown, whenever ANY digit is written using this system the various Locai are always numbered starting with 1 in the upper left most Locai and increasing the value by 1 for each Locai to the right, then continuing the count on the next row down until no Locai remain. Locai may be arranged in any method, so long as the values of the Locai follows this methodology. For the purposes of this article (and where you will see this again in later articles made by me and the D.E.C.I.D.E. crew) we will be using the below “boxxed Locai methodology” (unless otherwise noted):
Always remember that the upper-leftmost Locai always has the value of 1, and the lower-rightmost Locai will always have a value equal to the number of Locai represented. To keep this article fairly simple we will be sticking to the two Interval Systems enclosed in green bordered boxes. This brings me to the next point in the PVIS numbering system… the Interval System.
In the PVIS numbering system an Interval System is the closest a PVIS digit comes to having a base number. There are two things to note about Interval Systems. 1.) an Interval System’s value is always 1 higher than the number of Locai represented. 2.) the Rotational Symbol(s) of a Locai determines the position(s) of that Locai’s numerical value when the PVIS number is transcribed into a base system where the base number is equal to the Interval System’s value.
Shown to the left is the basics of the Rotational Symbols. When a Rotational Symbol is placed on a Locai, that Locai’s value becomes: the Locai value multiplied by the Intervail System raised to the power shown (or, as an alternative for the case of the nil and raising to the power of “0”, you may instead multiply by the values of 0 and 1 respectively). To show this simply allow that Locai = L; Interval System = S; and Rotational Symbol = R. For each Rotational Symbol you will figure out: L*S^R= ??? and add them together to determine the total value of the given digit
When inserting these Symbols there are two main rules to remember. 1.) no single rotation may be placed more than once in a digit (other than the nil value, which gets replaced by whatever Rotational Symbol(s) will go on that Locai). 2.) the rotations are placed as a stack, meaning you can have the rotations for ^5, ^7 and ^11 all on the same Locai. This merges the rotation’s appearance so it looks like one symbol on the Locai, but it’s the total shapes and angles that declare which Rotational Symbol(s) are there.
With this knowledge you should be able to read those two digits I gave at the beginning of the article. What? Still having difficulty reading the second digit? With the
? Remember the Interval System… this digit has an Interval of 2, so would be written in base 2 (aka, binary). Writing it as binary you get:11001001100110101000 which is the mathematical equivalent of:825,768 in base 10. Or you could just use the reading method and do the math.
The benefits of the PVIS system
Ok, so I have demonstrated a numbering system that does not use a base value like our current base system(s). When showing this numbering system to other people there are three questions I am always asked.
1.) Why would you try to make a new numbering system? (To see if I could)
2.) Why did you even try making a new numbering system? (For the challenge)
But most importantly is: 3.) What makes this new system better than our current system?
To properly address that final question though takes more than a simple sentence. First you have to understand that no numbering system is inherently “better” or “worse” than any other system. That is not to say that one numbering system may not be more efficient in one area than another. For example: In America the decimal system is king right? For human to human communication it certainly is… however in computers the language of mathematics (and of everything) is binary. Machines only recognize am I receiving enough power (yes =1, no =0) and they use that to communicate and translate information so that we can understand it.
So the more appropriate question would of course be “what are the Pros and Cons of the PVIS numbering system?”
The Pros
A.) It is more intuitive. (the “base”, or what is known as the Interval System in the PVIS is clearly shown rather than having to be declared or understood separately)
B.) The Interval System can easily reduce amount of space needed to write a number. i.e. which takes less space to write: 299,792,458,238,157,410
OR:
C.) using a single Locai you can describe a decimal number of up to 7 digits, or a binary number of 20 digits.
i.e.: the PVIS digit
(second image is zoomed in to show more precision) describes the binary number: 11111111111111111111 which is the decimal number: 1,048,575
D.) For those stuck in their ways, a PVIS digit can be described in Base System terms by noting that a PVIS digit would have a base of (Locai+1)^20, though doing this would be absolutely unnecessary and would be a complete waste of time.
Both a Pro and a Con
A.) A PVIS digit can be described as a negative number, HOWEVER the Interval System can only be positive numbers as it would be impossible to show a negative amount of Locai. This means that the issues of a “negative base” brought up in the article: Problems with the Current Base Mathematics System become irrelevant when using PVIS digits.
B.) Once a PVIS digit maxes out in value, you either have to increase the Interval System or put in a second PVIS digit (using the same Interval System) with a separator (I.e. a comma). Using a comma to separate two PVIS digits makes it clear that they are separate digits, which would reduce the chances of thinking they are one digit o a Higher Interval System (which would actually be a MUCH lower number than intended)… but can lead to people thinking of it as a base numbering system. (in fact the final Rotational Symbol of the first digit would have a ^19 demarcation, while the first Rotational Symbol of the second digit would start at a ^20 demarcation and the rotations would simply continue to go up accordingly from there). Fortunately, the size of numbers required to cause this issue is higher than most would see on a daily basis, and the above described remedies should minimize confusion.
The Cons
A.) PVIS Locai can become crowded and difficult to read if a large number of Rotational Symbols are present (though even at the minimal size shown on this site they are still identifiable)
B.) As a PVIS has its Interval System’s value increase, the amount of space it saves the writer diminishes accordingly. (it takes a much larger decimal number to have a PVIS be more efficient than it would if it were a decimal number instead)
C.) People are stuck in their ways and refuse to consider that a numbering system doesn’t require a base, and will try to force the issue. While the PVIS digits can be described using a base number (as shown in The Pros D. ) the numbering system itself negates the need of an identifying base number thanks to it’s internal and highly visible Interval System.
Ancillary Interests
A.) The PVIS system is purely visual. As such, it may show new patterns which would not be recognized or even considered in a standard base numbering system.
B.) There is a large community of people who still seek out intelligent life amongst the stars. Having more than one form of numbering system (seriously, read Seeking answers in other mathematical systems and it will show that EVERY other system on our planet is a base system of some form) may increase the odds of interplanetary life responding to us (after all, who says that other intelligent life forms would use a base system for their numbers? The more systems we can use and send out, the more likely another species will be able to understand what is sent)
C.) Symbology. There are some symbols used in certain belief systems (I.e. Constellations for Astrology, the Kaballah Tree of Life, Chakra, for a few examples) that may find benefit in using the PVIS numbering system in conjunction with their Symbology to further understand the importance certain numbers may or may not have within their belief structure.
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