Prime Number System

This is a short paper I created for a class where I was prompted to create a number system without being isomorphic to an existing one. Without being able to use normal base-x systems like the common base-10, I tried to be inventive in my notation. Basing a system on primes sounded intriguing and also probably impractical so that is the route I decided to explore. There are some fundamental issues with this system that I go into near the end. There are undoubtedly some changes that could be made to make this much more useful but I still think this is a fun little exercise that I hope you might enjoy.


Thematically I was very inspired by the Egyptian number system, especially the choice of a tadpole or froglet to represent a number. My system focuses in on that idea and represents numbers with the life cycle of a frog or other amphibian. In addition, each symbol in this system is a prime, furthering the distinctness of each of the phases of metamorphosis and offering a unique (to me) way of expressing numbers. The zero is an empty vessel, perhaps a pond, representing absence and lack of life. The one is a single egg, the first phase of life. The two is a tadpole, composed of a head and a tail. The three is a tadpole with its back legs developed. The five is an adult frog from the side, with visible head, legs, and body. The eleven is a pairing of two adult frogs and an egg. Twenty-three is similar to the representation for zero, but this time the strikethrough represents the vessel having been filled. The forward slash is an operator used in representing bigger numbers which will function similar to multiplication. The bullet will be analogous to addition. This is the list of symbols present and their representations in the base 10 Hindu-Arabic symbols as well as an arbitrary pronunciation borrowed from Solfège for ease of description.












The representation of other numbers in this system is similar in many ways to the Egyptian system as explored in class.

For example:

= la-sol

Would represent ‘16’ as the values of each symbol are simply summed for the total. This does mean that there exist numbers with more than one representation, the ‘16’ above could also be displayed as:

= mi-mi-mi-mi-mi-mi-mi-mi




I would establish the convention that the largest value symbol be used until needing to move on to the next smallest one and so on. This limits the number of symbols as much as possible.




I would further propose that more complicated numbers be possibly represented by establishing one of the numbers as a multiplier and then using a backslash and then the existing symbols to describe that number.

For example:




/ = ti-si-la-mi

= 299




As we have set the multiplier at 23 and in this case, 23*13 is 299

Is much more economical and easy to parse at a glance than,




= ti-ti-ti-ti-ti-ti-ti-ti-ti-ti-ti-ti-ti

= 299




A consequence of the symbols representing prime numbers is that representations of larger numbers are composed directly of their factors, up to 23.


/

=414


/

=529




This does present a significant issue when dealing with larger primes. As they cannot be expressed as the result of multiplying two numbers. For example: 521 is prime.



= 521




I would then propose one more piece of notation to rectify this.


/•= ti-si-la-la ut-la-mi-mi

= 521




This will function similarly to addition. However, I think it is also analogous to a remainder or a decimal when concerned with division. It is a ‘leftover’ from a composite number to make it prime.




Having established the conventions we can begin to judge this system by its merits as discussed in the lecture.

The economy of symbols is . Learning seven symbols and an additional two to describe more ‘complicated’ numbers is comparable to the Hindu-Arabic system, however with addition/multiplication fairly well defined, similarly to the Egyptian system, by these 9 symbols there is arguably less to learn to do the same operations. The number of symbols to express a number is fairly economical for small numbers. It is worse but still reasonable for certain composite numbers of fairly large size. However, large numbers, primes especially can be exceptionally lengthy to express ( not to mention exceptionally long and hard to parse). In the table below I have presented examples of each.












The system here is not positional, however understanding the rules of the operations does not leave room for ambiguity in what number an arrangement represents. This is non-trivial though, as an understanding of addition and multiplication is required to understand most numbers this is a significant hurdle to overcome for anyone who does not already have experience with math education.

It is also difficult to see at a glance what representation of a number is the most simplified however, so multiple representations can express the same number. I think some incorporation of a position and the zero quantity I established at the beginning would go a long way in simplifying the expressions and improving ease of understanding.

Overall, I think this system is considerably less versatile and easy to use than any of the other systems we have explored. I suppose there is a good reason not to use indivisible units as representation for numbers that I have demonstrated here. I do feel that some operations would be easier, such as division (especially long division) however the work to get numbers in this form is almost certainly not worth the effort. I believe there is an interesting idea buried in here somewhere and would not be surprised if some application existed for a system similar to this one that makes sense, but as is, It is not much more than an exercise in exploring a number system constructed of primes.