The Simi numeral system

It is superfluous to present the numerical system practically the only one used today in the world. It is also well-known that computing uses a same – binary – system and that any other system can be set up. If we want to evaluate the various numeral systems, we could propose some – somewhat subjective – requirements:

- be easy to understand and learn,

- be comfortable for adding numbers,

- be comfortable for multiplying numbers,

- be able to quickly identify certain properties of the number.

We might ask ourselves, how much this makes sense, since we do not really know a new alternative numeral system. However, in 2019 the situation changed: the Simi numeral system (SNS) was born!

The rule is that we write down the canonical form of the number (in descending order of the primes), but with all the smaller prime numbers (with a zero power), and then we sort the powers one after the other. This sequence is the number in the Simi numeral system. Eg. 20 = 5¹×3⁰×2². Thus, in the Simi system 20 is described: 102. The Fundamental theorem of arithmetic guarantees that in this form all natural numbers can be written by one and only one way.

Decimal	Simi	Decimal	Simi
1	0	11	10 000
2	1	12	12
3	10	13	100 000
4	2	14	1001
5	100	15	110
6	11	16	4
7	1000	17	1 000 000
8	3	18	21
9	20	19	10 000 000
10	101	20	102

What's the point, what would be the advantage of this numeral system? Cannot overlook some extraordinary, very useful features of the new system. A huge, unmatched advantage: multiplying the numbers written in this system (which, let's face it, especially if the numbers are very large, is a macerated task), can be done here by simple addition. Remaining in the table above: multiples of 4 and 5 are 20, in the Simi system the same is 2 plus 100, which is 102. One can imagine how pleased computers would be to multiply millions of times per second for more serious tasks.

There is a natural consequence of this advantage: raising a number to a square or any power becomes a simple multiplication. Decimal: 4 on the second 16, on the Simi system 2 on the second 4. True, adding here is a harder matter. Continuing with the previous example, the sum of 2 and 100 is 20! Why, who knows?

But the Simi system has another incredible advantage. Since primes have appeared, mathematicians have been overly depressed by their ignorance of the regularity of the appearance of primes, and we have no formula that gives us the 100th prime, if desired. It is now resolved in this form forever and for all primes. The hundredth prime is simply 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

Here is the moment when skeptics can speak up: what do we know from the above inscription on the hundredth prime? Why, we could answer that, and what would we know if the same thing were written in decimal? (The debate is by no means uninteresting, but this is not the moment).

Let's see another interesting chart.

Decimal	Simi	Decimal	Simi
2	1	1	0
4	2	10	101
8	3	100	202
16	4	1 000	303
32	5	10 000	404
64	6	100 000	505
128	7	1 000 000 000	606
256	8	10 000 000 000	707
512	9	100 000 000 000	808
1024	(10)	1 000 000 000 000	909

The calculation of the smallest common multiple and the largest common divisor is also very simple: an operation similar to traditional addition. We write the two numbers one under the other, and and choose for the result in the first case the largest, in the second the smallest digit. Incidentally, in this system, divisibility by a number is no longer an issue.

The Simi system seems to have a serious weakness. How should one describe a number that has a prime with greater power the 9? Undoubtedly, this interferes with the incredible convenience of the numerical system, but the problem is not insolvable. A possible solution would be to put a power greater than 9 in parentheses. According to this, the infamous Googol number, 10¹⁰⁰, would be described in Simi as (100)0(100).

Otherwise, this technique can also simplify the writing of the hundredth prime. It would look like this: 1((99) 0).

* * *