Inventing “New” Math

Mathematics is very interesting and a fundamental building block of our world, both for man-made constructs and natural constructs.

I’ve always wondered how it was like to invent or “discover” math, and I’ve recently been looking at the many ways certain fields of math were born, like Calculus, Linear Algebra, as well as more current discoveries (while somewhat lacking) that have recently been proven or are still in the process of being proven, like the Riemann Hypothesis.

Riemann Sums
Riemann Sums (matplotlib)

I feel it would be amazing if I could ever get such an opportunity, for my own sense of development as well as providing a benefit to the rest of humanity. However, while little concepts may have been floating around in my head, it can be very hard to make a breakthrough that would actually be useful in a specific scenario, let alone general scenarios.

Many theorems have already been found, and the idea of “breaking the rules” to find new discoveries can be very hard to tackle with, mainly due to the hard separation of truth vs lunacy as well as coincidental outcomes. But on the other hand, trying to stay completely safe while inventing can often lead to no progress being made.

However, new math doesn’t always happen from looking at a unsolved problem and making a new formula or construct to find a solution. A lot of new math is also made for providing shortcuts to existing problems, a little bit larger than simple mnemonics or method loopholes.

The more complex topics of math would be the host of most potential in the long term, but I don’t have much time and wanted to make something small but still technically new. I saw exponents and how they were repeated multiplication, and thought “hey, how about repeated division?”

Now I know that sounds kind of ignorant as division is just the inverse of multiplication, and this kind of pattern is not very common in the world, but I wanted to test myself and see if I could make anything given a base definition and expand towards rules and equations, no matter the usefulness. This was just a test. The test of divisience.

Yes I called it divisience (like the word “exponentiation”) with the construct being called a divisial (like the word “power”) comprising of a base and a divisient (like the word “exponent,” okay you get the point). The primitive definition is shown below.

Formal divisial definition (vanilla division and modified reciprocal product)

So this is what I started with for the concept. I decided to put the “n” on the bottom-left because top-right is for exponents, bottom-right is for subscripts, and top-left is for tetration, so this seemed like the only spot available. Yes there is also pentation and other hyper-exponentiation forms, but those are very uncommon. In the case that pentation or pochammer symbol is also used, to make a divisial stand out a little more, it can be written as “x D n” or you can include brackets around the divisient. The name doesn’t exactly roll off the tongue and it may take some getting used to, but it represents the concept behind the name very well.

I said that I wanted to come up with rules and interactions, and while pure division was technically the most representative at the time, larger calculations would get quite annoying later on, so I also went on to make an exponential definition. I had noticed that general results form proper exponents of the same base but reciprocal and with a difference, which turned out to be 2 after more testing. So it can also be written like so below (along with some examples).

Divisial examples
Divisial examples (natural and exponential representations)

Before moving forward, let’s denote some special values here, just like what powers have. A divisial to the divisient of 1 is just the base itself, a divisient of 2 is like the base over the base (so that would just be 1), and a divisient of 3 is the divisient of 2 over the base, which gives the reciprocal of that base. Now the next few are a a little more strict. Any base to the divisial of 0 is defined as 0 (representing the absence of anything to divide), and this is kind of the fuzziest one which is subject to change in the future with any roadblocks for rules and other representations. Any 0 to a divisient has the branch of 0 (for n being 0 or 1), and it will be undefined (NaN) for anything greater (just a bunch of indeterminate forms on top of each other). Next there is 1 to any divisient, and that results in 1 (for n greater than 0), and finally, just like powers, even divisials give positive answers and odd divisials give negative answers (for negative bases).

Special values
Special divisial values

Now comes the fun and “useful” part of making rules and laws that allow quick simplification of certain expressions with same base divisials. As this is in the beginning and there haven’t been many revisions or advancements, there are quite a few restrictions present, but that’s just life for the mathematician in the “alpha” stage of their invention. First, we start off with the product law, probably the easiest of the bunch. There is no “direct” way to work with divisials, especially in terms of identities where exponential form would be the best. It’s just a little bit of manipulation to get the proof done, and the only restrictions are that this doesn’t work with n and/or m as 0.

Product rule
Formal product rule proof (using exponential form)

Naturally, next would be the quotient rule, but this one has a lot more restrictions to hold onto. I’ve tried to find a single formula that works for any case of n and m, but I have been unsuccessful so there are two branches (one for n greater than m and one for n less than m, both can’t be 0). The proof goes by similarly to the one for the product rule, but we don’t get an end proof through pure exponential manipulation and simplification. There is a general gateway result, but more cases have to be tested analytically with the inequalities. First of all, the divisients have the n and m difference switch places for each restriction (to keep away from negatives), but the magnitude is the same, so both can be absolute value to simplify those conditions. Next, the base seems to result with the reciprocal of the opposite case (this was tested by expanding to vanilla division and cancelling out common terms on the fractions, ending up with either 1/n or 1/1/n (which is just n)), so this is what resulted in the separation to 2 branches.

Formal quotient rule proof (using exponential form)

These rules seem good enough for now, so just like with special values before, I’ll show some special expressions and evaluations that include these rules. Any divisials multiplied with any 0 on the top result in 0 (0x equals 0, 0/x = 0), but if any divisial is divided by 0, the value is undefined (NaN). The last expression is a format specification for negative bases, where similar to exponents, if no bracket is put around the negative x, it gets treated as -1 to the divisient of n times x, resulting in positive or negative x (depending on whether the divisient is even or odd).

Special expressions
Special divisial expressions

Another series of expressions we can look at is infinite sums and products of divisials. Again, they cannot really be represented in any way outside of exponential form here, so that will have to do, even if that may not be the “truest” evaluation of its infinite sums and products. Unsurprisingly, the product converges towards 0 (due to approaching 1/∞) and the sum is just a convergent geometric sequence for x values with a magnitude greater than 1, but they are still interesting expressions to look at.

Infinite sums and products of divisials (exponential form)

With these rules and expressions we can now do some basic calculations with divisials in groupings. Here are some examples. This first one is just a simple combination of multiplication and division of divisials, a nice demonstration of the product rule and quotient rule. The next one shows some interesting details with order of operations (using 2 different orders and ending up with the same answer; conventionally, the exponent should be applied to the entire divisial). The third one shows a property present in powers, which is divisial distribution into both the numerator and denominator of fractional bases (this also works with distributing to products). Finally, the last one shows the even divisient principle for negative bases.

Divisial expression examples
Basic divisial examples (with rules included)

You might might say that I’m missing something, specifically nested divisials, and you’re right. Don’t assume I forgot about that. There is a rule for it which I will show the proof of below, but this one is much larger in terms of calculations and there are more numbers present (very similar to matrix determinant formulas) so it can hard to remember. The derivation here is more important to generally know, as it can help generate and simplify a formula for greater than 2 “nests,” as well as if the divisients are both the same for example, almost like “squaring” them. While it may have seemed like the proof was over pretty quickly, there is the consideration that x and the divisient cannot equal 0 (specifically where the exponential form does not work, and also for negative divisients which have not been defined (yet)) and with some testing for critical values, it has been found that the result needs to be the divisial’s reciprocal to avoid that (again hinting to potential changes with the result of x to the divisient of 0 in the future, if more inconsistencies arise).

Nested divisial rule
Formal nested divisial rule proof (using exponential form)

Since this rule can be a little harder to work with, here are some examples with just this rule in use. The first one is a direct usage with a positive base, nothing much to do except simplify, convert to exponential for evaluation and move on. Next is an instance with a negative base, so the even-odd divisient principle needs to be considered here. Since the inner divisient is odd, the result will be negative and that means the minus sign can be take out within the bracket. It can then be further taken outside the nested divisial due to negative 1 being in a divisial with an odd divisient, and the nested rule is plugged and simplified to get the answer once again. The last example is a special case with the two divisients having the same value (as well as an irrational base, but that doesn’t really change much). The formula changes and simplifies to n² - 4n + 6 which shows why the understanding of the derivation is important, and this is also a precursor to further expansions of nested divisials that end up with higher order divisients.

Nested divisial examples
Nested divisial examples

I think this is a very good start for now. All that may be left is extending the domain of divisients to negative numbers and fractions (possibly even complex bases if we want to be adventurous). I may write another article about extending the domain later on.

Overall, I hope this was a nice demonstration of the method behind creating new concepts in math or simply creating anything scientifically inclined. Lay out your postulates and definitions, expand by finding specific cases of certain behaviours with your construct, and find a way to generalize them to multiple scenarios.