Stay away from dividing by zero, is

the advise

I think the main question is not "Can we define division by zero?" (I have seen papers before that claim to do this in a consistent way), but "Is it useful to define division by zero?" Does your definition allow you to solve problems that could not be solved without the definition?

Regular and respected math texts

all indicate this as a no no.

Most importantly, I need to add that you are doing really well to think about these things deeply in high school, even more so to write up your findings :)

James Tuite

Thank you so much, Dr. Tuite,

I really appreciate your valuable feedback and I fully agree with your point about the importance of asking the right questions to produce significant work. Reflecting on the usefulness of research is indeed crucial.

Sincerely,

Ziad Rabea

Now,if you ask the difference between

0 and an infinitesimal, thats valid stuff

Dear Ziad Rabea,

Division by Zero has been mainly investigated by

Prof. Saburou Saitoh and his followers. A recent

paper is added.

Everyrhing can be investigated. The only important thing

that only precise definitions should be used.

Philosophycal arguments lead to stupidities.

With best regards,

Arpad Szaz

Mathematicians tend to be quite pragmatic - they'll allow any definition so long as it turns out to be useful. For example, eventually the square root of minus one was accepted (despite most people 'knowing' that it 'obviously' wasn't 'real'), because it made so much mathematics simpler and more elegant and provided a powerful tool for proving theorems (e.g. the prime number theorem amongst many others). So the question is: if division by zero can be defined rigorously, why has it not been adopted into mainstream mathematics? In my opinion, because it turns out not be a useful idea – I haven’t seen any problems solved using such a new definition. But I would be glad to be proved wrong :)

Arpad Szaz

Dear Dr. Szaz,

Thank you for your feedback and for sharing the paper. I have great respect for the work of Prof. Saitoh and his colleagues. I agree that precise definitions are essential in mathematical investigations, as philosophical interpretations can sometimes lead to contradictions. However, it is also important to acknowledge that some foundational mathematical concepts, such as infinity, have their roots in philosophical thought.

Best regards,

Ziad Rabea

James Tuite

Dear Dr. Tuite,

Thank you for your insightful feedback. I appreciate your perspective on the pragmatism of mathematicians and the importance of utility in adopting new definitions. In my paper "Zerodivics: Where Math Goes Beyond Infinity & Below None," I propose reconceptualizing zero and infinity through the concept of Progression Rate (PR). This framework aims to provide a rigorous foundation for division by zero and address various mathematical paradoxes, such as Zeno's paradox and indeterminate forms like 0/0​ and ∞−∞.

I understand that the utility of a new mathematical concept is crucial for its acceptance. In my work, I present several examples where PRs offer a novel way of resolving these paradoxes and provide a framework to solve indeterminate forms, I am aware that my approach is still in its early stages and may appear more intuitive than conventionally rigorous. However, I also believe that with further development and collaboration, this framework can be refined to demonstrate its practical utility more clearly.

I truly understand that no one starts perfect, and I am willing to further learn and develop the paper.

Thank you again for your constructive comments.

Best Regards,

Ziad Rabea

Ziad Rabea Regarding the origin of infinity, it seems to me that it is more mathematics influencing philosophy. Going back to ancient times, probably the unending natural numbers were mankind's first introduction to infinity? But in modern times Cantor's mathematical study of infinite sets were if anything hindered by people's ideas of what infinity 'should be'. What do you think?

Ziad Rabea I am not sure that I understand your definition of a progression rate. Can you give a short definition of it? Am I correct that you are thinking about the sequence in place of the limit?

For clarity, how would your definition circumvent 'arguments' such as "2* infinity = infinity, so 2 = 1", or "infinity+infinity = infinity, so infinity = 0"?

Some deep thought about the definitions of math may help:

• `i´ is to get seen defined using the old and wrong rules (at √-1 = i no calculation takes place!). A little correction of the rule of the square-root makes it also calculable.

•••

Old version:

half the minimum of the absolute value of the sum of all possible pairs of factors,

which generate the radicand as their product

•••

New version:

half the minimum of the sum of the absolute values of all possible pairs of factors,

which generate the radicand as their product

•••

But `i´ doesn't exist by reformed math.

[www.mathe-neu.de]

• Math-equations base on equivalence-relations. Left side the problem (task), right side its calculated solution.

So addition as well as subtraction is defined on combination of two real and independent acting quantities (numbers at the deepest level of understanding). The same for multiplication and division.

A quantity depends on three parts: number (of amount), measure (in square brackets) and unit.

If the number of amount is `zero´, no combination (calculation) could take place. The needed second partner for doing so misses.

Dear all, think about!

James Tuite

Hello Dr. Tuite, Thank you for your questions and for giving me the opportunity to clarify the concept of Progression Rate (PR).

PR Philosophy :

Progression Rate (PR) is a representation of how a function diverges to infinity or converges towards 0, it originates from the classic continuity paradox, if you have a unit and you halve it for an infinite amount of time, will it ever reach Zero? in fact it would approach Zero and akin to Zeno's paradox after an infinite amount of times it will reach Zero, it's just about the nature of Zero when it comes to multiplication and division.

PR definition :

Progression Rate is defined as the ration between coefficients of the nominator or denominator when you divide by infinity or Zero.

Addressing the Paradoxes:

2*infinity != infinity (in the framework of PRs).

In the framework of PRs, 2×∞ ≠ ∞ Instead, consider infinity as the behavior diverging by a factor of 1. Thus 2*infinity represents diverging by a factor of 2, at any given point, 2 infinity is always double the value of infinity, however, this doesn't contradict standard math, because what happens eventually is that the value diverges beyond finite boundaries.

infinity + infinity != infinity (in the framework of PRs).

∞+∞ = ∞ (1 + 1) = ∞ * 2 = 2∞

===========================

I think I should've used a different notation for the PR instead of coefficients notation, I would like to hear your suggestion regarding that point.

Dear Ziad,

before discussion could take place standards are to accept. It is not allowed to take such symbols, as `∞´ is, for mathematical calculations.

More in conflict between best introduced `experts´ the same on `0´.

Think on zero isn't any number (of amount) and see paradoxes blown away.

No combination for a pure number!

Math begins at the definition for the subjects. A number is ...? A combination is? A function is? Limits are?

James Tuite — Cantor's second diagonal-argument: refuted !

Its simple. Infinity is not a number

It implicates all kinds of paradoxes.

In your paper, you discuss the 'paradox'

1+1 = (0*infinity)+(0*infinity)=(0+0)*infinity=0*infinity=1

and try to resolve it by saying that we can think of 0+0=2*0. But surely we must be able to write 0+0=0? Or, by the same logic, we can also think of 0 as n*0.

Or, put differently, being able to write 0+0=0 with a clean conscience is surely more important than being able to divide by zero and multiply by infinity?

If one doesn't reach the level, we should say that! He doesn't.

Peter Kepp Yes, but remember Ziad is in high school - he isn't going to learn anything if we just tell him that he's wrong :)

We did! And?

Well I hope that Ziad can continue his mathematics studies and that he feels this discussion is constructive. Even if they do not lead anywhere, conducting a research project like this is great experience. And you have plenty of curiosity, enthusiasm and imagination Ziad Rabea well done - all very necessary ingredients for mathematics research! Make sure that you get a good grounding in all the basics - it is easier to do original research when you thoroughly understand what has already been done in that direction. Keep it up.

James Tuite

Dear Dr. Tuite,

Words fail to express how grateful I am for your invaluable feedback, constructive comments and for the enriching conversation we had.

It was truly one of the most rewarding experiences of my academic journey, and I will never forget it.

With sincere appreciation,

Ziad Rabea

James Tuite

J. T.: Yes, but remember Ziad is in high school - he isn't going to learn anything if we just tell him that he's wrong :)

But this isn't any high school helping platform, this is for researchers. Those which want to get one may read here. Acting should depend on an accepted published paper.