Imaginary Numbers: My Number System & Research Paper

[Abdul Wadood]

Well I have developed a number system which allows the existence of imaginary numbers.
Please visit it at : http://www.scribd.com/doc/46064105/Math-Paper.
An intro of these ideas is presented at :http://www.scribd.com/doc/46117043/Introduction-to-My-Research-Paper
Please provide me feedback. Am I thinking on the right track?
Thanks for your time and (mental) effort.

 

[micromass]

Hmm, I don't see many imaginary numbers in your paper?

Your paper is certainly interesting, but I have some comments. It's meant to be constructive:

Anyway, you seem to like to work with infinity in your paper. But this poses a lot of problems. Infinity is not a standard number, so you'll have to define rigorously what you mean with $$\infty$$, I don't see you doing this anywhere. When writing a math paper it is VITAL to define everything well, this is something that is lacking in your paper.

From looking at your paper, I kind of guess that you look at the integers in a new way, but what are the advantages of looking at the integers in this way? You should include something motivational there...

I'd like to discuss this with you, and to show you how you present thing rigorously. So I hope you take this comments well!

 

[Abdul Wadood]

Thank you very much for such a positive response.
1. Infinity in this discussion would be any arbitrary number that you choose. You can consider this as the largest number you would require in a given problem. Just don’t go beyond it. That number, which is practically large to suffice for all our needs in a situation, would be infinity. Once you have chosen it, it would be the largest number in your number system. And its reciprocal would be the smallest number in your system. And then all the numbers would be multiples of this smallest number.
2. When I discussed the operations, it rose out in this number system that when we would multiply two numbers of the same sign, the product would be of the same sign as well. So two negative numbers will have a negative product. And the product of two numbers with different signs will have the sign of the larger number. So imaginary numbers are possible.
3. Motivational……………. I guess allowing imaginary numbers is pretty motivational for me. And regarding the choice of infinity for our problems, I just had a hunch (no logical grounds whatsoever) that for our present Quantum mechanics we could use Planck length and the Planck time as the smallest numbers and their reciprocals as the infinities?

Thanks again for the reply. I hope this discussion continues.

 

[ramsey2879]

Thank you very much for such a positive response.
1. Infinity in this discussion would be any arbitrary number that you choose. You can consider this as the largest number you would require in a given problem. Just don’t go beyond it. That number, which is practically large to suffice for all our needs in a situation, would be infinity. Once you have chosen it, it would be the largest number in your number system. And its reciprocal would be the smallest number in your system. And then all the numbers would be multiples of this smallest number.
2. When I discussed the operations, it rose out in this number system that when we would multiply two numbers of the same sign, the product would be of the same sign as well. So two negative numbers will have a negative product. And the product of two numbers with different signs will have the sign of the larger number. So imaginary numbers are possible.
3. Motivational……………. I guess allowing imaginary numbers is pretty motivational for me. And regarding the choice of infinity for our problems, I just had a hunch (no logical grounds whatsoever) that for our present Quantum mechanics we could use Planck length and the Planck time as the smallest numbers and their reciprocals as the infinities?

Thanks again for the reply. I hope this discussion continues.

What would be the sign of -3*3 in your system? How would one multiply (a + b)*(a - b)?
Without knowing their magnitudes, you couldn't just write a^2 - b^2, could you?
Without knowing the purpose of your system, I couldn't say whether or not you were on the right track.

 

[Abdul Wadood]

-3*3 would be a 9 which would be neutral, like zero. It would neither be positive nor negative because its positive and negative parts would be the same.
In this system, (2+2) is different from 4 because the PARTS of (2+2) are ¥ more than parts of 4.¥ would be the largest number you would choose for your situation. The same goes for other operations (bit different in multiplication and division). So a^2 would be of the same sign as a but would be different than the square of a. (e.g. 3^2 is different than 9 because the parts of 3^2 are ¥ times greater than 9)
We just have to consider the parts of a number (or the sum or product or difference or quotient). If positive part is greater, then the number is positive and if the negative part is greater, then the number is negative.
What this implies is that
1. An operation changes the nature of a number.
2. Imaginary numbers are allowed.
3. The Math deals with finite numbers.
I have to admit that I really had no Mathematical paradox or problem to solve for so that I developed this system. These ideas just came to my mind and I developed it. So I thought I would discuss them. But they MIGHT have some application, as is the case with such theoretical games. The idea about Planck length and time are simply the hunches of a novice.
I don’t really have an advanced background to apply these ideas, but I am searching for applications (just like many Mathematics concepts, these may just be concepts).
If you know of a place to apply these, do tell me. A computer scientist tells me they deal with confined number systems in some place in their field.
Comments are welcome.

 

[dimension10]

-3*3 would be a 9 which would be neutral, like zero.

$$-3\times 3=-9\neq 9\neq 0$$

 

[Abdul Wadood]

exactly. -9 would be neutral but not equal to zero. all the treatment depends on the parts of numbers.

 

[HallsofIvy]

Can we take it then that you titled this "Imaginary Numbers" only because you had no idea what "imaginary numbers" meant?

 

[ramsey2879]

exactly. -9 would be neutral but not equal to zero. all the treatment depends on the parts of numbers.

So 2*-18 would be -36 while 12*-3 would be +36 but 6*-6 would just be 36 if I understand what you are saying. Does this mean that -36 equals +36 equals 36 (something like the absolute value being the size of a number, while the sign says something about the history of operations that obtain the number). Looks too weird to have a useful purpose but then the same was true for boolean math.

 

[agentredlum]

Thank you very much for such a positive response.
1. Infinity in this discussion would be any arbitrary number that you choose. You can consider this as the largest number you would require in a given problem. Just don’t go beyond it. That number, which is practically large to suffice for all our needs in a situation, would be infinity. Once you have chosen it, it would be the largest number in your number system. And its reciprocal would be the smallest number in your system. And then all the numbers would be multiples of this smallest number.
2. When I discussed the operations, it rose out in this number system that when we would multiply two numbers of the same sign, the product would be of the same sign as well. So two negative numbers will have a negative product. And the product of two numbers with different signs will have the sign of the larger number. So imaginary numbers are possible.
3. Motivational……………. I guess allowing imaginary numbers is pretty motivational for me. And regarding the choice of infinity for our problems, I just had a hunch (no logical grounds whatsoever) that for our present Quantum mechanics we could use Planck length and the Planck time as the smallest numbers and their reciprocals as the infinities?

Thanks again for the reply. I hope this discussion continues.

You claim in 1 "And its reciprocal would be the smallest number in your system." What would happen if you multiplied this number by itself? You would get a number SMALLER which is not in your system. If you put this new number into your system and multiply it by itself again you get ANOTHER number not in your system, so your infinity is rapidly increasing without bound and your reciprocals are quickly approaching ZERO.

This is the kind of trouble you run into when you define infinity as some finite number.

 

[SubZir0]

Quater-Imaginary is a number system that Donald Knuth made that can represent complex numbers.

There is a base -1+i that can model the Dragon Curve fractal.

People who were interested in what this thread might have been, may like to come help me with my puzzle: https://www.physicsforums.com/showthread.php?t=511758

 

[dimension10]

Well I have developed a number system which allows the existence of imaginary numbers.
Please visit it at : http://www.scribd.com/doc/46064105/Math-Paper.
An intro of these ideas is presented at :http://www.scribd.com/doc/46117043/Introduction-to-My-Research-Paper
Please provide me feedback. Am I thinking on the right track?
Thanks for your time and (mental) effort.

Isn't the paper just describing a new algebra? There are an infinite number of algebras.

What I mean is:

Let us say there is some set which happens to be a function of a parameter t (which you call time). At any time t, you define $x_1(t)$ and $x_k(t)$ (which you call the "smallest" number" and "largest number"), where $k=\mbox{n}\left(S(t)\right)$, such that any element in S(t) $x_j(t)$, $x_1(t)\leq x_j(t)\leq x_k(t)$ and $x_1(0)=x_k(0)=\mbox{indeterminate}$. Then you define the operations + and × as you do in your paper. Then, the object (S(t),+,×) can be called R(t). Now, R(t) is the algebra you introduce in your paper.

But then, you realize that there are an infinite number of R(t)'s at a given point in t, depending on how you define $x_1(t)$ and $x_k(t)$. So you then try to scope down R(t) by restricting $x_1(t)$ and $x_k(t)$ such that $\forall t,\;R(t)$ is always a ring.

What I just said was a description of the algebra you are defining.

But note that no matter what you do, you can't possibly bring imaginary numbers into question. They already are elements of the imaginary and complex number rings. Oh, and you should really discard the "neutral" numbers. They don't make sense, at least to me. You really need to redefine the multiplication you introduce.

P.S. I wish there where Euclid fonts in the list of PF fonts.

 

[meldraft]

Quater-Imaginary is a number system that Donald Knuth made that can represent complex numbers.

There is a base -1+i that can model the Dragon Curve fractal.

People who were interested in what this thread might have been, may like to come help me with my puzzle: https://www.physicsforums.com/showthread.php?t=511758

Interesting, but how is this connected to this thread?

Anyway, this looks like an interesting number system. One area of application you might want to look into, is complex potential theory. There we make use of bounded infinity in numerical applications, and the positive, negative, and neutral distinction might yield some interesting results when defining electrical charges or solid bodies in the field.

 

[Dickfore]

1. Infinity in this discussion would be any arbitrary number that you choose. You can consider this as the largest number you would require in a given problem. Just don’t go beyond it. That number, which is practically large to suffice for all our needs in a situation, would be infinity. Once you have chosen it, it would be the largest number in your number system. And its reciprocal would be the smallest number in your system. And then all the numbers would be multiples of this smallest number.

But, this is in direct violation of Peano's axioms for natural numbers. Namely, what is the successor of the largest number you imagine?

 

[micromass]

This thread is too old and too crackpot to revive.