Symmetrical Math Calculator

[Phil]

Where to begin? I'm hoping to build a simple calculator application that is unlike most other calculators I have ever used. Since there are already hundreds of other calculators around I thought I'd attempt to build one that might revolutioninze the world of mathematics as we have known it.

For the longest time I have been of the opinion that our system of mathematics, although quite functional, is seriously flawed by a very simple assumption that we made several centuries ago when we figured out a way to incorporate negative numbers in our math. It's always bothered me that the square root of -1 is said to be an imaginary or complex, if you prefer, number. That is the result of an adaptation that had to be made when we declared that 'minus times minus is plus' ... because of that simple 'rule' the square root of any negative number cannot be represented as a 'real' number.

It turns out that I am not to first person to be 'bothered' by this notion. A fellow named Derek Nalls has formulated a symmetrical system of mathmatics where -1 times -1 is -1 just like +1 time +1 is +1. Makes a lot of sense and his symmetrical operations on the number line to the left of zero are mirror image of the operations to the right of zero. He is also a mathematician and some friends of his have developed a simple calculator application for him to demonstrate his system of symmetrical math. He has developed the proof that his system is consistent although the results are quite different than what we're used to seeing. The implications of symmetrical math are enormous and I hope to develop a slightly more extensive version of his calculator and make it as pretty and pleasant to use as it can be. I think I can do that. His paper and proof are beyond my full understanding at this point but I think his system is fundamentally sound. I had previously written some small C programs to check many of the 'rules' developed in his system and everything I tested was consistent.

I also happen to believe that the beauty and wonderous curves, shapes, and contrasts that we see in our wanderings near the boundry of the Mandelbrot Set aren't of an imaginary origin. I think what we notice there is quite natural and as real as it could considering that it's just a simulation of our current system of mathematics in the imaginary number plane. In the future, I hope to re-create the Mandelbrot set using Derek's system or a derivitive thereof to demonstrate that the boundry that tends to wander infinately thru the complex number plane near the origin quite naturally occurs in a system of symmetrical math where the negative meets the positive. That task is beyond my abilities at the moment but I think I can find the energy to approach it if I start with a fairly simple calculator that includes the symmetrical root and trig functions. I have attached a very simple minded console app that produces the Mandelbrot Set in both our current system of math and Derek's symmetrical math .... However, there are futher reductions that need to be made in order to prove or disprove my assumption that the boundry in the Mandelbrot Set occurs where negative meets positive. Anyway, the output produced by this program is possibly quite similar to what Mandelbrot himself might have looked at when he imagined that there was a deeper level of meaning in what he was seeing!



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[Eóin]

Phil, I understand your dislike at the lack of symmetry in the number line (or number plane when including imaginaries), but why should it be symmetrical?

Classical multiplication contains the notions of an 'identity', a number which when you multiply it by another number doesn't affect it. So i.e. given a number A and the identity I we have A*I = A. For us one is the identity of multiplication. Addition also has an identity, zero, because A+0 = A.

I bring this up because this symmetrical math seems to have two multiplicative identities, 1 and -1. And this begs the obvious question; what do you get if you multiply 1 by -1? It would seem that the answer should be both 1 and -1 which should really be a contradiction. Various other things about the calculator make no sense, i.e. 1 * -2 = 0.5 but -2 * 1 = -2.

Actually I tell a lie, its not that these thing make no sense, it's just that his 'revised multiplication' is not commutative. Which is fair and a quick read of the paper would suggest that this is intentional. Also even classical math has the notions that Sqrt(1) is equal to both 1 and -1.

This is not a new and revolutionary form of mathematics, its is existing maths with some new operators defined and irresponsibility named as being 'revised' versions of existing operators even though they alter some of the most basic properties of those operators.

Classical math is the way it is because ultimately it tries to model and simulate the real world. Multiplication is defined as it is because this is how multiplication in the real world seems to work.

As an intellectual puzzle and hobby, symmetrical math is perhaps as justifiable as any other, but I'd be wary about viewing it as anything more. Personally it strikes me a very narrow minded and vain attempt to force the very human (or biological) concept of symmetry into mathematics with no justification of why it should even belong there.

[Phil]

Phil, I understand your dislike at the lack of symmetry in the number line (or number plane when including imaginaries), but why should it be symmetrical?

We have symmetry on the number line for both addition and subtraction and those operators are very easy for our children to understand. Multiplication and division are also easy to understand as a series of repeated additions or subtractions. To me, the difficulty arises when we consider the sign of the result. When we multiply two positives we end up with a larger positive and move further to the right on the number line if both quantities are greater than 1. When we multiply two negatives our conventional math proclaims that the sign of the product will be positive. Likewise, our conventional math says that the product of any two numbers of opposite signs will be negative so conceptually we get an unusual and, in my opinion, unnatural effect of bouncing to the left and right of zero on the number line as we multiply a sequence of numbers with opposite signs.

With symmetrical math everything that you can do with two positive numbers produces an identical negative result when you operate on two negative numbers. I'd guess that our universe is actualy governed by very simple rules which are symmetrical in nature. Complexity, beauty, and even chaos arise near boundries of dissimilarity.

I bring this up because this symmetrical math seems to have two multiplicative identities, 1 and -1. And this begs the obvious question; what do you get if you multiply 1 by -1? It would seem that the answer should be both 1 and -1 which should really be a contradiction. Various other things about the calculator make no sense, i.e. 1 * -2 = 0.5 but -2 * 1 = -2.

What happens when an positron and electron have a head on collision? If our guess is correct they both cease to exit and they are instantly converted to energy! The concept of nil isn't built into Derek's model, but my guess is that -1 times 1 produces nil, or nothingness. In Derek's model the sign of the result is determined by the sign of the first operand, or 1 times -1 equals 1 and -1 times 1 equals -1 for your example(s). I believe it will be possible to show that the boundry of the Mandelbrot Set is precisely located at this boundry of nil in a system of symmetrical math. If I can manage to do that then I think more people will be inclined to consider the possibility of applying symmetrical math to re-examine other problems.

The second point you made is that multiplication is not commutative in symmetrical math for numbers of opposite signs. In Derek's model, the sign of the first operator determines the sign of the result. When numbers of opposite signs are multiplied, the second operator is converted to its inverse reciprocal and consequently the same sign as the first before the multiplication. So to make sense of your examples:

Code Select Expand

1 * -2   becomes 1 * -(-1/2) which is 1 * 1/2 and the product is 1/2 or 0.5
-2 * 1 becomes -2 * -(1/1) which is -2 * -1 and the product is -2


As an intellectual puzzle and hobby, symmetrical math is perhaps as justifiable as any other, but I'd be wary about viewing it as anything more. Personally it strikes me a very narrow minded and vain attempt to force the very human (or biological) concept of symmetry into mathematics with no justification of why it should even belong there.

Yes, it's a puzzle an certainly non-professional pursuit for me. Thanks for you note of caution. Derek's paper is very long and much of it is beyond my full understanding at this point but I certainly see it as a complete and self-consistent alternative model of mathematics. It's probably not the holy grail of mathematics and it's probably not the ultimate model that will allow us to reduce string theory to 5 or 6 dimentions in order to develop an understandable Theory Of Everything, but I think his model is a giant step in the right direction just to help us think more clearly about conventional math and also some of the symmetries that do exist in the natural universe that we attempt to model with our systems.

I don't want to give anyone the wrong impression here. I'm a fairly bright guy at times but I'm certainly not really that bright! I can *almost* understand 4 dimensions and the concept of space-time but it's clear to me that I need to be applying some of my energies to resolving the operations on our 1 dimensional number line so they really make sense to me ... I'm not quite there yet but, for me, thinking about our number line in terms of symmetrical math seems to help me see things more clearly. Sort of like RPN and my Hp15-c helped me understand conventional math a little better. It was hell learning to use the thing because it was missing the '(', ')', and '=' keys. After a while I began to realize that I could do my own operator scheduling and pressing the keys in precisely the right order began producing the correct results! Learning to use it helped me understand the problems I was trying to solve just a little better. Amazing! Sure do miss it ... but the $9 Casio fx-260 I'm currently using seems to be producing the right answers most of the time. Darned thing won't work under candle light the way my Hp15-C would but at least I don't have to buy batteries for it!

[Eóin]

Hi Phil,

I'd like to make one last point before bowing out- Going back to multiplication; I just don't see the advantages. The rules of the revised multiplication are at least twice as complicated as regular multiplication, and the result is that we lose the whole new world of math that negative numbers opened up originally. The only plus from the whole endeavour is that you get an aesthetic symmetry, and again I ask why pursue this symmetry?

To correct you on points, first its wrong to suggest that 'minus times minus is plus' is just an assumption that could be wrong. But it's not, these things are very rigorously defined and you really familiarise yourself with these basics. If nothing else it's very interesting.

Secondly, subtraction isn't really an operator (or at least not a wholly independent one), A - B is really A + B^-1. To give a quick background (this is not necessarily wholly inclusive, or using the correct notation)-

For the addition operator, '+', we have an identity written 'I' and the inverse of an element A written  A^-1. And we have the following rules.

A + I = A
A + A^-1 = I

With our regular numbers, where A is any number n we have I = 0, and A^-1  = -n.

Now with addition we also have the rules

A * I = A
A * A^-1 = I

But here when A = n, I = 1 and A^-1 = ¹/_n (here I mean the fraction as opposed to 1 divided by n).

Like subtraction being defined in terms of addition of an inverse, division is defined as multiplication by an inverse. So A ÷ B is really just A * B^-1.

Why do I point this all out? Well this is why there is a symmetry for addition, its because  –n is the additive inverse of n. Multiplication also has its own symmetry, it just involves fractions. Why not come up with a 'revised addition' where n + ¹/_n = 1?

The thing is that none of this gets rid of imaginaries, because classical multiplication is still there. This 'revised multiplication' is a separate extra entity, not a replacement.

As I said, here I bow out though, because there isn't anything wrong with these 'revised operators' and there may well be some very interesting properties with them which will open up new areas, or shed light on existing ones. But, and this is a huge but, you really should give yourself a very good grounding in math and the 'axiom' approach to math and proofs because its very hard to discover something new without understanding what's already there.


[EDIT]P.S. Just rereading my post they sound a bit condescending so if i've sounded arrogant anywhere here, or if you've felt I perhaps was a bit too personal once or twice well then I do appologies and certainly didn't mean it as such :U .

[Phil]

Eóin, Thanks for taking an interest in this thread and for your analysis and feedback. It's certainly true that I need to extend my understanding of axiomatic systems. Much of the paper that Derek published seems to be a rigorous proof that his system is self-consistent and complete. I understand about 30-40% of what I've read and I wrote a small C program to test the axioms in his system to prove to myself that it works correctly. I had also built a tiny Mandelbrot program that maps our coventional algebraic handling of complex numbers into his symmetrical system of algebra according to my understanding of his axioms.

I figure the best way to proceed now is to get on with the implementation of a calculator for his 'proven' symmetrical system of math and get it working up to the trigonometric functions. That will extend my understanding of his definitions to about 50-60% of what he has defined.

Before I had discovered his paper on the net I had attempted without success to reproduce the Mandelbrot Set with a symmetrical system where both the additive and multiplicitive identities are 0. I was hoping to prove that in such a system the boundry of the Mandelbrot set occurs precisely at points in the calculations where a negative value is being multiplied by the same positive value. In my mathematically naive mind it makes sense that adding 'nothing' to 'something' is still 'something'. The same idea makes sense to me when you multiply 'something' by 'nothing' or vice-versa the result should still be the same 'something' you started with instead of the big 'zero' that is defined as the correct answer within our conventional system of mathematics. Derek had assured me that what I am attempting to do is 'impossible' to do for reasons that I do not fully understand but I did manage to prove to my own satisfaction that his system works well enough to reproduce the Mandelbrot Set using symmetrical math where there are no complex numbers. I figured its best for me to start with a symmetrical model that works. His multiplicative identities are 1 and -1 as you pointed out earlier but I have not detected any contradictions in his system as it is defined.

Finally, I'd like to anwer your basic question about why I want to pursue a symmetrical system at all considering that we already have a fully functional conventional system of math that almost always produces the correct result. Here are some of the advantages as I see them:

a. Every number would have exactly 1 n^th root and every root would have the same sign as the number.
b. There is no need for a complex number plane because all operations on any real number produce a real result.
c. The non-linearity that manifests itself when dealing with complex variables in our conventional system of math will be much more understandable when it is defined with the axiomatic rules you spoke of within a system of symmetrical math.
d. I believe it is true that using a symmetrical system to solve real problems will be much more difficult, at first, than solving the same problems using our conventional system because we have come to know, understand, and trust it so dearly. Still, it's going to be fun for me to 'play with' just to see if it really works up to the pre-calculus level. I think, in time, I'll be able to accomplish that and, perhaps, even supply a couple of panes in the calculator window that simplify the mapping of conventional algebra into their symmetric equivalents.
e. Once I have it working as defined I'll be a lot closer to the 'axiomatic' understanding of simple number theory that will be required to change the multiplicitive identity to '0' in order to prove that A * -A is nil, and therefore undefined just as division by zero is in our conventional system. This is where I think the boundry condition exists in the Mandelbrot Set but we can't see it because it's tucked away and masked from clear view in our conventional complex number plane. (I know this is one of those 'impossible' quests but that shouldn't stop me from pursuing it, at least long enough to understand 'why' ... I'm proceeding with the 'why not' attitude at the moment.)
f. It's not really a very practical thing to be doing and it seems fall squarely within my area of expertise! (I'm just the technician!)
g. It's fun to think and talk about ... Also, removing the necessity for the complex number plane in solving real world problems that deal with both negative and positive quantities would actually reduce the number of dimensions required to think about the really complex nature of our universe.
h. A very nice Symmetrical Calculator might be a very nice 'gag' gift to pass along to a friend, or enemy. Sort of like the 'melt' screen that Leo had inspired Stephen Coy to do on the Amiga so many years ago!
i. Two wrongs do not make anything right and, I contend, it would be beneficial for humanity to have convenient access to a symmetrical system of math where minus times minus also produces a minus!  :wink

[Eóin]

Hello Phil, I can understand your reasons, and I do recommend you pursue this area.

Perhaps just keep one thing in the back of your mind, complex numbers are a wonderful addition to real numbers and they have allowed an awful lot of problems to be solved which just aren't solveable with real numbers alone. The world would be worse off if they never existed :8) .

[Phil]

Okay, finally I have something for test and comments. It doesn't do much except look pretty, thanks exclusively to artwork adapted from Andy Thomas' DreamCalculator³ ... I used paint to bend and reshape it slightly but, I really liked the colors and contrast so I took the liberty to 'borrow' his artwork. After I finish creating this post I'll send him a request for permission to use the changes I've made to his original bitmap in exchange for telling everyone I know about how wonderful his graphing calculator appears to be! Its marketed as shareware and has an affiliate program so promoters can help spread the word about his work so I think maybe he will allow me to use the modifications that I have made.

Meanwhile, I haven't tried his calculator or user guide because I want to stay focused on building a symmetrical calculator. I've just taken a very small step here and I'd appreciate any comments and suggestions. I'm hoping that I can manage to come up with a fairly clean example that will be useful to others who might like to use the code for their own calculator. I'm sure not everyone is going to like an RPN Symmetrical Calculator!

Anyway, it seems like it took me forever just to get this part off the ground so I'll attach the zip and quiet down!


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[Phil]

My work is progressing rather slowly. Here's my most recent version that uses a jpg file instead of a bmp thus reducing the executable size significantly. I've also discovered how to be lazy and just set the image as a background brush to greatly simplify the program. I'm also reading a lot of posts here and in the Win32AsmCommunity board.

Edit:

Thanks to Eddie Murphy for the image library in Hutch's example9\CAR example9 program and to Hutch for including the functionality in the masm32 library.

Here's the latest template. Still doesn't do anything except recognize left mouse button clicks. Please let me know if you notice anything that I'm doing wrong. I'm especially interested in knowing if it doesn't work on earlier versions of Windows. I'm hoping to create something that will work on Windows 95 and above. I think I've got it right so far ... but, it's better to post it a step at a time. I was glad to get things down to a single window instead of loading the bitmap into a static control.


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[bao_ho]

 :bg i want to thank you Phil for starting this post here otherwise it could be alot longer before i reach this material. i am writing this response immediately after reading your, Phil and Eóin, conversation. (I am not picking sides) I respect the way the universe works and all subjects that are in it. I want to share a short story of mine. It is about a few events during my development years and ideas that i had came to me at the time. I was an observer of all things natural that surroungs me. Numbers and mathematics was very natural to me. In exams i received 100% whenever i paid extra attention. It was up until year 12 of secondary school that mathematics ceased to make sense (to me). I blamed that i was just being more lazy than others. I tried realy hard like never before but i couldnt get a grip on what was being taught in class. And the reason for that was, on my own i was mentally working with a different kind of numbering system and set of tools. Now i know the name of that tool "Symmetrical Math". This is a big relief to me to know that i'm not that crazy afterall. I'm surely going to take upon that road again which i've been hiding from for the last 6years for sake of family.

anyhow, here's stuff that i think may have a chance in contributing to the 'shedding light' about the truth of numbers to advance the human processing engine.


1. In the natural world, one cannot take oranges and add/mul with pears to get only oranges. a new unit of interpretation is introduced and should be IDentified properly. There is no such thing as -1*-1. I can see it as ink and paper, i cannot see it elsewhere.
it is fair to say, "He eats two apples in 3 days." It is utta-nonsense to eat one apple in 10 apples. "in" being a multiplicative of the number of apples and number of days. Don't let the flexibility of the english language fool you here.  :wink

[Phil]

bao ho: Thanks for your interest and reply. Just don't take the Symmetrical Math too seriously. As others have pointed out we do have a fine system of math that works very well with most of the problems we encounter and I, for one, certainly need to spend more time studying it as well!

I'm hopeful that I can actually succeed at making a nice little toy that can be used to more thoroughly explore the properties of a system of Symmetrical Math. It's rough going for me at the moment but I am still plugging away at it! Thanks again for tuning in!

[bao_ho]

how serious is too serious? will i enter self destruct mode? :lol

if i could help in anyway let me know. i could spare an hour a day. optmising algorithms is what i like doing.  :green

[Eóin]

Some more mathsey ramblings follow...

I'd like to give a different, geometric way, of viewing negatives and even complex numbers. But first I'd like to highlight something. Number are not things, they are values. This probably seems obvious, but its still important to always remember it. 3 pears isn't a value or a number. It's a thing, specifically, its 'pears' and the number 3 tells us how many.

Bao_ho mentions that in the natural world taking 'oranges and add/mul with pears to get only oranges' makes no sense. True, but math wouldn't give you just oranges anyway. You can't multiply 2 pears * 3 oranges, because neither is a value. Only if pears and oranges take on the idea of being units, can we start to make sense of this, i.e. 2 oranges * 3 pears could be said = 6 (orange*pears).

Regardless though my point is that positive and negative numbers are not different in this way, they both represent a value. I find it interesting how willingly people accept negative number and not imaginary ones. Of course there's a complexity issue but largely speaking negative numbers make no more sense than imaginary ones, you can't have -2 pears no more than you can 3i pears. Both are values, and while -2 pears could be taken to represent a quantity of pears that you owe to someone maybe, that's still just a human concept with no place in the real world.

Anyway, I perhaps made things even less clear after all that. As mentioned though here is a different way of looking at negatives and imaginaries.

Negatives first, and this is very simple, think of the minus as indicating a unit describing a change of direction, so -2 + -3 = -5 because its really 2*- + 3*- = (2+3)*- = 5*-, or finally -5. When you do -2 * -3 it goes 2*- * 3*- = 2*3*-*- = (2*3)*(-*-) = 6*(+) = 6. Why does the -*- become + ? Because don't forget minus is a change of direction and if there's only two directions to choose form and I change twice I end up going back in the initial direction.

Imaginaries expand that idea onto the plane, here minus (a change to the opposite direction) becomes rotation by 180 degrees. And all of a sudden it makes perfect sense that rotation by half of that, i.e. 90 degrees can't lie on the number line and instead must point perpendicularly away from it. Imaginary i then becomes rotation by 90 degrees. Try it yourself, draw out a number plane, pick any random complex number and multiply it by I, if you plot point and connect then to the origin you'll see the angle between them is 90 degrees.

[Phil]

P.J. Nahin wrote An Imaginary Tale: The Story of i [The square root of -1] that covered a lot of the history concerning how we came to accept and incorporate negative numbers and their roots into our system of conventional math. It's probably available at your libraries and would be a very interesting read if you're interested in complex numbers. You don't have to be a mathematician to understand most of the text. I probably grasped about 60-70% of what he talked about. I'm posting the link here because I don't want to let us wander too far into the imaginary number plane in our discussions yet, it is very important to understand.


In the diagram above, the square roots of -1 at i and -i on the left represents the 90^o rotation that you spoke of. In fact, when you multiply any real or complex number by i it causes this rotation to occur. The next two examples show the points on the unit circle where the cube and fifth roots of -1 occur. As you can see, the rotations are different here and create an equilateral triangle and pentagon within the unit circle because we are considering the third and fifth roots, respectively.

Now, on to symmetry. Forget about the sign for the moment and just imagine that a number has an origin, magnitude, and direction. There are no negative numbers to contend with. All operations on any two numbers with the same origin produce a resultant number with the same origin. In order to incorporate the sign we need only flip the direction by 180^o. On the real number line we could consider anything to the left as negative and anything on the right as positive. Notice that direction doesn't have to be one-dimensional. It could be an x,y,z or x,y,z,time direction vector. In Derek's paper on Symmetrical Math I think this is what he refers to as 'rays' although there are many areas where I do not yet fully appreciate what he is describing.

With symmetry all numbers are just magnitudes with direction and origin. It wouldn't matter if the number line ran horizontally, vertically, or at any other arbitrary angle thru x,y,z space. Any operations on two numbers with the same 'direction' and 'origin' would produce a resultant number with the same 'direction' and 'origin'. Operations on numbers with different 'directions' or 'origins' will produce the complex nature that is only visible today with our conventional math in the complex number plane.

My i's are crossing and my t's are dotting so I'd better finish this post and come back again later to see if I can finish the thought. It's no wonder that things are progressing slowly for me ... I can't believe how long it's taken just to post this little bit!

[bao_ho]

i dont believe in fairy tales nor do i believe in IFs. it's not a matter of 'if' but a matter of when.
i've never eaten an (orange*pear) before nor have i seen it fabricated.

a value is not a thing. but a value is a number of things.

geometry is the mapping of objects in space. and space is a thing.

a value is not a thing. therefore numbers on it's own has no use. nor should it have transformation.

[Mark Jones]

"I cannot say for certain over-unity is possible. All the laws of physics seem to indicate that it is not. But these laws are only laws in the minds of man; the universe knows no such law. Being only but a man, I can't say but what I have observed, and that entails incomprehensibly little of the universe." MCJ 2003