Beyond + and - tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/22/2012, 10:07 PM since we want x/x = +1 we let +1 have the complex value 1. the 3 3rd roots of unity are the complex solutions to x^3 = 1 and they happen to lie on a circle in the complex plane with radius one and seperated by equal angles ( i dont know how much you are familiar with complex numbers ). going counterclockwise § is the first 3rd root of unity plz dont confuse - and - of course since we use both the complex and the new -. i will write '-' for the new. ( for now ) § = 1^(1/3) = exp(1/3 * 2pi i) = 1/2 * (-1 + sqrt(3) i ) '-' = 1^(2/3) = exp(2/3 * 2pi i) = 1/2 * (-1 - sqrt(3) i ) ( notice they are complex conjugates of each other ) +1 = 1^(3/3) = exp(3/3 * 2pi i) = 1 ( you could express these numbers different with sine and cosine etc etc but that is irrelevant at the moment ) now the multiplication with +1 ( complex or ' new' ) is trivial. hence we only need § * '-' , §*§ and '-' * '-'. from the complex roots of unity above it follows § * § = '-' '-' * '-' = § § * '-' = + and our multiplication table is complete. my apologies if i said something trivial to you but i dont know how much you know about complex numbers and group theory. im trying to be clear. as said before , now we can investige the properties ( asso distri etc ) and the polynomials. i think we can prove the number of solutions for a polynomial of degree n by induction on the first 2 or 3. which is almost identical to claiming that the pattern is "simple". " simple " because if we use induction and we are restricted to rising positive integers we have integer recursion and a short definition of it. so " simple " is not a formal term but to me it means more or less not more complicated than fibonacci type. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/22/2012, 10:20 PM if x^3 = 1 and x=/= 1 then x^4 = x this implies that x^2 =/= x and also that x^2 =/= 1 hence let y =/= x and =/= 1 x^2 = y y^2 = needs to be x since y^2 = x^4. x^2 = y y^2 = x x*y = 1 follows another way to arrive at the multiplication table. more my style i guess but perhaps harder to follow. seems i wanted to share it. regards tommy1729 Benny Junior Fellow Posts: 8 Threads: 1 Joined: Feb 2012 08/23/2012, 12:08 PM (This post was last modified: 08/23/2012, 12:10 PM by Benny.) I don't know much about group theory or complex numbers, so I appreciate you explaining it in detail. Given that you expressed § and '-' as complex numbers, does this mean that we could in theory express the same with complex numbers? It seems that equations with addition don't work with those definitions. So your equalities of complex numbers with "-"/§ are more analogies which mean they act as if they were equal to it (with respect to multiplication)? With your multiplication table we now have "nice" squareroots of all numbers (while with i we get a mess taking squareroots). Actually now I get a massive idea. If we define a sign that is associated to a number between, say, 0 and 1, let's call it °x°, then, as far as I can see, we recover the usual + and - (°0° and °0.5°), + - and § (°0° and °1/3° and °2/3°), and all imaginary numbers (expressed as x*°y° with y representing the angle; addition of complex numbers can be expressed in terms of multiplication of these numbers). Only that we now have nice addition (very important!) and neat looking roots. Addition works as with + - and §. Multiplication works as usual (with respect to the values), and means that we add the sign value (taking it to be cylic group). Squaring means multiplicating the sign value with 2, taking the square root means dividing the sign value through 2 (third power means multiplicating with 3 etc...). To formalize it °x°*°y°=°x+y° °x°^y=°x*y° But what happens if we get °°0.5°°. Do we go into the third dimension? I would really like to see a plot of some functions of these numbers (should work similar to a plot of complex functions), and fractals? Any good programmers here? I think as soon as we draw a really great fractal people will be convinced that the concept makes sense . tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/23/2012, 02:12 PM (This post was last modified: 08/23/2012, 02:14 PM by tommy1729.) (08/23/2012, 12:08 PM)Benny Wrote: I don't know much about group theory or complex numbers, so I appreciate you explaining it in detail. Given that you expressed § and '-' as complex numbers, does this mean that we could in theory express the same with complex numbers? It seems that equations with addition don't work with those definitions. So your equalities of complex numbers with "-"/§ are more analogies which mean they act as if they were equal to it (with respect to multiplication)? yes analogies with respect to multiplication. Quote:With your multiplication table we now have "nice" squareroots of all numbers (while with i we get a mess taking squareroots). yes however the abc formula for solving 2nd degree polynomials may fail because of lacking certain properties of these numbers and the amount of solutions =/= 2. and thus also the concept of discriminant might need revision. i stick with my story ; main questions are what properties and how many zero's for a certain degree polynomial. i think considering polynomials first and other functions later makes a lot of sense. Quote:Actually now I get a massive idea. If we define a sign that is associated to a number between, say, 0 and 1, let's call it °x°, then, as far as I can see, we recover the usual + and - (°0° and °0.5°), + - and § (°0° and °1/3° and °2/3°), and all imaginary numbers (expressed as x*°y° with y representing the angle; addition of complex numbers can be expressed in terms of multiplication of these numbers). Only that we now have nice addition (very important!) and neat looking roots. Addition works as with + - and §. Multiplication works as usual (with respect to the values), and means that we add the sign value (taking it to be cylic group). Squaring means multiplicating the sign value with 2, taking the square root means dividing the sign value through 2 (third power means multiplicating with 3 etc...). To formalize it °x°*°y°=°x+y° °x°^y=°x*y° But what happens if we get °°0.5°°. Do we go into the third dimension? i dont think any living being on this planet understood that. dont you mean variable when you say sign ? are you talking about operators or logaritms ? i didnt understand a word , number or symbol of that. i doubt if its a good idea , but im sure it must be explained better , in fact im not sure if that is even "math talk". Quote:I would really like to see a plot of some functions of these numbers (should work similar to a plot of complex functions), and fractals? Any good programmers here? I think as soon as we draw a really great fractal people will be convinced that the concept makes sense . since the signs have no superposition ( they cancel each other ) you have 3 lines starting from the origin ( = 0 ) rather than 3 dimensions ( that requires superposition and between 3 and 6 variables ). fractals would thus be hard to draw. and btw fractals might convince most ppl , but not necc most mathematicians. regards tommy1729 Benny Junior Fellow Posts: 8 Threads: 1 Joined: Feb 2012 08/23/2012, 03:33 PM (08/23/2012, 02:12 PM)tommy1729 Wrote: Quote:Actually now I get a massive idea. If we define a sign that is associated to a number between, say, 0 and 1, let's call it °x°, then, as far as I can see, we recover the usual + and - (°0° and °0.5°), + - and § (°0° and °1/3° and °2/3°), and all imaginary numbers (expressed as x*°y° with y representing the angle; addition of complex numbers can be expressed in terms of multiplication of these numbers). Only that we now have nice addition (very important!) and neat looking roots. Addition works as with + - and §. Multiplication works as usual (with respect to the values), and means that we add the sign value (taking it to be cylic group). Squaring means multiplicating the sign value with 2, taking the square root means dividing the sign value through 2 (third power means multiplicating with 3 etc...). To formalize it °x°*°y°=°x+y° °x°^y=°x*y° But what happens if we get °°0.5°°. Do we go into the third dimension? i dont think any living being on this planet understood that.Sorry, the concept of using an meta-operator that is assigned a sign to produce an infinite numbers of operators is quite new, so that's why it might be confusing. As we added a new direction from + to + and - and from + and - to + and - and §, we can obviously do that again and again (always using the same simple addition rules based on "exlusive direction"). Then we get the 4th roots of unity, the 5th roots of unity, etc... So we can extend the concept to infinity, but then we can't use unique signs, because we can only define a limited amount of them. So we simply use a number within the "meta-operator" (°°) to express which operator we mean. Really it doesn't matter which numbers we use, so I suggested 0 to 1. (with °0°=°1°=+). Then °0°=°1°=+, °0.5°=-, °1/3°='-', °2/3°=§. Given that we then have infinitely many directions, we can use that to map them onto the 2-d plane, so we have a description of a point on it using "°operator-value° number-value". We could convert it to an angle and a length, or to complex numbers. Of course we still have the advantage of addition (or rather °°-tion) working the same for all angles (in contrast to imaginary numbers were -1+1=0 and i+1!=0) and having nice roots. Actually, maybe we could extend that to the third (or n-th) dimension using a meta-operator with two arguments (°x,y°) - x representing the angle with respect to + in the second dimension, y representing the angle with respect to + in the third dimension. Or maybe even with nested operators °x°y°°? tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/23/2012, 04:12 PM ah i see. but i think we have to start small and investigate our 3 signs first. dont run before you can walk. ( how ironic since i cant do either at the moment ) regards tommy1729 Benny Junior Fellow Posts: 8 Threads: 1 Joined: Feb 2012 08/23/2012, 07:41 PM Yeah, you might be right. Just found the idea so cool that I had to write it down. hixidom Junior Fellow Posts: 20 Threads: 3 Joined: Feb 2014 05/28/2014, 01:40 AM (This post was last modified: 05/28/2014, 03:09 AM by hixidom.) I tried to think on something similar recently: I was thinking about a number "line" that is 2-dimensional, but I couldn't imagine anything that wasn't equivalent to the complex plane. The more I think about them, the more I am made uneasy by the idea of negative numbers: Why should a "sign" flip at some arbitrary point on the number line? Why can't the numbers just decrease in value without ever reaching zero? Are the properties of our number system determined by the way that humans naturally think, or is the converse true? The more I try to push beyond the current number system, the more I realize that I am a slave to ideas like plus and minus, for whatever reason. Regarding the topic of the OP, if we can think of + and - as states of a number, then perhaps a number could be a superposition of those 2 states (a.k.a. the physicist's approach). This can perhaps be expressed by replacing every single-valued number with a 2D vector, where the i and j components correspond to the + and - components of the number. Then we would have to redefine the basic arithmetic operations on these 2D numbers. I am not sure what would happen if the value of one of the components changes sign; Does it somehow contribute to the other component? How do you express vector operations with vectors for which one component can spill over into another? Edit: So I think that the 3-sign system could be expressed in terms of complex numbers. Say we define a complex number, c, with a fixed magnitude. Then decreasing the real component requires an increase in the imaginary component. Now throw in an additional type of imaginary number (call it "j") and thus complex numbers give us the needed functionality if we interpret +i as -1 and +j as §1 (where I'm using "§" to represent the 3rd sign). In this system +1, -1, and §1 point in different directions. Sure + and - are no longer opposite directions, but I'm not sure you can have opposite directions in a 3-sign system. JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 05/28/2014, 04:33 PM (This post was last modified: 05/28/2014, 04:34 PM by JmsNxn.) I used to think about trying to generalize arithmetic operations like negation and addition. I never came to three but I did come up with the following operator that I thought was very interesting: $x \box y = \ln(e^x + e^y)$ which is holo in $x$ and $y$ by doing a little calculus that I'm too lazy to do atm ^_^. It has the cool property $(x \box y)+z = (x+z) \box (y+z)$ and $x + y \box x = x+ (y \box 0)$ Then we define a nice metric: $||x|| = |e^{x}| < e^{|x|}$ so that $||x \box y|| \le ||x|| + ||y||$ Now we see we can start talking about calculus even because this operator is continuous. I always liked the box derivative: $\frac{\box}{\box x} f(x) = \lim_{h \to -\infty} [f(x\box h) \box (f(x) + \pi i)] - h$ $\frac{\box}{\box x}[ f(x) \box g(x)] = [\frac{\box}{\box x} f(x)] \box [\frac{\box}{\box x} g(x)]$ $\frac{\box}{\box x} nx = \ln(n) + (n-1)x$ and general box analysis ^_^ This may be a little off topic, this thread just reminded me of this. hixidom Junior Fellow Posts: 20 Threads: 3 Joined: Feb 2014 05/28/2014, 04:46 PM Quote:It has the cool property $(x \box y)+z = (x+z) \box (y+z)$ That is interesting. That's a property that I would expect zeration to have. « Next Oldest | Next Newest »