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
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