Are natural numbers isomorphic to complex numbers?
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From: Akira Bergman on 31 May 2010 18:59 On Jun 1, 3:25 am, Stephen Montgomery-Smith <step...(a)math.missouri.edu> wrote: > Akira Bergman wrote: > > Euler's identity contains all the goodies; > > > e^(i*Pi) = -1 > > > 'e' is implied in the natural numbers (N) by the density of the primes > > (P). > > > How are 'i' and Pi included in N (if they are)? > > How about Euler's other identity > > sum 1/n^2 = pi^2 / 6. Yes this is also Zeta(2). I know about Zeta related pi sums, but they involve the powers of pi. I do not know how to interpret this. Maybe the pi values in the functional equation of Zeta; pi^(s-1)*sin(pi*s/2) are easier to interpret.
From: Akira Bergman on 31 May 2010 19:14 On May 31, 11:10 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Don Stockbauer wrote: > > > Are natural numbers isomorphic to complex numbers? > > > They are to me, but then, I never was too good at math. > > An isomorphism is (among other things) a map that preserves structure. > What structure do N and C have? > > An isomorphism is (among other things) a map that is onto. What map > from N to C is onto? > > -- > I can't go on, I'll go on. N is a simple feedback summation structure with the initial value set to 0; N(n+1) = N(n) + 1 N(0) = 0 C has all the structures of N, Z and R, and on top of that, R is rotated by pi/2 to make the polarized complex plane. It is a union of the square with the circle. The decision involved in the solution of the equation; x^2 = -1 x = {+i,-i} enforces a spin onto the C.
From: porky_pig_jr on 31 May 2010 22:11 On May 31, 7:14 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > C has all the structures of N, Z and R, and on top of that, R is > rotated by pi/2 to make the polarized complex plane. It is a union of > the square with the circle. The decision involved in the solution of > the equation; > > x^2 = -1 > x = {+i,-i} > > enforces a spin onto the C. My head is spinning. Just from reading that.
From: Akira Bergman on 31 May 2010 22:20 On Jun 1, 12:11 pm, "porky_pig...(a)my-deja.com" <porky_pig...(a)my- deja.com> wrote: > On May 31, 7:14 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > > > > C has all the structures of N, Z and R, and on top of that, R is > > rotated by pi/2 to make the polarized complex plane. It is a union of > > the square with the circle. The decision involved in the solution of > > the equation; > > > x^2 = -1 > > x = {+i,-i} > > > enforces a spin onto the C. > > My head is spinning. Just from reading that. Which part is spinning? The pork or the pig?
From: Transfer Principle on 1 Jun 2010 01:40
On May 31, 4:26 am, Gerry <ge...(a)math.mq.edu.au> wrote: > On May 31, 7:21 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > Euler's identity contains all the goodies; > > e^(i*Pi) = -1 > > 'e' is implied in the natural numbers (N) by the density of the primes > > (P). > Huh? It appears that Bergman is seeking a connection between the three fundamental mathematical constants e, i, pi, and the set N of natural numbers. He finds it interesting that the constant e is related to the set N via the Prime Number Theorem, which states that the number of primes less than a given number n is approximately n/log_e(n). And so Bergman seeks uses of the constants i and pi in number theory as well. It's possible that he wants to define e, i, pi using purely number theoretic definitions, rather than the standard definitions of these constants. > > P/log(P) has a uniform density with fluctuations. > > Can these fluctuations be used to explain 'i' and Pi? > > I did some formant analysis on P/log(P) with Praat, and found 4 > > distinct patterns, probably corresponding to the four cycle nature of > > 'i' > Huh?? > What is there to explain? It appears that, in searching for a purely number theoretic appearance of i, Bergman notes that i^4 = 1 and thus the function f(n) = i^n is 4-periodic. And so he seeks 4-periodicity in some function related to N. Since he has already used the Prime Number Theorem for e, he might as well use it for i as well. Now how Bergman proceeds is mysterious to me. It appears that he looks at the function g(n) = n/ln(n) and somehow notices 4-periodic behavior in that function. But it escapes me what pattern Bergman found that would be related to 4-periodicity. But here's a suggestion for Bergman: As he already notes, the multiplicative group {1,i,-1,-i} is isomorphic to the additive group Z/4Z. So perhaps Bergman can look for other groups that are isomorphic to Z/4Z as well. We notice that there are four natural numbers n such that phi(n) = 4, namely 5, 8, 10, and 12. For two of these values, namely 5 and 10, the multiplicative group (Z/nZ)x is isomorphic to Z/4Z. This is more noticeable for 10, since we use a decimal base. Every prime (with only finitely many exceptions) ends in 1, 3, 7, or 9, and we can say that 3 corresponds to i, 7 to -i, 9 to -1, and of course 1 to itself. (The above isn't mathematically significant, but might be the sort of thing that Bergman is looking for.) > > But I am not confident with his one. I have no idea how to explain Pi > > in N. Maybe through the Zeta function? Now Bergman seeks a connection between our final constant pi and the natural numbers. He notes that zeta(2) = pi^2/6, and so this gives a connection between the sums of the reciprocals of the perfect squares (hence number theory) and pi. But then again, if Bergman is going to allow infinite sums, then why not replace "perfect squares" by "factorials" above, which would then give us e directly and much more simply than the Prime Number Theorem? (Of course, infinite sequence of rationals is ever going to converge to an imaginary number like i.) |