Can This Definition of Pi Work in More Abstract Algebras?

Gerenuk
Messages
1,027
Reaction score
5
I quite like the following definition of pi and I wonder which minimal algebraic rules are needed to make this definition work?

\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=1
\therefore |a|=2\pi k

(For example, are there algebras more general than complex numbers, where this works?)
 
Physics news on Phys.org
Well, in the quaternions e^{2\pi u} = 1 where u is any quaternion with u^2 = -1
 
So quaternions probably also work.

Any more abstract algebra? Hmm, what do I need? I need addition, multiplication, scalar division, limiting process and modulus, right? What has to be fulfilled as to yield pi then?
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Dfns group action & group, written in terms of the other
Replies
26
Views
382
I Definition of minimal ideal using symbolic logic notation
Replies
3
Views
564
I Meaning of "passage to the quotient"?
Replies
3
Views
445
I Meaning of "identify" & variations in different math context
Replies
5
Views
507
I Base Pi Integers: Isomorphism with Rationals?
Replies
33
Views
4K
I Proving inequality about dimension of quotient vector spaces
Replies
25
Views
3K
A Anti-dual numbers and what are their properties?
Replies
1
Views
2K
I Meaning of the notations: ##\mathbb{Z}[\frac{1}{a}]##
Replies
1
Views
450
B Is Super Commutativity Essential in Defining a Super Lie Module?
Replies
4
Views
1K
A What physical meaning can the “determinant” of a divergency have?
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top