The complex numbers

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 17 May 2011

Complex numbers

The complex numbers is the set = { a+bi | a,b }, with i2=-1, so that addition is given by (a+bi) + (c+di) = (a+c) + (b+d) i , and multiplication is given by (a+bi) (c+di) = (ac-bd) + (ad+bc) i .

Define the absolute value on by |x| : 0 z |z| by |z| = a2 + b2 if z= a+bi.

Define the distance on , d:× 0 by d(x,y) =|y-x| .

Let ϵ >0 . The ϵ-ball at x is ϵ(x) = { y | d(x,y) <ϵ }.

Let E be a subset of . The set E is open if E is a union of ϵ-balls.

Complex conjugation is the function P : given by a+bi = a-bi .

(a) The set with the operations of addition, multiplication and open sets as in (1.1) is a topological field.
(b) If f: is a function such that
(a) If x,y then f(x+y) = f(x) + f(y) ,
(b) If x,y then f(xy) = f(x) f(y) , and
(c) If x then f(x) = x ,
then f is either the identity function or complex conjugation.

as an -algebra

The complex numbers is the -algebra
= -span {1,i} ={ x0+x1i | x0,x1 }
with product and topology determined by
i2=-1 and 2, respectively.
The norm N: 0 and absolute value |x|: 0 are given by
N(x0+x1i) = x02 + x12 = |x|2 .
N(xy) = N(x) N(y).
The conjugate P : is given by
x0+x1i = x0-x1i and xy = y x .
The conjugate is the unique automorphism of the topological field that is not the identity. Then
GL1() =× ={x | x0} | U1() ={x | xxt=1 } = {x0+x1i | x02 + x12 =1 }
× (×) × U1() x |x| x|x|
Here (×) =>0 is the connected component of the identity in the Lie group GL1() =×. This polar decomposition is an example of the Cartan decomposition G=K(exp𝔭) (see Segal Theorem 4.1 and/or Knapp Prop. 1.2), where K is a maximal compact subgroup of G, and 𝔤=𝔨𝔭 with 𝔤=Lie(G) and 𝔭 orthogonal to 𝔨 =Lie(K) with respect to the Killing form.

Notes and References

The reference [Ch. VIII § 1, BouTop] provides a thorough introduction to the complex numbers. Viewing as an -algebra helps to make the construction of the Hamiltonians feel natural.


[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????

page history