The complex numbers $ℂ$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 17 May 2011

Complex numbers $ℂ$

The complex numbers $ℂ$ is the set $$ℂ=\{a+bi|a,b\in ℝ\},\text{with}{i}^2=-1,$$ so that addition is given by $$(a+bi)+(c+di)=(a+c)+(b+d)i\text{,}$$ and multiplication is given by $$(a+bi)(c+di)=(ac-bd)+(ad+bc)i\text{.}$$

Define the absolute value on $ℂ$ by $$\begin{array}{rrcl}\left|\phantom{x}\right|:& ℂ& ⟶& {ℝ}_{\ge 0}\\ & z& ⟼& \left|z\right|\end{array}\text{by}\left|z\right|=\sqrt{a^2+b^2}$$ if $z=a+bi$.

Define the distance on $ℂ$, $$d:ℂ\times ℂ\to {ℝ}_{\ge 0}\text{by}d(x,y)=|y-x|\text{.}$$

Let $ϵ\in {ℝ}_{>0}$. The $ϵ$-ball at $x$ is $${ℬ}_{ϵ}\left(x\right)=\{y\in ℂ|d(x,y)<ϵ\}\text{.}$$

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

Complex conjugation is the function $$\bar{\phantom{P}}:ℂ\to ℂ\text{given by}\overline{a+bi}=a-bi\text{.}$$

(a) The set $ℂ$ with the operations of addition, multiplication and open sets as in (1.1) is a topological field.

(b) If $f:ℂ\to ℂ$ is a function such that

(a) If $x,y\in ℂ$ then $f(x+y)=f\left(x\right)+f\left(y\right)$,

(b) If $x,y\in ℂ$ then $f\left(xy\right)=f\left(x\right)f\left(y\right)$, and

(c) If $x\in ℝ$ then $f\left(x\right)=x$,

then $f$ is either the identity function or complex conjugation.

$ℂ$ as an $ℝ$-algebra

The complex numbers is the $ℝ$-algebra

$ℂ=ℝ\text{-span}\{1,i\}=\{x_0+x_{1}i|x_0,x_1\in ℝ\}$

with product and topology determined by

$i^2=-1\text{and}ℂ\simeq {ℝ}^2,\text{respectively.}$

The norm $N:ℂ\to {ℝ}_{\ge 0}$ and absolute value $\left|\phantom{x}\right|:ℂ\to {ℝ}_{\ge 0}$ are given by

$N(x_0+x_{1}i)=x_0^2+x_1^2={\left|x\right|}^2$.

Then

$N\left(xy\right)=N\left(x\right)N\left(y\right)$.

The conjugate $\bar{\phantom{P}}:ℂ\to ℂ$ is given by

$\overline{x_0+x_{1}i}=x_0-x_{1}i\text{and}\overline{xy}=\bar{y}\bar{x}$.

The conjugate is the unique automorphism of the topological field $ℂ$ that is not the identity. Then

$\begin{array}{cl}{\mathrm{GL}}_1\left(ℂ\right)& ={ℂ}^{\times }=\{x\in ℂ|x\ne 0\}\\ \cup |& \\ U_1\left(ℂ\right)& =\{x\in ℂ|x{\bar{x}}^t=1\}=\{x_0+x_{1}i|x_0^2+x_1^2=1\}\end{array}$

and

$\begin{array}{ccc}{ℂ}^{\times }& \stackrel{\sim }{⟶}& {\left({ℝ}^{\times }\right)}^{\circ }\times U_1\left(ℂ\right)\\ x& ⟼& \left|x\right|\cdot \frac{x}{\left|x\right|}\end{array}$

Here ${\left({ℝ}^{\times }\right)}^{\circ }={ℝ}_{>0}$ is the connected component of the identity in the Lie group ${\mathrm{GL}}_1\left(ℝ\right)={ℝ}^{\times }$. This polar decomposition is an example of the Cartan decomposition $G=K\cdot \left(\exp 𝔭\right)$ (see Segal Theorem 4.1 and/or Knapp Prop. 1.2), where $K$ is a maximal compact subgroup of $G$, and $𝔤=𝔨\oplus 𝔭$ with $𝔤=\mathrm{Lie}\left(G\right)$ and $𝔭$ orthogonal to $𝔨=\mathrm{Lie}\left(K\right)$ with respect to the Killing form.

Notes and References

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

References

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

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