Complex Algebra

Introduction

This article introduces the complex algebra as an algebraic structure, without yet discussing its representations. The goal is to define the algebra precisely, establish its basic properties, and describe the distinguished real vector subspaces that arise from the natural conjugation.

The treatment is mathematically honest: every claim is either proved or stated as a definition. No physics is invoked. The complex algebra is defined algebraically, and its identification with the plane is not covered here.

The Complex Numbers

Definition

The complex algebra $\mathbb{C}$ is the two-dimensional real algebra with basis

$$ 1, \qquad i, $$

and multiplication rules

$$ i^2 = -1. $$

A general complex number is written in developed form as

$$ A = a\,e_0 + a'\,e_1, \qquad a, a' \in \mathbb{R}, $$

or, more compactly, as

$$ A = a + i a', \qquad a, a' \in \mathbb{R}. $$

The real number $a$ is the real part, and the real number $a'$ is the imaginary part. We also write

$$ A = a + i a', \qquad a = \operatorname{Re} A, \quad a' = \operatorname{Im} A. $$

The notation $i$ is chosen deliberately. In some of the older literature, the imaginary unit is written $j$ or $\sqrt{-1}$, which collides visually with the index $j$ or with the square root sign. This collision is a persistent source of confusion, especially when the complex algebra is tensored with other algebras. Using $i$ for the imaginary unit and reserving $j$ for indices avoids the collision entirely. This is the notation we use throughout the blog.

Basic Properties

Commutative. Complex multiplication is commutative: $A B = B A$.

Associative. Complex multiplication is associative: $(A B) C = A (B C)$.

Division algebra. Every nonzero complex number has a multiplicative inverse, and there are no zero divisors. The conjugation is

$$ \bar A = a - i a', $$

and the inverse is built from it with the norm. The norm is a form and a distance, a structure of the layer above this one, and the inverse formula belongs to the companion article Complex Norm and Invertibility; the algebra layer records only that the inverse exists.

Frobenius theorem. The complex algebra is one of only three finite-dimensional associative real division algebras, the others being $\mathbb{R}$ and $\mathbb{H}$. In fact, $\mathbb{C}$ is the only one that is commutative but not ordered.

The Imaginary Part and $\mathbb{R}^2$

The pair $(a, a')$ can be identified with a vector in $\mathbb{R}^2$. Under this identification, the product of two complex numbers is

$$ A B = (a b - a' b') + (a b' + a' b) i, \qquad A = a + i a', \quad B = b + i b'. $$

This is the formula for the product in the algebra, written in real coordinates. There is no dot product and no cross product in this expression, because $\mathbb{C}$ is commutative and two-dimensional. The multiplication is the only operation of the algebra; the reading of it as a rotation and a scaling of the plane is a geometric one, made in the companion article Rotations and Reflections in the Complex Plane.

Complex Algebra

Definition

The complex algebra is the field $\mathbb{C}$ considered as a two-dimensional real vector space equipped with its field multiplication. As a real vector space, $\mathbb{C}$ has dimension $2$. As a ring, it is a field. A general element is written in developed form as

$$ A = a\,e_0 + a'\,e_1, \qquad a, a' \in \mathbb{R}, $$

or, more compactly, as

$$ A = a + i a', \qquad a, a' \in \mathbb{R}. $$

We write

$$ A = a + i a', $$

where $a$ is the real part and $a'$ is the imaginary part. The tilde is absent because $A$ is an element of the algebra $\mathbb{C}$, not a vector in some larger space.

Multiplication

The product of two complex numbers is defined by extending the real multiplication bilinearly:

$$ A B = (a + i a')(b + i b') = (a b - a' b') + (a b' + a' b) i. $$

In developed form,

$$ A B = \sum_{\mu=0}^{1} \sum_{\nu=0}^{1} A_\mu B_\nu \, e_\mu e_\nu, $$

where the products $e_\mu e_\nu$ are those of the complex algebra.

Conjugations

There are two natural conjugations on $\mathbb{C}$, and they form the two-element group generated by complex conjugation:

Complex conjugation $\bar A$:

$$ \bar A = a - i a'. $$

Identity conjugation $\operatorname{id}$:

$$ \operatorname{id}(A) = A. $$

Each conjugation is an involution: applying it twice returns the original complex number. Each has a fixed-point set, which is a real vector subspace of $\mathbb{C}$. The two subspaces are described in the following sections.

The Two Fixed-Point Subspaces

Each of the two conjugations has a fixed-point set, i.e., a set of complex numbers left invariant by the conjugation. Each fixed-point set is a real vector subspace of $\mathbb{C}$. The two subspaces are described below.

The Real Subspace

The fixed points of complex conjugation are the complex numbers satisfying $\bar A = A$. In developed form,

$$ a - i a' = a + i a'. $$

Comparing the coefficients of $1$ and $i$:

• Coefficient of $1$: $a = a$, always satisfied.
• Coefficient of $i$: $-a' = a'$, so $a' = 0$.

The fixed points are complex numbers with vanishing imaginary part:

$$ A = a, \qquad a \in \mathbb{R}. $$

This is the real subspace $\mathbb{R}_{\mathbb{C}}$, a copy of the real number line embedded in $\mathbb{C}$ as the real axis. It is a real vector space of dimension $1$. It is a subalgebra of $\mathbb{C}$ (isomorphic to $\mathbb{R}$), and it is the only one of the two fixed-point sets that is an ordered field.

The Complex Subspace

The fixed points of the identity conjugation are all complex numbers, since $\operatorname{id}(A) = A$ for every $A$. In developed form they are

$$ A = a + i a', \qquad a, a' \in \mathbb{R}. $$

This is the complex subspace $\mathbb{C}_{\mathbb{C}}$, a copy of the complex plane embedded in $\mathbb{C}$ as the whole thing. It is a real vector space of dimension $2$. It is a subalgebra of $\mathbb{C}$, and it is a division algebra.

Complex Decomposition

The real subspace $\mathbb{R}_{\mathbb{C}}$ and its imaginary translate $i \mathbb{R}_{\mathbb{C}}$ are the two eigenspaces of complex conjugation. Every complex number decomposes uniquely as the sum of a real part and an imaginary part:

$$ A = a + i a', \qquad a \in \mathbb{R}, \quad a' \in \mathbb{R}. $$

The real and the imaginary parts are obtained from the complex conjugation:

$$ a = \frac{1}{2}(A + \bar A), \qquad a' = \frac{1}{2i}(A - \bar A). $$

Indeed, $a$ is fixed by complex conjugation, so it lies in $\mathbb{R}_{\mathbb{C}}$, and $a'$ is real, so $i a'$ is negated by complex conjugation.

This gives the direct sum decomposition

$$ \mathbb{C} = \mathbb{R}_{\mathbb{C}} \oplus i \mathbb{R}_{\mathbb{C}}, $$

where $i \mathbb{R}_{\mathbb{C}}$ is the set of complex numbers of the form $i a'$ with $a' \in \mathbb{R}$. Both are real vector spaces of dimension $1$, and their direct sum is the full algebra $\mathbb{C}$ of real dimension $2$.

This is the complex decomposition of a complex number. It expresses $A$ as a real number plus $i$ times another real number. It is the natural decomposition when we think of $\mathbb{C}$ as the complexification of $\mathbb{R}$.

Conjugate Decomposition

The complex conjugation $\bar{\cdot}$ is an involution, so it has eigenvalues $+1$ and $-1$. Its eigenspaces are the real subspace $\mathbb{R}_{\mathbb{C}}$ (eigenvalue $+1$) and the imaginary subspace $i \mathbb{R}_{\mathbb{C}}$ (eigenvalue $-1$). Every complex number decomposes uniquely as

$$ A = A_+ + A_-, \qquad A_+ = a, \quad A_- = i a'. $$

This is the same decomposition as above, written in terms of the eigenspaces of the conjugation. It is the algebraic analogue of writing a real number as the sum of its even and odd parts under a reflection.

Quadratic Forms and Inner Product

The quadratic form of the algebra and the inner product are a form and a distance, structures of the topology layer, which this layer may name but cannot measure with. They are developed in the companion article Complex Norm and Invertibility, which develops the norm $N(A) = A\bar{A}$, its multiplicativity, the Hermitian form, the Euclidean norm and the inner product $\langle A, B\rangle = \bar{A}B$. Nothing is proved with them here, and the absence of that development is itself the layer statement of this article: the algebra carries its multiplication, its single nontrivial involution and its two subspaces, and the form enters at the next layer.

Summary

The complex algebra $\mathbb{C}$ is the real algebra of dimension $2$ with basis $1$, $i$ and the single relation $i^2 = -1$. With its multiplication it is a field: commutative, associative and unital, with every nonzero element invertible. A general element is written as $A = a + i a'$.

Complex conjugation is the nontrivial $\mathbb{R}$-linear involution; its fixed subspace is the real subspace $\mathbb{R}_{\mathbb{C}}$, on which it acts as the identity, and its anti-fixed subspace is the imaginary subspace $i\mathbb{R}_{\mathbb{C}}$, which it negates. These are the eigenspaces for the eigenvalues $+1$ and $-1$, and every complex number decomposes uniquely in each of the two ways the article records: as a real part plus an imaginary part, and as the sum of the $+1$ and $-1$ eigencomponents.

The quadratic form of the algebra and the inner product — the norm $N(A) = A\bar{A}$, the Hermitian form and the inner product $\langle A, B\rangle = \bar{A}B$ — are a form and a distance, structures of the topology layer, and they are developed in the companion article Complex Norm and Invertibility, not here.

Summary of Notation

Further Reading

• William Rowan Hamilton, Lectures on Quaternions (1853), for the origin of the complex algebra in the quaternion program.
• Carl Friedrich Gauss, Theoria residuorum biquadraticorum (1831), for the geometric interpretation of complex numbers.
• Edmund Landau, Grundlagen der Analysis (1930), for the axiomatic treatment of the complex field.
• Walter Rudin, Real and Complex Analysis (McGraw-Hill, 1987), for the standard modern treatment.
• John H. Conway and Derek A. Smith, On Quaternions and Octonions (A K Peters, 2003), for the classification of real division algebras.
• Charles C. Pinter, A Book of Abstract Algebra (Dover, 2010), for the representation theory of fields.