List of Number Systems by Property

Introduction

This article lists the number systems of the corpus as a table of properties: for each system it records whether the multiplication is commutative, associative, alternative and power-associative, whether there are zero divisors, whether the system is a division algebra, and whether it is a field. The list is organised along the doubling chain, and at each step it names the property that is lost.

Every entry points to the article that introduces the number system and states its properties. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a number system that fails a property the previous one had — the loss of order at $\mathbb{C}$, of commutativity at $\mathbb{H}$, of associativity at $\mathbb{O}$, of the multiplicative norm at the sedenions — with the failure named and the article that records it.

The Ordered Number Systems

The chain begins with the systems that are ordered: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$. Each is commutative, associative, alternative and power-associative; $\mathbb{N}$ is not a group under addition, $\mathbb{Z}$ is an integral domain but not a field, $\mathbb{Q}$ and $\mathbb{R}$ are fields, and only $\mathbb{R}$ is complete. None has zero divisors.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{N}$	commutative, associative, power-associative; no zero divisors; not a division algebra and not a field; ordered	The Natural Numbers
$\mathbb{Z}$	commutative, associative; an integral domain, no zero divisors; not a division algebra and not a field; ordered	The Integers
$\mathbb{Q}$	commutative, associative; a field, so a division algebra; no zero divisors; ordered	The Rational Numbers
$\mathbb{R}$	commutative, associative; a field and a division algebra; no zero divisors; ordered and complete	The Real Numbers
$\mathbb{Q}$ as a prime field	the smallest field of characteristic $0$; the prime field embedding	Fields
Non-example: $\mathbb{N}$ as a field	fails: it has no additive inverses and no multiplicative inverses beyond $1$	The Natural Numbers
Non-example: $\mathbb{Z}$ as a division algebra	fails: only $\pm 1$ are units	The Integers

The Complex Numbers: the Loss of Order

The doubling of $\mathbb{R}$ gives $\mathbb{C}$, which retains commutativity, associativity, alternativity and power-associativity, remains a field and a division algebra, and has no zero divisors. What it loses is the order: $\mathbb{C}$ is not orderable, and its only field automorphisms fixing $\mathbb{R}$ are the identity and complex conjugation.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{C}$	commutative, associative; a field and a division algebra; no zero divisors; not orderable	The Complex Numbers
$\mathbb{C}$ as $\mathbb{R}[t]/(t^2+1)$	the algebraic closure of $\mathbb{R}$; every polynomial splits	Splitting Fields and Algebraic Closure
Complex conjugation	the nontrivial $\mathbb{R}$-automorphism; the only non-identity one	The Complex Numbers
Non-example: $\mathbb{C}$ as an ordered field	fails: no ordering is compatible with the field operations	Algebraically Closed Fields
Non-example: $\mathbb{C}$ as a real algebra of dimension $1$	fails: its real dimension is $2$	The Complex Numbers

The Quaternions: the Loss of Commutativity

The doubling of $\mathbb{C}$ gives the quaternions $\mathbb{H}$, which retain associativity, alternativity and power-associativity, remain a division algebra and have no zero divisors, but lose commutativity: $ij = k$ while $ji = -k$. The quaternions are a division ring but not a field, and their centre is $\mathbb{R}$.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{H}$	associative, alternative, power-associative; a division algebra, no zero divisors; not commutative, not a field; centre $\mathbb{R}$	Quaternion Algebra
Quaternion units	$i^2 = j^2 = k^2 = ijk = -1$; $ij = k = -ji$	Quaternion Algebra
$\mathbb{H}$ as a division ring	every nonzero element invertible, with $x^{-1} = \bar x/N(x)$	Quaternion Algebra
$\mathbb{H}$ as an algebra over its centre	dimension $4$ over $\mathbb{R}$, the centre	Division Algebras
Non-example: $\mathbb{H}$ as a field	fails: multiplication is not commutative	Quaternion Algebra
Non-example: $\mathbb{H}$ as an ordered field	fails: an ordered field is commutative, and $\mathbb{H}$ is not even a field	Quaternion Algebra

The Octonions: the Loss of Associativity

The doubling of $\mathbb{H}$ gives the octonions $\mathbb{O}$, which retain alternativity and power-associativity, remain a division algebra and have no zero divisors, but lose associativity: the associator $(xy)z - x(yz)$ is nonzero. The octonions are the last normed division algebra, and they are neither commutative nor associative.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{O}$	alternative and power-associative; a division algebra, no zero divisors; not associative, not commutative, not a field	Octonion Algebra
Octonion basis $e_0,\dots,e_7$	$e_0 = 1$, $e_i^2 = -1$ for $i \geq 1$; the Fano-plane multiplication	Octonion Algebra
The associator	$(e_1e_2)e_4 = e_7$ but $e_1(e_2e_4) = -e_7$; the failure of associativity made explicit	Normed Division Algebras and the Hurwitz Theorem
Artin's theorem	the subalgebra generated by two elements is associative, so $\mathbb{O}$ is alternative	Octonion Algebra
Non-example: $\mathbb{O}$ as an associative algebra	fails: the associator is nonzero	Octonion Algebra
Non-example: $\mathbb{O}$ as a field	fails: it is not commutative and not associative	Octonion Algebra

The Sedenions: the Loss of Division

The doubling of $\mathbb{O}$ gives the sedenions $\mathbb{S}$, of dimension sixteen; the norm is no longer multiplicative, the system has zero divisors, and it is therefore not a division algebra. The sedenions are still power-associative, and the doubling continues indefinitely as a sequence of algebras.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{S}$	power-associative; has zero divisors; not a division algebra, not alternative, not a field	Octonion Algebra
The failure of the norm	$N(xy) \neq N(x)N(y)$ in dimension $16$; the multiplicative norm stops at $\mathbb{O}$	Normed Division Algebras and the Hurwitz Theorem
Non-example: $\mathbb{S}$ as a division algebra	fails: it has zero divisors	Octonion Algebra
Non-example: a normed division algebra of dimension $16$	does not exist: Hurwitz' theorem stops the normed division algebras at $8$	Normed Division Algebras and the Hurwitz Theorem

The Split and Degenerate Number Systems

Beside the division chain the corpus develops the split and degenerate systems, which are commutative or associative but have zero divisors and are not division algebras: the split-complex numbers, the dual numbers, the split-biquaternions, the split-octonions and the biquaternions. These systems show that the properties are independent, and that losing order, commutativity or associativity is not the only way to fail to be a field.

Number system	Commutative, associative, alternative, power-associative; zero divisors; division algebra; field	Introduced in
$\mathbb{D}$ split-complex	commutative, associative; has zero divisors, $\cong \mathbb{R}\times\mathbb{R}$; not a division algebra, not a field	Split-Complex Algebra
$\mathbb{D}'$ dual numbers	commutative, associative; has nilpotents, local; not a division algebra, not a field	Dual Numbers Algebra
Split biquaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	associative; has zero divisors, $\cong \mathbb{H}\oplus\mathbb{H}$, and no nonzero nilpotents; not a division algebra, not a field	Split-Biquaternion Algebra
Split-octonions	alternative; has zero divisors; not a division algebra, not associative	Normed Division Algebras and the Hurwitz Theorem
$\mathbb{B}$ biquaternions	associative; has zero divisors, $\cong M_2(\mathbb{C})$; not a division algebra, not a field	Biquaternion Algebra
Non-example: $\mathbb{D}$ as a field	fails: $(1-t)(1+t) = 0$	Split-Complex Algebra
Non-example: the split biquaternions as a division algebra	fail: they have zero divisors, though they have no nonzero nilpotents	Split-Biquaternion Algebra

The Property Table

The properties are independent, and the table records which system has which; each entry reads yes or no.

System	Commutative	Associative	Alternative	Power-associative	No zero divisors	Division algebra	Field
$\mathbb{N}$	yes	yes	yes	yes	yes	no	no
$\mathbb{Z}$	yes	yes	yes	yes	yes	no	no
$\mathbb{Q}$	yes	yes	yes	yes	yes	yes	yes
$\mathbb{R}$	yes	yes	yes	yes	yes	yes	yes
$\mathbb{C}$	yes	yes	yes	yes	yes	yes	yes
$\mathbb{H}$	no	yes	yes	yes	yes	yes	no
$\mathbb{O}$	no	no	yes	yes	yes	yes	no
$\mathbb{S}$	no	no	no	yes	no	no	no
$\mathbb{D}$	yes	yes	yes	yes	no	no	no
$\mathbb{D}'$	yes	yes	yes	yes	no	no	no
$\mathbb{H}_{\mathbb{D}}$	no	yes	yes	yes	no	no	no
split-octonions	no	no	yes	yes	no	no	no
$\mathbb{B}$	no	yes	yes	yes	no	no	no

Summary

The list gathers the number systems of the corpus by their properties. The ordered systems are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$, which are commutative and associative and have no zero divisors. The doubling chain then loses one property at each step: $\mathbb{C}$ loses order, $\mathbb{H}$ loses commutativity, $\mathbb{O}$ loses associativity and the sedenions lose the multiplicative norm and acquire zero divisors. $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are the four normed division algebras, and only $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields. The split and degenerate systems — the split-complex numbers, the dual numbers, the split-biquaternions, the split-octonions and the biquaternions — are commutative or associative but have zero divisors and are not division algebras. The final table records the seven properties for all thirteen systems. The non-examples — $\mathbb{N}$ and $\mathbb{Z}$ as non-fields, $\mathbb{Z}$ as a non-division algebra, $\mathbb{C}$ as non-orderable, $\mathbb{H}$ as non-commutative, $\mathbb{O}$ as non-associative, $\mathbb{S}$ as having zero divisors, the split-complex numbers as a non-field and the split-biquaternions as non-division-algebras — each name the failure.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$	naturals, integers, rationals, reals
$\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$	complex, split-complex, dual numbers
$\mathbb{H}$, $\mathbb{O}$, $\mathbb{S}$	quaternions, octonions, sedenions
$\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	split biquaternions, of real dimension $8$
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	biquaternions
$i,j,k$	quaternion units
$e_0,\dots,e_7$	octonion basis; $e_0,\dots,e_{15}$ the sedenion basis
$N(x)$	norm