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**ℍ**: Number systems

#### Natural numbers

- - The set of
**Natural numbers**/**Non-negative integers**. forms a well-ordered commutative semiring.

#### Integers

- - The set of
**Integers**, a superset of that can be constructed from equivalence classes of ordered pairs of natural numbers given the equivalence relation , the first is called the minuend and the second the subtrahend. forms a totally ordered integrally closed domain.

#### Integers modulo

- - The set of
**integers modulo**. forms a commutative ring and for prime forms a finite field.

#### Rational numbers

- - The set of
**Rational numbers**, a superset of constructed from equivalence classes of ordered pairs of integers (), the first is called the numerator and the second is strictly nonzero and is called the denominator. forms a totally ordered field and the field of fractions over the ring of integers.

#### Real algebraic numbers

- - The set of
**Real algebraic numbers**, a superset of constructed by finding all real numbers that are roots of a non-constant polynomial with rational coefficients. forms a totally ordered field.

#### Gaussian integers

- - The set of
**Gaussian integers**, a superset of constructed by adjoining the complex number^{[1]}. forms an integrally closed domain and forms a two-dimensional algebra over the ring of integers.

#### Gaussian rationals

- - The set of
**Gaussian rationals**, a superset of constructed by adjoining the complex number to the rationals. forms a field and is the field of fractions over the ring of Gaussian integers. forms a two-dimensional unital associative algebra over the field of rational numbers.

#### Algebraic numbers

- - The set of
**Algebraic numbers**, a superset of and constructed by taking the algebraic closure of . forms an algebraically closed field.

#### Lipschitz integers

- - The set of
**Lipschitz integers**, a superset of constructed by adjoining the quaternionic elements , , and to the integers. forms a noncommutative ring and forms a four-dimensional unital associative algebra over the ring of integers.

#### Gravesian integers

- - The set of
**Gravesian integers**, a superset of constructed by adjoining the seven non-unity octonionic basis elements to the integers. forms an eight-dimensional algebra over the ring of integers.

#### -adic integers

- - The set of
**-adic integers**for prime , a superset of constructed from taking the inverse limit of the integers modulo . forms an integral domain.

#### Ordinals

- , - The proper class of
**Ordinals**, a superclass of . forms a totally ordered proper class-sized left near-semiring and forms a well-ordered proper class-sized left near-semiring.

#### Surintegers

- , - The proper class of
**Surintegers**/**Omnific integers**, a superclass of and . forms a commutative proper class-sized ring.

### Hypercomplex numbers

#### Real numbers

- - The set of
**Real numbers**, a superset of , the Cauchy completion of with respect to Euclidean metric. forms a complete, totally ordered field. where is the identity function and is square function forms a unarion normed division composition algebra over itself.

#### Complex numbers

- , - The set of
**Complex numbers**, a superset of and constructed by considering the algebraic closure of . forms a complete algebraically closed field. where is the complex conjugate and is the absolute value squared forms a binarion normed division composition algebra over the field of real numbers and where is the identity function and is square function forms a unarion normed division composition algebra over itself. Basis is typically taken to be where squares to -1.

#### Dual numbers

- , - The set of
**Dual numbers**/**Parabolic complex numbers**, a superset of . forms a commutative ring and forms a two-dimensional unital associative algebra over the field of real numbers. Basis is typically taken to be . is nilpotent, squaring to 0.

#### Split-complex numbers

- , , - The set of
**Split-complex numbers**/**Hyperbolic complex numbers**/**Double numbers**^{[2]}^{[3]}/**Duplex numbers**/**Perplex numbers**^{[4]}/**Bireal numbers**/**Motors**^{[5]}, a superset of . forms a commutative ring. where where is the split-complex conjugate and is the split-complex modulus squared forms a binarion split-composition algebra over the field of real numbers. Basis is typically taken to be where squares to 1. Alternatively, the basis can be taken to be where is the idempotent and is its split-complex conjugate . The set of all split-complex numbers such that is denoted ; the set of all split-complex numbers such that is denoted ; and the set of all split-complex numbers such that is called the**light cone at**or just the**light cone**, denoted . A light cone at any arbitrary split-complex number is the set of all split-complex numbers such that .

#### Quaternions

- - The set of (real)
**Quaternions**, a superset of . forms a noncommutative division ring. where is the quaternionic conjugate and is the absolute value squared forms a quaternion normed division composition algebra over the field of real numbers. Basis is typically taken to consist of the unit vectors such that , alternatively denoted as . Any arbitrary unit versor that squares to -1 can be taken as equal to the complex number , but it’s traditionally that is taken as .

#### Octonions

- - The set of (real)
**Octonions**, a superset of . where is the octonionic conjugate and is the absolute value squared forms octonion normed division composition algebra over the field of real numbers. Basis is typically taken as , alternatively notated as .

#### Split-quaternions

- , - The set of (real)
**Split-quaternions**, a superset of , , and . forms a noncommutative ring. where is the split-quaternionic conjugate and is the split-quaternionic modulus squared forms a quaternion split-composition algebra over the field of real numbers. Basis is typically taken as where and , alternatively notated as . The nilpotent can be taken to equal to the dual number .

#### Ternions

- - The set of (real)
**Ternions**, a superset of and and a subset of . forms a three-dimensional unital associative algebra over the field of real numbers. Basis is typically taken to be . forms a noncommutative ring.

#### Split-octonions

- , - The set of (real)
**Split-octonions**, a superset of and . where is the split-octonionic conjugate and is the split-octonionic modulus squared forms an octonion split-composition algebra over the field of real numbers. Basis is typically taken as such that if and if , alternatively notated as . The nilpotents and can be taken to equal to and respectively.

#### Quintonions

- - The set of (real)
**Quintonions**, a superset of and a subset of . forms a five-dimensional unital nonassociative alternative algebra over the field of real numbers. Basis is typically taken to be .

#### Sextonions

- - The set of (real)
**Sextonions**, a superset of and a subset of . forms a six-dimensional unital nonassociative alternative algebra over the field of real numbers. Basis is typically taken to be .

#### Bicomplex numbers

- , - The set of (real)
**Bicomplex numbers**/**Complex complex numbers**/(real)**Tessarine numbers**(as an algebra over the real numbers), a superset of and . forms a commutative ring. where is the bicomplex conjugate and is the bicomplex modulus squared forms binarion split-composition algebra over the field of complex numbers. form a four-dimensional unital associative algebra over the field of real numbers. Basis is typically taken as either and every bicomplex number is written as or such that and every tessarine number is written as .

#### Biquaternions

- , - The set of (real)
**Biquaternions**/**Complex quaternions**/(real)**Octarine numbers**, a superset of and . forms a noncommutative ring. where is the biquaternionic conjugate and is the biquaternionic modulus squared forms a quaternion split-composition algebra over the field of complex numbers. form an eight-dimensional unital associative algebra over the field of real numbers.

#### Bioctonions

- , - The set of (real)
**Bioctonions**/**complex octonions**/(real)**Hexadecarine numbers**, a superset of and . where is the bioctonionic conjugate and is the bioctonionic modulus squared forms an octonion split-composition algebra over the field of complex numbers. forms a sixteen-dimensional unital nonassociative alternative algebra over the field of real numbers.

#### Sedenions

- - The set of (real)
**Sedenions**, a superset of constructed from applying the Cayley-Dickson construction to the octonions. forms a sixteen-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Trigintaduonions

- - The set of (real)
**Trigintaduonions**/**Pathions**^{[6]}, a superset of constructed from applying the Cayley-Dickson construction to the sedenions. forms a thirty-two-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Tricomplex numbers

- - The set of (real)
**Tricomplex numbers**/**complex bicomplex numbers**, a superset of . forms a commutative ring. forms a two-dimensional unital associative algebra over the ring of bicomplex numbers, forms a four-dimensional unital associative algebra over the field of complex numbers, and forms an eight-dimensional unital associative algebra over the field of real numbers.

#### Tetracomplex numbers

- - The set of (real)
**Tetracomplex numbers**/**complex tricomplex numbers**, a superset of . forms a commutative ring. forms a two-dimensional unital associative algebra over the ring of tricomplex numbers, forms a four-dimensional unital associative algebra over the ring of bicomplex numbers, forms an eight-dimensional unital associative algebra over the field of complex numbers and forms a sixteen-dimensional unital associative algebra over the field of real numbers.

#### Pentacomplex numbers

- - The set of (real)
**Pentacomplex numbers**/**complex tetracomplex numbers**, a superset of . forms a commutative ring. forms a two-dimensional unital associative algebra over the ring of tetracomplex numbers, forms a four-dimensional unital associative algebra over the ring of tricomplex numbers, forms a eight-dimensional unital associative algebra over the ring of bicomplex numbers, forms sixteen-dimensional unital associative algebra over the field of complex numbers and forms a thirty-two-dimensional unital associative algebra over the field of real numbers.

#### Split-biquaternions

- , - The set of (real)
**Split-biquaternions**/**split-complex quaternions**/**elliptic biquaternions**/**Clifford biquaternions**/**dyquaternions**, a superset of and . forms a noncommutative ring. forms a four-dimensional unital associative algebra over the ring of split-complex numbers and forms an eight-dimensional unital associative algebra over the field of real numbers.

#### Dual quaternions

- - The set of
**Dual quaternions**a superset of and . forms a noncommutative ring. forms a four-dimensional unital associative algebra over the ring of dual numbers and forms an eight-dimensional unital associative algebra over the field of real numbers.

#### Dual complex numbers

- - The set of
**Dual complex numbers**, a superset of and . forms a commutative ring. forms a two-dimensional unital associative algebra over the ring of dual numbers, forms a two-dimensional unital associative algebra over the field of complex numbers and forms a four-dimensional unital associative algebra over the field of real numbers.

#### Dual octonions

- - The set of
**Dual octonions**a superset of and . . forms a four-dimensional unital associative algebra over the ring of dual numbers and forms an eight-dimensional unital nonassociative algebra over the field of real numbers.

#### Parabolic quaternions

- - The set of (real)
**Parabolic quaternions**, a superset of constructed similarly to the quaternions but the non-unity basis elements all square to 0 instead of -1. forms a four-dimensional nonassociative nonalternative algebra over the field of real numbers.

#### Hyperbolic quaternions

- - The set of (real)
**Hyperbolic quaternions**, a superset of constructed similarly to the quaternions but the basis elements all square to 1 instead of -1. forms a four-dimensional nonassociative nonalternative algebra over the field of real numbers.

#### Parabolic octonions

- - The set of (real)
**Parabolic octonions**, a superset of constructed similarly to the octonions but the non-unity basis elements all square to 0 instead of -1. forms an eight-dimensional nonassociative nonalternative algebra over the field of real numbers.

#### Hyperbolic octonions

- - The set of (real)
**Hyperbolic octonions**, a superset of constructed similarly to the octonions but the basis elements all square to 1 instead of -1. forms an eight-dimensional nonassociative nonalternative algebra over the field of real numbers.

#### Split-sedenions

- , - The set of (real)
**Split-sedenions**, a superset of and . forms a sixteen-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Split-trigintaduonions

- , - The set of (real)
**Split-trigintaduonions**, a superset of and . forms a thirty-two-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Sexagintaquattuornions

- - The set of (real)
**Sexagintaquattuornions**/**Chingons**^{[7]}, a superset of constructed from applying the Cayley-Dickson construction to the trigintaduonions. forms a sixty-four-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Centumduodetrigintanions

- - The set of (real)
**Centumduodetrigintanions**/**Routons**^{[8]}, a superset of constructed from applying the Cayley-Dickson construction to the sexagintaquattuornions. forms a one-hundred-twenty-eight-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

#### Ducentiquinquagintasexions

- - The set of (real)
**Ducentiquinquagintasexions**/**Voudons**^{[9]}, a superset of constructed from applying the Cayley-Dickson construction to the centumduodetrigintanions. forms a two-hundred-fifty-six-dimensional unital nonassociative nonalternative algebra over the field of real numbers.

### Extended hypercomplex numbers

#### Affinely extended real numbers

- - The set of
**Affinely extended real numbers**. Has two new elements and that close the real line, creating a structure topologically equivalent to a line segment called the extended real line.

#### Projectively extended real numbers

- - The set of
**Projectively extended real numbers**, the projective line over the field of real numbers. Has a new element that closes the real line, creating a structure topologically equivalent to a circle called the real projective line.

#### Extended complex numbers

- - The set of
**Extended complex numbers**, the one-point compactification of and projective line over the field of complex numbers. Has a new element that closes the complex line, creating a sphere called the Riemann sphere. is taken to be . The Riemann sphere can be embedded in as a projection of the complex line onto the unit sphere .

#### Extended dual numbers

- - The set of
**Extended dual numbers**, the projective line over the ring of dual numbers. The structure created is an infinite tube. The dual projective line can be embedded in as a projection of the dual line onto the infinite tube .

#### Extended split-complex numbers

- - The set of
**Extended split-complex numbers**, the projective line over the ring of split-complex numbers. can be embedded in as a projection of the onto the hyperboloid of one sheet . The split-complex projective line can be embedded in .

#### Extended quaternions

- - The set of
**Extended quaternions**, the one-point compactification of and projective line over the ring of quaternions. Has a new element that closes the quaternionic line, creating a structure topologically equivalent to a tetrasphere called the quaternionic projective line. The quaternionic projective line can be embedded in as a projection of the quaternionic line onto the unit tetrasphere.

#### Extended octonions

- - The set of
**Extended octonions**, the one-point compactification of and projective line over the octonions. Has a new element that closes the octonionic line, creating a structure topologically equivalent to an octasphere called the octonionic projective line. The quaternionic projective line can be embedded in as a projection of the octonionic line onto the unit octasphere.

#### Extended split-quaternions

- - The set of
**Extended split-quaternions**, the projective line over the ring of split-quaternions. - , - The set of
**Extended bicomplex numbers**/**Extended tessarines**, the projective line over the ring of bicomplex numbers.

#### Extended biquaternions

- , - The set of
**Extended biquaternions**, the projective line over the ring of biquaternions.

#### Extended tricomplex numbers

- - The set of
**Extended tricomplex numbers**, the projective line over the ring of tricomplex numbers.

#### Extended tetracomplex numbers

- - The set of
**Extended tetracomplex numbers**, the projective line over the ring of tetracomplex numbers.

#### Extended pentacomplex numbers

- - The set of
**Extended pentacomplex numbers**, the projective line over the ring of pentacomplex numbers.

#### Extended split-biquaternions

- , - The set of
**Extended split-biquaternions**, the projective line over the ring of split-biquaternions.

#### Extended dual quaternions

- - The set of
**Extended dual quaternions**, the projective line over the ring of dual quaternions.

#### Extended dual complex numbers

- - The set of
**Extended dual complex numbers**, the projective line over the ring of dual complex numbers.

### -adic hypercomplex numbers

#### -adic numbers

- - The set of
**-adic numbers**for prime , a superset of and , the Cauchy completion of with respect to the -adic metric. forms a complete field.

#### Algebraic closure of -adic numbers

- - The set of all elements of the algebraic closure of . forms an algebraically closed field. Unlike the algebraic closure of , , which has a degree of 2 (the degree of a field extension is the minimum number of elements of the field needed to describe elements of the field extension, two real numbers are needed to describe a complex number) and is Cauchy complete, the degree of is and is not Cauchy complete.

#### -adic complex numbers

- - The set of
**-adic complex numbers**, a superset of , the Cauchy completion of . forms an algebraically closed field.

### Surhypercomplex numbers

#### Surreal numbers

- - The proper class of
**Surreal numbers**, a superclass of and . forms a totally ordered proper class-sized field. where is the identity function and is square function forms a unarion division composition algebra over itself.

#### Surcomplex numbers

- / - The proper class of
**Surcomplex numbers**, a superclass of and constructed by considering the algebraic closure of . forms an algebraically complete proper class-sized field. where is the surcomplex conjugate and is the surcomplex absolute value squared forms a binarion division composition algebra over the field of surreal numbers.

### Others

- - The set of all elements of a non-unital D “normed division algebra” over the field of real numbers. An example of which is the set of all sequences of real numbers such that the sum of all elements of the sequence converge (). The absolute value over is defined to be and multiplication is defined as . Elements of and be nicknamed “
**apeironions**”.

## References

- ↑ I will be using to represent the classical complex imaginary unit, to represent the classical dual imaginary unit, and to represent the classical split-complex imaginary unit to differentiate them a little more from other similarly notated objects.
- ↑ https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/other/doubleNumbers/index.htm
- ↑ https://pdfs.semanticscholar.org/a93b/3e9b0eda86c95ed40a032d7bf48b15935563.pdf
- ↑ https://aapt.scitation.org/doi/abs/10.1119/1.14605
- ↑ https://en.wikipedia.org/wiki/Motor_variable
- ↑ https://arxiv.org/abs/math/0207003
- ↑ https://arxiv.org/abs/math/0207003
- ↑ https://arxiv.org/abs/math/0207003
- ↑ https://arxiv.org/abs/math/0207003