## Functions of a hypercomplex variable

#### Constant zero function

• Complex constant zero function (complex map / 3D plot / sliders / domain colouring)
• • is elementary
• is complex analytic
• is entire
• Dual constant zero function (dual map / 3D plot / sliders)
• Split-complex constant zero function (split-complex map / 3D plot / sliders)
• Quaternionic constant zero function (vectors / sliders / 3D rotations)
• Octonionic constant zero function (vectors / sliders)

#### Constant one function

• Complex constant one function (complex map / 3D plot / sliders / domain colouring)
• • is elementary
• is complex analytic
• is entire
• Dual constant one function (dual map / 3D plot / sliders)
• Split-complex constant one function (split-complex map / 3D plot / sliders)
• Quaternionic constant one function (vectors / sliders / 3D rotations)
• Octonionic constant one function (vectors / sliders)

#### Identity function • Complex identity function (complex map / 3D plot / sliders / domain colouring / contour plot)
• • is an involution
• is elementary
• is complex analytic
• has a simple zero at • has a simple pole at • is conformal
• is entire
• Dual identity function (dual map / 3D plot / sliders)
• Split-complex identity function (split-complex map / 3D plot / sliders)
• Quaternionic identity function (vectors / sliders / 3D rotations)
• Octonionic identity function (vectors / sliders)
• Split-quaternionic identity function (vectors / [ sliders])
• Split-octonionic identity function (vectors / [ sliders])
• Tessarine (bicomplex) identity function (vectors / [ sliders])

### Arithmetic

• Complex addition (complex map / 3D plot / [ sliders] / domain colouring / contour plot)
• • forms an abelian group with identity element • • is elementary
• is complex analytic
• has a simple zero at • has a simple pole at • is conformal
• is entire
• Dual addition ([ dual map] / [ 3D plot] / [ sliders])
• Split-complex addition ([ split-complex map] / [ 3D plot] / [ sliders])
• Quaternionic addition ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic addition ([ vectors] / [ sliders])

#### Multiplication

• Complex multiplication (complex map / 3D plot / [ sliders] / domain colouring / contour plot)
• • forms an abelian group with identity element • • is elementary
• is complex analytic
• has a simple zero at • has a simple pole at • is conformal except if • is entire
• Dual multiplication (dual map / 3D plot / [ sliders])
• Split-complex multiplication (split-complex map / (Spacetime globes: Position-time spacetime globe / Time-position spacetime globe) / 3D plot / [ sliders])
• Quaternionic multiplication (vectors / [ sliders] / 3D rotations)
• Octonionic multiplication (vectors, showcasing noncommmutativity / vectors, showcasing nonassociativity / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])
• Split-quaternionic multiplication (vectors / [ sliders])
• Ternionic multiplication (vectors / [ sliders])
• Split-octonionic multiplication (vectors, showcasing noncommmutativity / vectors, showcasing nonassociativity / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])
• Quintonionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])
• Sextonionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])
• Tessarine (bicomplex) multiplication (vectors / [ sliders])
• Octarine (biquaternionic) multiplication ([ vectors, showcasing noncommmutativity] / [ sliders, showcasing noncommutativity])
• Hexadecarine (bioctonionic) multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity])
• Sedenionic multiplication (vectors, showcasing noncommmutativity / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])
• Trigintaduonionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])
• Tricomplex multiplication ([ vectors] / [ sliders])
• Tetracomplex multiplication ([ vectors] / [ sliders])
• Pentacomplex multiplication ([ vectors] / [ sliders])
• Split-biquaternionic multiplication ([ vectors, showcasing noncommmutativity] / [ sliders, showcasing noncommutativity])
• Dual complex multiplication ([ vectors] / [ sliders])
• Dual quaternion multiplication ([ vectors] / [ sliders])
• Parabolic quaternionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])
• Hyperbolic quaternionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])
• Parabolic octonionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])
• Hyperbolic octonionic multiplication ([ vectors, showcasing noncommmutativity] / [ vectors, showcasing nonassociativity] / [ vectors, showcasing nonalternativity] / [ sliders, showcasing noncommutativity] / [ sliders, showcasing nonassociativity] / [ sliders, showcasing nonalternativity])

#### Negation

• Complex negation (complex map / 3D plot / sliders / domain colouring / contour plot)
• • is an involution
• is elementary
• is complex analytic
• has a simple zero at • has a simple pole at • is conformal
• is entire
• forms an algebraically closed ring
• Dual negation ([ dual map] / 3D plot / sliders)
• Split-complex negation ([ split-complex map] / 3D plot / sliders)
• Quaternionic negation (vectors / sliders / 3D rotations)
• Octonionic negation (vectors / sliders)
• Split-quaternionic negation (vectors / [ sliders])
• Split-octonionic negation (vectors / [ sliders])
• Tessarine (bicomplex) negation (vectors / [ sliders])

#### Reciprocal

• Complex reciprocal (complex map v1 / complex map v2 / 3D plot / sliders / domain colouring / contour plot)
• • is an involution
• is elementary
• is analytic on • is meromorphic
• has a simple zero at • has a simple pole at • is conformal
• forms an algebraically closed field
• Dual reciprocal (dual map / 3D plot / sliders)
• Split-complex reciprocal (split-complex map / 3D plot / sliders)
• Quaternionic reciprocal (vectors / sliders / 3D rotations)
• Octonionic reciprocal (vectors / sliders)
• Split-quaternionic reciprocal (vectors / [ sliders])
• Split-octonionic reciprocal (vectors / [ sliders])
• Tessarine (bicomplex) reciprocal (vectors / [ sliders])