The defining relation, and commutativity
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<notatio-cell value="i_1 ^ 2" />
<notatio-cell value="i_2 * i_1" />Hypercomplex Algebras
A hypercomplex number is an element of a finite-dimensional algebra over the reals — you take ℝ, adjoin some units that are not real numbers, and say what happens when you multiply them. The complex numbers are the smallest interesting case: one unit, i, with i² = −1. Everything else on this page is what you get by making other choices. Let a unit square to +1 instead and you have the split-complex numbers, which contain zero divisors and are not a field. Let it square to 0 and you have the dual numbers, where f(a + ε) carries f'(a). Adjoin several units and decide whether they commute: keep them commuting and you climb the multicomplex tower ℂ₁ ⊂ ℂ₂ ⊂ ℂ₃ …; let them anticommute and you get the Clifford algebras, among them the quaternions.
They are all the same construction with different dials, which is why one implementation covers them: @enumeratio/hypercomplex adds them to the shared engine so that ordinary arithmetic — + - × ÷, integer powers, Norm, Conjugate, division — just works on them, with exact coefficients.
Hypercomplex units
A generator is fixed by just two facts — what it squares to, and whether it commutes — so the families are the whole 3 × 2 grid and there is nothing else to add:
| commuting | i_k — multicomplex ℂₙ | j_k — split / perplex | ε_k — dual |
| anticommuting | f_k — Clifford Cl(0,n), and ℍ | e_k — Clifford Cl(n,0) | θ_k — Grassmann |
Read a column as "a square root of that", and a row as whether the roots commute with each other. i_k is the familiar one; f_1 and f_2 together are the quaternions; θ_k are the fermionic generators. All the anticommuting generators anticommute across families too, as they must in a superalgebra.
A generator is an ordinary subscripted symbol, so the notation costs nothing — compute-engine's LaTeX parser already reads 1 + 2i_1 - 3i_1i_2 as one sum over three basis units. What the package adds is the algebra: + - × ÷ and integer powers reduce to a canonical form, and Norm, Conjugate and division come along with it.
Everything else is untouched: x_1 is still a variable, e is still Euler's number, and compute-engine's own i is still its native complex unit.
These units are not only formal — ℤ/m is full of them, one per CRT channel. Finite: ℤ/m and the places follows that all the way down.
Multicomplex: the commuting tower
Each generator squares to −1, and distinct generators commute — so this is the commutative corner of the hypercomplex world, not the quaternions.
The signature is mixed, which is the whole story of ℂₙ for n ≥ 2: the blade i_1i_2 squares to i_1²i_2² = (−1)(−1) = +1, so ℂ₂ contains a square root of +1 that is not ±1 — and with it zero divisors and idempotents that ℂ₁ has none of.
A blade that squares to +1
Sign follows the popcount of the index: (−1) to the power of the number of units in the blade.
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<notatio-cell value="(i_1 * i_2) ^ 2" />
<notatio-cell value="(i_1 * i_2 * i_3) ^ 2" />Products expand over the basis
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<notatio-cell value="(1 + i_1) ^ 2" />
<notatio-cell value="(1 + i_1) * (1 + i_2)" />
<notatio-cell value="(a + b * i_1) ^ 2" />Coefficients are kept as expressions rather than floats, so rationals stay rational and free symbols ride along — (a + b i_1)² comes back as a² − b² + 2ab·i_1.
Conjugation, and why the norm is a determinant
\overline{z} sends every generator to its negative, so odd-grade blades flip sign.
Conjugation
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<notatio-cell value="OverBar(1 + i_1 + i_2 + i_1 * i_2)" />For the Gaussians z·conj(z) is the norm. Above n = 1 it is not even a scalar:
z·conj(z) keeps a blade
The i_1i_2 part is 2(ad − bc) — it only vanishes by accident.
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<notatio-cell value="(1 + 2i_1 + 3i_2 + 4 * i_1 * i_2) * OverBar(1 + 2i_1 + 3i_2 + 4 * i_1 * i_2)" />So Norm is the algebra norm: the determinant of multiplication-by-z on the 2ⁿ-dimensional space. It is computed through the tower ℂₙ = ℂₙ₋₁[iₙ]/(iₙ²+1), where z = u + iₙ·v gives N(z) = N(u² + v²) one level down — n squarings, not a 2ⁿ × 2ⁿ elimination. At n = 1 it collapses to the Gaussian a² + b².
The algebra norm
25 = 3²+4². And Norm(1 + 2i_1 + 3i_1i_2) = 160 is the determinant of [1 −2 0 3; 2 1 −3 0; 0 −3 1 −2; 3 0 2 1].
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<notatio-cell value="Norm(3 + 4i_1)" />
<notatio-cell value="Norm(1 + 2i_1 + 3 * i_1 * i_2)" />Units, zero divisors, idempotents
An element is invertible exactly when its norm is — and division is exact.
Inverses
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<notatio-cell value="1 / (1 + i_1)" />
<notatio-cell value="i_1 ^ (-1)" />
<notatio-cell value="(1 + 2i_1 + 3 * i_1 * i_2) * (1 + 2i_1 + 3 * i_1 * i_2) ^ (-1)" />1 + i_1i_2 has norm 0. It is a genuine zero divisor, so division by it is left standing rather than answered with something false.
A zero divisor, and the idempotent beside it
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<notatio-cell value="Norm(1 + i_1 * i_2)" />
<notatio-cell value="(1 + i_1 * i_2) * (1 - i_1 * i_2)" />
<notatio-cell value="1 / (1 + i_1 * i_2)" />
<notatio-cell value="((1 + i_1 * i_2) / 2) ^ 2" />Split (perplex) units: j_k² = +1
The same machinery with the sign flipped. The norm becomes a² − b², which is indefinite — so the split algebra is full of zero divisors, and (1 ± j_1)/2 are the two orthogonal idempotents that resolve the identity.
Split-complex arithmetic
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<notatio-cell value="j_1 ^ 2" />
<notatio-cell value="Norm(3 + 4j_1)" />
<notatio-cell value="(1 + j_1) * (1 - j_1)" />
<notatio-cell value="(1 + j_1) / 2 * ((1 - j_1) / 2)" />Families mix freely — i_1j_1 squares to (−1)(+1) = −1.
Mixing families
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<notatio-cell value="(i_1 * j_1) ^ 2" />Dual units: ε_k² = 0
Nilpotent generators give the dual numbers, where a cube carries its own derivative: (2 + ε)³ = 8 + 12ε, and 12 is d/dx x³ at 2.
Dual numbers differentiate
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<notatio-cell value="epsilon_1 ^ 2" />
<notatio-cell value="(2 + epsilon_1) ^ 3" />
<notatio-cell value="(2 + epsilon_1) ^ (-1)" />Clifford units: e_k anticommute
e_1e_2 = -e_2e_1, and getting that sign out of compute-engine takes some care: Multiply is declared commutative, so canonicalisation sorts its operands before any handler runs, and by then the transposition sign is gone beyond recovery.
Juxtaposition escapes it, because juxtaposition is not Multiply yet — it parses to InvisibleOperator, one step upstream of the sort, where a product whose sign is at stake is rerouted onto the ordered head. So e_2e_1 written out reads correctly. An explicit \times or \cdot has no such step: it parses straight to Multiply, already sorted. Rather than return a sign it cannot justify, × declines there and leaves the product inert — write ⊗ (or NonCommutativeMultiply) instead.
A separate head is the standard answer to this, not a workaround. Wolfram puts the non-commutative product on its own head — NonCommutativeMultiply, infix ** — and matrix multiplication on another, Dot, infix ., for exactly the same reason; compute-engine's own Dot is likewise declared commutative: false, which is why Dot(A,B) and Dot(B,A) stay apart while Multiply would collapse them. So the ordered product here is NonCommutativeMultiply, with GeometricProduct as an alias for the geometric-algebra reading. Both work on the commuting families too, where they just agree with ×.
The ordered product
Juxtaposition agrees with it; explicit × is the one that has to decline.
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<notatio-cell value="e_1 ^ 2" />
<notatio-cell value="NonCommutativeMultiply(e_1, e_2)" />
<notatio-cell value="NonCommutativeMultiply(e_2, e_1)" />
<notatio-cell value="GeometricProduct(e_1, e_2, e_1, e_2)" />
<notatio-cell in-form="latex" value="e_2e_1" />
<notatio-cell in-form="latex" value="e_1e_2e_1" />
<notatio-cell value="e_2 * e_1" />Quaternions and Grassmann: the anticommuting half
f_k squares to −1 and anticommutes, which is Cl(0,n) — and Cl(0,2) is exactly the quaternions. Take i = f_1, j = f_2, k = f_1f_2: all three square to −1, and ijk = −1.
⟨f_1, f_2⟩ is ℍ
k² = −1, and ij·k = −1 — the quaternion relations.
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<notatio-cell value="f_1 ^ 2" />
<notatio-cell value="CircleTimes(f_1, f_2, f_1, f_2)" />
<notatio-cell value="CircleTimes(f_2, f_1)" />Anticommuting nilpotents have a use too: θ_k are the fermionic / Grassmann generators, and their blades span the exterior algebra Λ(ℝⁿ). Any repeated generator kills the blade, which is why Λ(ℝⁿ) stops at grade n.
Grassmann generators
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<notatio-cell value="theta_1 ^ 2" />
<notatio-cell value="CircleTimes(theta_2, theta_1)" />
<notatio-cell value="CircleTimes(theta_1, theta_2, theta_1)" />There is a second route to an imaginary anticommuting generator, since i commutes: (i·e_1)² = i²e_1² = (−1)(+1) = −1. So i e_k behaves like f_k; f_k is just the direct spelling.
Named algebras
An algebra here is nothing but an ordered list of generators — the families already carry the squares and the commutation rules — so naming one is a way to say "these generators" and get its basis, dimension and signature back, in the spirit of Wolfram's CliffordAlgebra.
CliffordAlgebra(p, q) takes p generators squaring to +1 and q squaring to −1; MulticomplexAlgebra(n), SplitAlgebra(n), DualAlgebra(n) and GrassmannAlgebra(n) name the commuting corners. The ones with names of their own can be written that way: Quaternions (or \mathbb{H}), BicomplexNumbers, TricomplexNumbers, SplitComplexNumbers and DualNumbers.
ℍ is Cl(0,2)
Basis order is grade then generator, so ℍ comes back as (1, i, j, k).
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<notatio-cell value="Basis(Quaternions)" />
<notatio-cell value="AlgebraSignature(Quaternions)" />
<notatio-cell value="AlgebraDimension(CliffordAlgebra(2, 1))" />
<notatio-cell value="Basis(BicomplexNumbers)" />No element constructor is needed: Dot already threads a tuple of scalars over a basis, so an algebra's elements are one composition away.
Elements from a coefficient tuple
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<notatio-cell value="Dot([1, 2, 3, 4], Basis(Quaternions))" />And because they name generator sets, the algebras work as sets: containment asks whether every unit occurring in an element belongs to the algebra. A free symbol gets no answer rather than a presumptuous one.
Containment
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<notatio-cell value="f_1 * f_2 in $\mathbb{H}$" />
<notatio-cell value="f_3 in $\mathbb{H}$" />
<notatio-cell value="i_1 in BicomplexNumbers" />
<notatio-cell value="i_3 in BicomplexNumbers" />These units already live inside ℤ/m
A split unit is an x with x² = 1 and x ≠ ±1 — and ℤ/m is full of them. By CRT, ℤ/m ≅ ∏ ℤ/pᵢ^aᵢ, a square root of 1 is a root in every channel independently, and an odd prime power has exactly the two roots ±1. So for odd m the roots of 1 are the 2^ω(m) spectral sign vectors, ω(m) = the number of distinct primes.
The split units of a modulus
15 = 3·5, so ω = 2 and there are 2² = 4 roots of 1: the trivial ±1 pair plus 4 and 11.
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<notatio-cell value="PowerModList(1, 1 / 2, 15)" />
<notatio-cell value="PowerModList(-1, 1 / 2, 65)" />Sending j_1 ↦ 4 is a ring homomorphism ℝ[j]/(j²−1) → ℤ/15, so the identities the symbolic algebra proves come back as facts about ℤ/15. That, and the place-by-place story behind it, is the subject of Finite: ℤ/m and the places.
Things worth knowing
Norm depends on the ambient algebra. It is taken over the subalgebra generated by the units that actually occur, and adjoining a generator squares the determinant (N at level n+1 is N at level n, squared). So N is multiplicative on a fixed unit set — that is the theorem — but the two cells below are 2 and 2 while their product is 16, not 4.
The norm is relative to a level
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<notatio-cell value="Norm(1 + i_1)" />
<notatio-cell value="Norm((1 + i_1) * (1 + i_2))" />compute-engine's i stays a scalar. It is a number literal, not a symbol, so it lives inside coefficients and native complex arithmetic is unchanged. i_1 is therefore a separate commuting square root of −1 — which is the point: it is what makes ℝ[i_1, …, iₙ] the multicomplex tower rather than a re-spelling of ℂ.
Native complex arithmetic is untouched
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<notatio-cell value="(1 + i) ^ 2" />
<notatio-cell value="i * i_1 * i_1" />Anything opaque stays symbolic. A generator under a head the package cannot see through — \sin(i_1), a symbolic exponent — is left alone rather than guessed at.
Declining to answer
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<notatio-cell value="Sin(i_1)" />Design considerations, recorded
The infix symbol for the ordered product is unsettled. ** is out: epsil already takes it for exponentiation (alongside ^), and compute-engine's own LaTeX parser rejects it regardless. × / xx would read best — a Cartesian product is inherently ordered, so the connotation is right — but \times and bare × are both already compute-engine's trigger for Multiply, and repointing them would change every multiplication in the language. Too much to take unilaterally, so \otimes (⊗) carries it for now: it had no definition at all, and ⊗ connotes an ordered product anyway. Worth revisiting with upstream, because a possibly-non-commutative × that reduces to the commutative case is an attractive design.
⊗ is not a Clifford-only spelling, either, which is part of why it fits: a tensor product of two scalars is just their product (the implied identity factor), so 2 ⊗ 3 is 6, and on the commuting families ⊗ simply agrees with ×. It is total, and non-commutative only where the algebra actually is.
Dot is a weak precedent, despite being the one that exists. The scalar product genuinely is commutative, so a head named Dot reading as matrix multiplication (which is not) is a conflation on compute-engine's side. What it does establish — and the part worth leaning on — is that CE is willing to declare a product head commutative: false and keep it apart from Multiply.
Two things to raise upstream. (1) Multiply being declared commutative means canonicalisation destroys operand order before any evaluate handler runs, so an extension cannot implement an anticommuting product on × even in principle. (2) Redeclaring an operator silently drops its type handler unless the caller copies it, which widens Add's result from number to value and then breaks unrelated operators — that reads like surface that has not been built yet rather than a decision.
The measurement-units system is a tantalising neighbour, and currently closed. A unit written as an identifier already parses as Multiply(3, m) — structurally the same thing as a generator here, just without a square or a commutation rule. So these units are the units system with two extra facts attached. But Quantity / UnitConvert / Dimension have no public registry to extend (there is no ce.declareUnit), so adding a dimension — let alone an imaginary or non-commutative one — is not reachable from outside today. Worth pursuing: the quaternions contain a 2-sphere's worth of anticommuting square roots of −1, one per direction in 3-space, which is a genuinely unit-like way to think about them.
Not built yet. A Norm that can name its ambient level rather than inferring it from the units present; a Clifford norm and reverse; and modular coefficients, so the algebra could run over ℤ/m directly instead of only mapping into it.