Essay 04 · Division
When division
fails
Why division in higher-dimensional myrions is controlled by a locus rather than by zero alone.
In ordinary and complex arithmetic, division has one familiar exception: zero. Higher-dimensional myrions have a larger exceptional set.
Once a third myrion dimension is introduced, some non-zero values can multiply a non-zero value to give zero. Other division equations can have several solutions — or none. The exceptional values form the singular division locus.
Σₙ = {x ∈ Mₙ : det Lₓ = 0}
Here Lₓ is the linear transformation produced by multiplication by x. Outside Σₙ, every division equation has exactly one solution. On it, unique division fails.
Division as an equation
To divide a value z by x, seek a myrion y satisfying x ⋆ y = z. For fixed x, this is the linear system Lₓ(y) = z.
- If Lₓ is invertible, every z has exactly one solution.
- If it is singular, some targets may have no solution.
- If it is singular, targets that do have solutions can have more than one.
For a finite matrix, invertibility is detected by its determinant. This is why the determinant of the multiplication operator, rather than whether x is literally zero, governs myrion division.
The singular division locus
If x ∉ Σₙ, then Lₓ is invertible and x ⋆ y = z has the unique solution y = Lₓ⁻¹(z). Setting z = 1 gives a unique multiplicative inverse.
If x ∈ Σₙ, the operator is singular. This does not say every equation has no solution. It says division by x is not uniquely available for every target: there may be no solution or a whole family of them.
“Singular division locus” describes where the multiplication operator is singular. It is not a claim about singular points in the algebraic-geometric sense.
The complex case: zero is the whole locus
The two-dimensional myrion algebra is exactly the complex plane. For z = a + bi, multiplication is represented by L_z = [[a, −b], [b, a]], whose determinant is a² + b².
Over the reals that determinant vanishes only when a = b = 0, so Σ₂ = {0}. The familiar instruction not to divide by zero is the two-dimensional instance of the more general rule:
Unique division is possible precisely outside the singular division locus.
What changes in three dimensions?
A three-dimensional myrion is x = a + bi + cj. In the basis 1, i, j, multiplication by x has matrix:
Lₓ = [[a, −b − c, −b], [b, a, c], [c, 0, a]]
Its determinant is the cubic Δ₃(a,b,c) = a³ + ab² + 2abc − bc² − c³. Thus Σ₃ is the zero set of that polynomial. It contains non-zero points and, because the polynomial is homogeneous, scaling a point on the locus keeps it on the locus.
A non-zero product that becomes zero
The new behaviour appears in one short calculation:
(j − i) ⋆ i = j ⋆ i − i ⋆ i = −1 − (−1) = 0.
Neither factor is zero. Thus j − i and i are zero divisors in M₃. This is exactly what singularity means: if x ∈ Σₙ, then Lₓ has a non-zero kernel vector, and conversely.
Σₙ = {0} ∪ {non-zero zero divisors of Mₙ}
Every higher-dimensional myrion includes i and j − i, so non-zero zero divisors occur for every n > 2.
No inverse or many inverses
A singular element with no inverse
For x = j − i, attempting to solve (j − i) ⋆ y = 1 gives incompatible coefficient conditions. The inverse equation has no solution.
A singular element with many inverses
For x = i, both i ⋆ (−i) = 1 and i ⋆ (−j) = 1. More generally, i ⋆ (−i + t(j − i)) = 1 for any real t, because i ⋆ (j − i) = 0.
An element on the locus may therefore have no inverse, or may have many candidates that multiply to one; neither case provides unique division.
Cancellation depends on the locus
For ordinary numbers, xy = xz implies y = z when x ≠ 0. In myrions, the corresponding law holds when x ∉ Σₙ: if x ⋆ y = x ⋆ z, then Lₓ(y − z) = 0, and invertibility forces y = z.
On the locus, cancellation can fail: (j − i) ⋆ i = (j − i) ⋆ 0 = 0, although i ≠ 0. In a non-associative algebra the safe justification is the linear-map argument, not blindly inserting an inverse and changing brackets.
Exact singularity and numerical instability
The equation det Lₓ = 0 identifies exact failure. But the magnitude of the determinant alone is not a scale-independent measure of numerical safety. Near the locus, Lₓ can remain invertible while being poorly conditioned.
The smallest singular value σ_min(Lₓ) is zero exactly on the locus; small positive values indicate proximity to singular behaviour. The condition number κ(Lₓ) = σ_max(Lₓ) / σ_min(Lₓ) grows large as an invertible operator approaches singularity.
What is proved, and what remains open
- Outside Σₙ, every division equation has a unique solution.
- Σₙ is zero together with the non-zero zero divisors.
- det Lₓ is a homogeneous polynomial of degree n.
- Σ₂ = {0}, while every Σₙ for n > 2 contains non-zero elements.
The detailed geometry — including its components, dimension, connectedness and singular strata — remains a research problem. Computer plots can suggest useful conjectures but do not prove those properties.
The broader rule
More than “do not divide by zero”
For real and complex arithmetic, division requires x ≠ 0. Myrion arithmetic replaces that isolated exclusion with the geometric condition x ∉ Σₙ.
The singular division locus identifies exactly where unique inversion and binary cancellation stop being reliable. It is the natural structure governing division throughout the myrion family.