Exact complex numbers

Construct complex values

The canonical imaginary generator is available from .Exact and .Complex:

Runnable RiXRun cell

z := 3 + 4~{i}; w := .Complex.FromParts(1, -2); {: z, w }

Complex numbers are exact expressions in i, whose relation is i^2 + 1 = 0.

Division by complex expressions

Division uses algebraic inversion rather than floating approximation:

Runnable RiXRun cell

inverseI := 1 / .Exact[:i]; quotient := (1 + .Exact[:i]) / (1 - .Exact[:i]); {: inverseI, quotient }

The results are -i and i.

Conjugation

Use the namespace operation or the receiver method:

Runnable RiXRun cell

z := 3 + 4~{i}; byNamespace := .Complex.Conjugate(z); byMethod := z.Conjugate(); {: byNamespace, byMethod }

Conjugating twice returns the original exact value.

Real and imaginary parts

Re and Im preserve exact coefficients, including other real generators:

Runnable RiXRun cell

z := .Exact[:pi] + .Exact[:sqrt2] * .Exact[:i]; {: .Complex.Re(z), .Complex.Im(z) }

The result is (pi, sqrt2), still exact.

Norm squared

Magnitude would require a square root. NormSquared avoids that decision:

Runnable RiXRun cell

z := 3 + 4~{i}; {: z.NormSquared(), .Complex.NormSquared(z) }

Both entries are exactly 25. The next lesson introduces an exact magnitude through Cayley polar form. Arg remains deferred until RiX's real-number and trigonometric policies are ready.

Current boundary

RiX supports division in a single registered algebraic extension. A denominator mixing independent generators, or a multi-term transcendental denominator such as 1 + pi, is not automatically inverted.