Guide · 09

Complex numbers

Complex numbers are first-class. Push two doubles, tap ->COMPLEX, and you have a complex card — with arithmetic, trig, transcendentals, and three display modes for the way you want to read the result.

Constructing a complex

Rectangular. Real part, then imaginary part, then ->COMPLEX:

3.0 4.0 ->COMPLEX

1: 3+4i

Polar. Magnitude, then angle, then ->POLAR. The angle is interpreted in the engine’s current angle mode — degrees by default, radians under RAD-MODE, gradians under GRAD-MODE:

5.0 53.13_° ->POLAR        # produces the same value as 3.0 4.0 ->COMPLEX

COMPLEX-> pushes the real and imaginary parts back onto the stack as two separate doubles, in reverse construction order — useful for un-packing or for piping into another word.

Display modes

The same complex value displays three different ways depending on the engine mode:

Word	Mode	`3+4i` displays as
`MATH-MODE`	Math (default)	`3+4i`
`POLAR-MODE`	Polar	`5∠53.13`
`ENGR-MODE`	Engineering	`3+4j`

Switching modes is a display change only — the underlying value is unchanged and can be deparsed losslessly regardless of mode. The Engineering mode substitutes j for i to match the convention used in electrical engineering.

Arithmetic

The standard arithmetic words (+, -, *, /, ^) work on complex values directly. Cross-type with Double and Integer is automatic — the other operand is treated as having zero imaginary part:

3.0 4.0 ->COMPLEX  2.0 *           # → 6+8i
3.0 4.0 ->COMPLEX  1.0 0.0 ->COMPLEX  +     # → 4+4i
2.0 0.5 ->COMPLEX  ^                # → 3+4i ^ (2+0.5i)

Negation is CHS (change sign) — same word as for any number. Magnitude is ABS (which on a complex returns its MAG).

Extractors

Word	Pushes	Notes
`RE`	The real part as a `double`
`IM`	The imaginary part as a `double`
`MAG`	The magnitude `\|z\|` as a `double`	Same as `ABS`
`ARG`	The argument (angle) as a `double`	In the engine’s current angle mode

All four leave the original complex on NOS untouched if you use DUP first; otherwise they consume it.

Transcendentals, trig, hyperbolics

The full complex math library is registered: SIN, COS, TAN, EXP, LN, LOG, SQRT, SINH, COSH, TANH, and their inverses (ASIN, ACOS, ATAN, ASINH, ACOSH, ATANH). Each dispatches automatically — push a complex, tap the word, get the right result.

Trig honours the angle mode the same way ARG does: under DEG-MODE, SIN of a complex with a “degree-shaped” real part returns what you’d expect; under RAD-MODE, the argument is treated as radians.

0.0 1.0 ->COMPLEX  EXP             # → cos(0) + i·sin(0) = e^i
3.0 4.0 ->COMPLEX  SQRT            # → principal square root

Equality

== is strict — 1 1+0i == returns false. Same numerical real value but different types.

EQUIV? is the relaxed sibling — 1 1+0i EQUIV? returns true. Use it when you want to compare across the Complex/Double boundary and treat a Complex with zero imaginary as equivalent to its real component.

Detail card

Tap a complex card to open its detail sheet. You’ll see:

The real and imaginary parts, with their double-precision values
The magnitude (\|z\|) and argument (arg z) — in the current angle mode
The polar form r∠θ
A quick-pick to switch the display mode for that card without changing the global setting

The detail card is a good place to pivot between rectangular and polar thinking without committing to one mode for the whole session.