[Richard777 (OP)]

Two forces are reciprocal. Reciprocation requires three conditions:
-   the forces act upon a common particle
-   have equal strength
-   act in opposite directions
Force may be represented as a vector. A 3D force vector may represent “instantaneous force”.
Reciprocal forces may be represented as “reciprocal vectors”.
Polar co-ordinates of force: Fr, θ, φ
Special reciprocation requires one additional rule: φ = θ’ = ½π - θ
Does the result of this condition represent the complex constant (i) as a ratio of force components?
The math is attached.

[evan_au]

the forces act upon a common particle, have equal strength, act in opposite directions...
Does the result of this condition represent the complex constant (i) as a ratio of force components?

If you are looking at forces which directly oppose one another (ie their directions are 180° apart), then you can set up the problem as a 1-dimensional problem. You can then solve it with negative numbers, rather than complex numbers.
- You add the two forces (one of which is negative), and you come up with an answer (which may be positive or negative)

If, however, you are working with forces that operate in 2 dimensions (eg North-East and West, for example), then you need more than just negative numbers.
- As you say, you could use complex numbers. You add the two forces (at least one of one of which contains i), and you come up with an answer (which may be positive or negative, and may or may not contain i)
- Equivalently, you could break down the forces into x and y components (using sine & cosine), then you add the horizontal and vertical components separately. You come up with an answer (which may be positive or negative in the x component, and may or may not be zero in the y component)
- So, solving this problem using i is entirely optional. Solutions without i are equally feasible.

Fn = Fnxin + Fnyjn + Fnzkn

The attachment uses 3-dirmensional arithmetic, expressed in i, j and k.
These numbers are called Quaternions in 4 dimensions, in some ways a superset of imaginary numbers (in 2 dimensions), which is a superset of real numbers (positive and negative, in 1 dimension).

Like imaginary numbers, where i*i = -1, in Quaternions, i*i = j*j = k*k = -1

But if the forces are in opposite directions, you can solve it in 1 dimension. Don't make it harder for yourself than it needs to be.

It is true that Quaternions have been used in 3-dimensional computer graphics. As I understand it, the calculations are slightly more complex than processing the equivalent vectors of x, y and z components, so in practice, Quaternions are rarely used in computer graphics.
See: https://en.wikipedia.org/wiki/Quaternion

Two forces are reciprocal.

Editorial comment: I think you may be using the wrong word here,
- The definition of a reciprocal is that if b is the reciprocal of a, then a*b = 1 (eg the reciprocal of 2 is 1/2 = 0.5. 2*0.5 = 1)
        - This uses the Multiplicative inverse. Reciprocals involve division and multiplication
        - This is not the situation you are describing
- What you are talking about here is an opposing force, and forces add.
         - This uses the Additive inverse. Negation involves adding and subtraction..
See:
https://en.wikipedia.org/wiki/Multiplicative_inverse
https://en.wikipedia.org/wiki/Additive_inverse

[alancalverd]

"Reciprocal X" is used in navigation to signify X ± 180°, whichever value lies between 0 and 360, i.e. the opposite direction. Same as "reciprocating engine". This may be the source of confusion.

[evan_au]

Does the complex constant (i) represent a ratio of special reciprocity?

No. Multiplying by i is equivalent to a rotation by 90° in the complex plane.
To rotate by 180°, you need to multiply by -1=i*i (since 180° = 2 rotations of 90°).

Multiplying by i is also equivalent to a rotation by 90° in quaternion 4D space, but like I said, why make it more complex than you need to?.