Figure 1

Engineering problems involving vibrations are more easily solved using complex numbers. A complex number z can be written as z = a+i×b in which a and b are “ordinary” real numbers and i is a special number with the special propery i×i = -1. The same number can also be written as z = r ×(cosf + i sinf), with r² = a² + b² and tanf = ^b/_a, see figure 1, and this is equivalent to z = r e^if . For multiplication with a slide rule we need ln(z) = i f ln(r).

The real number a is called the real part of z: a = Re(z), and b is the imaginary part: b = Im(z). A complex number with the same real part as z, and an imaginary part with same size but opposite sign is called the complex conjugate of z: z = R (z) - i Im(z) = r e^-if 

It is easy to show that z + z = 2 Re(z) = 2 r cosf and z - z = 2 i Im(z) = 2 i r sinf

The trig formula's you probably learned by heart at highschool can easily be derived using complex numbers.

In 1893, Prof. Mehmke, of the Technische Hochshule Darmstadt, discussed^[1] several designs for complex number slide rules that were more or less realised in later years, as will shown below. As far as I know, only one of his designs did not become reality: the use of a four-legged pair (quartet?) of compasses on a single scale, used the same way as a Gunter scale for ordinary logarithms.

     Figure 2: Dumond Patent US1536574

The first complex slide rule Robert discusses is patented by J.W.M. Dumond^[2] in 1925 (US1536574). When multiplying using this rule, you move one set of scales relative to the other in two dimensions. This might remind you of a grid-iron slide rule, but a grid-iron is just a chopped-up linear slide rule, while the Dumond is truly two-dimensional. Reading a grid-iron is relatively easy, but reading the complex number slide rule is difficult, because its scales are rather ... complex (Figure 2).

Try an online simulation.

Around 1926, Michel G. Malti, an Electrical Engineering instructor at Cornell University, designed a complex-quantity slide rule consisting of a celluloid chart on which was pivoted a rotating [cursor]. He got his design copyrighted but not patented.^[3]

In 1952, German slide rule manufacturer Faber-Castell applied for a German patent for a complex number slide rule (DE1067622) The patent was granted in 1960. The slide rule is flat with a single sheet of curves and a ruler-like pointer.^[4]

Figure 4: Whythe's slide rule

A "Fuller-type"  slide rule, designed by D.J. Whythe for the BBC, was produced by Stanley.^[5] The 1962 Stanley sales catalog lists it for a price of £21 (which corresponds to about $500 in 2008). See figure 4.

Figure 5: Sander's prototype

Arnold Sander published a cylindrical slide rule in 1956.^[7] His design, of which he made a prototype, had a transparent complex number chart wrapped around another complex number chart.
 

J.H. Hetherington proposed a cylindrical complex number slide rule with a single plot of the complex plane and two cursors.^[6]. It works more or less like the Fuller-type (figure 6). A working variation of this type of slide rule, with scales in another direction, has recently been made by Tina Cordon.
 

Robert Étienne François Béraud patented (FR1441506) a slide rule consisting of two rotating cylinders with identical complex plane plots. One of the cylinders  could also be shifted. A sliding cursor facilitates reading (figure 7). This devices is much easier to use than a Dumond. I don't know if it was ever manufactured.
 

If you combine Hetherington's cursor with Béraud's two cylinders, and align the cylinders along a common axis, you end up with a Otis King-like arrangement. This idea was patented by Maurice Bouix in 1953 (FR1024896). Unlike the Otis King cursor, the cursor Bouix used consisted of two parts, linked by a bracket (figure 8).

Figure 6: Hetherington's proposal
 

Figure 7: Béraud (Patent FR1441506)

Figure 8: Bouix (FR1024896)

Robinson circular slide rule, designed by R.B. Robinson, Sydney University, and made by AWA (Amalgamated Wireless Australasia) Ltd., Australia, from 1957 to 1962.

UPDATE: in the "Proceedings of the 18th International Meeting of Slide Rule Collectors", 2012, Robert Adams gives a detailed description and an exploded view of a Robinson circular complex number slide rule.

Figure 9: Blundell Vector slide rule

The Blundell Vector slide rule^[8] is used to convert a+ib to r e^if. It was designed by Erich Siegfried Friedlander, and a British patent was applied for it in 1951 (granted in 1954 to Friedlander and General Electric Company Ltd. as GB713655). Erich Siegfried Friedlander was an electrical engineer who had many electric motor patents.

The rule was made by Blundell Rules Ltd. for the General Electric Company from 1952 to 1953 and was sold for £16.80 in 1953 (which corresponds to about $500 in 2008)

More information on Vector slide rules in the traditional slide rule shape is given by Bill Robinson

The Spirule is a complex calculating aid for control systems. You can find more information, and a manual, on Nathan Zeldes' website

Negative numbers