Giant brains; or, Machines that think

by Edmund Callis Berkeley

ORIGIN AND DEVELOPMENT

In Bell Telephone Laboratories the telephone system of the country is continually studied. Their research produced the common type of dial telephone system: a masterly machine for selecting information.

Now when a telephone engineer studies an electric circuit, he often finds it very convenient to use numbers in pairs: like 2, 5 or-4,-1. Here the comma is a separation sign to keep the two numbers in the pair separate and in sequence. Mathematicians call numbers of this kind, for no very good reason, complex numbers; of course, they are far less complex than why the sun shines or why plants grow.

When Bell Laboratories test the design of new circuits, girl computers do arithmetic with complex numbers. Addition and subtraction are easy: each means two operations of addition or subtraction of ordinary numbers. For example, 2, 5, plus-4,-1 equals 2-4, 5-1, which equals-2, 4. And 2, 5 minus-4,-1 is the same as 2, 5 plus 4, 1; and this equals 2 + 4, 5 + 1, which equals 6, 6. Multiplication of two complex numbers, however, is more work. If a, b and c, d are two complex numbers, then the formula for their product is (a × c)-(b × d), (a × d) + (b × c). To get the answer, we need 4 multiplications, 1 subtraction, and 1 addition. Division of two complex numbers requires even more work. If a, b and c, d are two complex numbers, the formula for the quotient of a, b divided by c, d is:

[(a × c) + (b × d)] ÷ [(c × c) + (d × d)],

[(b × c) - (a × d)] ÷ [(c × c) + (d × d)]

For example,

(2, 5) ÷ (-4, -1) = [(2 × -4 = -8) + (5 × -1 = -5)] ÷ [(-4 × -4 = 16) + (-1 × -1 = 1)],

[(5 × -4 = -20) - (2 × -1 = -2)] ÷ [16 + 1] = -(¹³/₁₇), -(¹⁸/₁₇)

Thus, division of one complex number by another needs 6 multiplications, 2 additions, 1 subtraction, and 2 divisions of ordinary numbers—and always in the same pattern or sequence.