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. The Complex Computer About 1939, an engineer at Bell Telephone Laboratories in New York, Dr. George R. Stibitz, noticed the great volume of this pattern arithmetic. He began to wonder why telephone switching equipment could not be used to do the multiplications and divisions automatically. He decided it could. All that was necessary was that the _relays_ (see Chapter 2) used in regular telephone equipment should have a way of remembering and calculating with numbers. Regular telephone equipment would take care of the proper sequence of operations. Regular equipment known as _teletypewriters_ would print the numbers of the answer when it was obtained. A teletypewriter consists essentially of a typewriter that may be operated by electrical impulses. It has a keyboard that may produce electrical impulses in sets corresponding to letters; and it can receive or transmit over wires. Dr. Stibitz _coded_ the numbers: each decimal digit was matched up with a group of four relays in sequence, and each of these relays could be open or closed. If 0 means open and 1 means closed, here is the pattern or code that he used: