292 ANNUAL REPORT SMITHSONIAN INSTITUTION, 19 60 



The "logical design" in digital computers extends to much more 

 than the adder. Most of the other operations of the computer, in- 

 cluding storage and decision operations, can be expressed in logical 

 terms and hence be composed of the same electronic logical devices. 

 The logical design of a computer is indeed very complicated, and most 

 computers use many thousand logical elements. 



So far we have considered only binary arithmetic. However, 

 humans who put data into computers and read its answers are much 

 more familiar with decimal numbers than with binary. Thus it is 

 necessary to convert decimal numbers into some form of binary for 

 input to a computer and to convert binary to decimal for output. 

 The computer itself can do this converting. One way of doing this is 

 to use combinations of binary digits to represent decimal digits in 

 hinary-coded decimal systems. Table 2 can be considered as an ex- 

 ample of such a system. By such methods it is possible to use digital 

 computers as if they were true decimal devices, although in fact they 

 all are binary in some form or another. 



Siibtraction. — Subtraction can be, and sometimes is, done in a man- 

 ner comparable to addition, that is, the subtraction table is formed, 

 the logical equivalent determined, and the corresponding electronic 

 circuitry built. However, many computers use the adder to do 

 subtraction by representing negative numbers by a complement 

 notation. 



To understand this, consider table 5 where the left column gives 

 the normal sequence of numbers from +5 backward to —5. It in- 

 cludes the concepts of zero and negative numbers. A complement sys- 

 tem for representing these numbers is given in the right column. (For 

 convenience, we consider only four digit numbers.) Wlien the number 

 is negative, the complement representation is the same as if the number 

 were subtracted from 9999. The process of subtracting by adding the 

 complement obviously is dependent on the fact that the complement 

 can be obtained by a process simpler than subtraction, and indeed it 

 can be done electronically. 



The sequence of numbers in the right "counter" column is unusual 

 but is as valid as the usual sequence if a set of rules is used for manip- 

 ulation that differs somewhat from the usual set. Examples are 

 given in table 6. Note first that results are always obtained by adding. 

 Further, when the sum of the two numbers exceeds the four-digit 

 size assumed, then the carry is added back to the right end of the sum. 

 This last rule is a result of the way the sequence of numbers was 

 defined. It is due to the fact that 0000 is not given in this sequence 

 and that zero is represented by 9999. Again the reasons for this will 

 not be considered here, but it does simplify the computer design. 

 It is suggeested that the reader try other examples using complements, 

 such as adding zero (9999) to other numbers including itself. 



