284 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1960 



made them impracticable. For example, the value of pi can be cal- 

 culated to any number of decimal places by several different formulas. 

 William Shanks, 1812-82, an English mathematician, spent many 

 years calculating the value of pi to 707 places. The results were pub- 

 lished in 1854, and 92 years passed before a computer duplicated this 

 remarkable feat.^ Then in 1954 a computer calculated pi to over 3,000 

 places in 13 minutes computing time. At present it could be done in 

 considerably less time. Incidentally, the method of calculation was 

 the same as that used by Shanks. 



There are many other problems that can be represented mathemati- 

 cally, the solution of which required a tremendously large number of 

 operations. The solving of a large number of simultaneous linear 

 equations is one example. Others are the "monte carlo" and relaxation 

 methods of solvinsr the intricate mathematics associated with atomic 

 energy studies. 



However, to answer the question above, digital computers do have 

 advantages other than speed. They can perform logical operations as 

 well as arithmetic. This is a very important property, which we will 

 consider further after discussing some of the details of how a computer 

 works and of the particular arithmetic of a computer. 



HOW COMPUTERS WORK 



Mechanical calculators use a system of motors, wheels, levers, dials, 

 and other mechanical devices to perform the operations required of 

 these calculators. The motor and wheels accomplish various numbers 

 and types of mechanical operations, and the levers convey these opera- 

 tions to the dials for indicating the numbers. Each dial has 10 posi- 

 tions on it for the numbers through 9. 



Digital computers, on the other hand, are predominantly electronic 

 rather than mechanical devices, but the electronic operations are 

 analogous to many of the mechanical operations in calculators. Num- 

 bers are represented in computers by scries of electrical pulses travel- 

 ing from one part of the computer to another by wires. (These 

 pulses are much like those produced in dialing a telephone.) They 

 occur at such a high speed (hundreds of thousands or millions per 

 second) that mechanical devices cannot be used to produce, control, 

 or count them. Instead, electronic devices called gates^ which use 

 radio tubes or transistors, are used. Electronic -fiip-iiop devices (de- 

 vices which are in one condition or another, with no intermediate 

 positions) also are widely used for counting pulses in computers. 

 Many thousands of these gates and flip-flops may be used in any one 

 computer. 



» Shanks had verified hlg results to 500 places. The computer solution, however, showed 

 that Shanks had erred so that hla figures beyond the 527th place are Incorrect. 



