AbstractsMathematics

A systematic and rigorous derivation of the Boolean functions that represent the three operations of the ring of integers in the 1-2-4-5 code is developed from their corresponding tables. The same is done for numerical complementation of a number. The equations of the latter are combined with those for addition to illustrate a representative calculating section of a practical arithmetic unit. The addition equations are also combined with the subtraction equations to form a so-called adder-subtractor which is shown to be faster in operation than a corresponding so-called adder-complementor that uses numerical complementation for subtraction. The amount of physical devices for such a unit, judged from the equations, turns out to be significantly less than twice the amount required for just an adder. For the adder-subtractor Boolean functions to determine the eventual sign of a sum or of a difference of two integers of arbitrary signs are developed as well. The equations of the multiplier represent a scheme of multiplication suggested in "Synthesis of Electronic Computing and Control Circuits" (The Annals of the Computation Laboratory of Harvard University, v. 27, p. 198). When compared to those of the adder they indicate that the amount of physical devices that is required to mechanize them is comparable to that required by the adder. In reality these quantities depend on the number of digits these units are designed to process simultaneously. Simplifications of some of the derived equations are carried out by the Veitch diagram for six independent Boolean variables, by Samson and Mills' or Ouine's consensus and prime implicant scheme and by, what shall be called for the lack of an existing term, a judicious observation method.