If a programmer would like to implement arithemetic in “pure Lisp”, numbers should be represented as symbolic expressions, for example, 857 as (8,5,7) and then to implement algorithms such as those as used by humans when computing basic computing operations.
D.P. Jenkinks implemented 1960. or 1961. a Lisp on a computer TREAC in Royal Radar Establishment in Malvern. Jenkins and Phillip Woodward 1961.^234 described a program for addition of numbers. Numerals are symbols contained in a list tab that has value
(0,1,2,3,4,5,6,7,8,9).
Ingeres are represented as lists of tab sublists, written in reversed order. For example, number 857 is represented by the list
((7,8,9),(5,6,7,8,9),(8,9)).
Functions needed for list addition are
sum[x; y] = [null[y] → x; T → inc[x; y; tab]]
inc[x; y; b] = [eq[caar[y]; car[b]] →cons[car[x]; sum[cdr[x];cdr[y]]]; T → inc[step[x]; y; cdr[b]]] step[x] =[null[x] → cons[cdr[tab]; NIL]; null[cdar[x]] → cons[tab; step[cdr[x]]]; T → cons[cdar[x]; cdr[x]]].
McCarthy wrote later that even the original group at MIT made an attempt to represent numbers in similar fashion, but that this approach was abandoned due to slowness.^235 Author of this book did not found any document in which such attempts are described, and neither did the members of the original group with whom he consultated remember such attempts.
In Lisp I., and also in later Lisp 1.5, numbers were already implemented as special type of atoms. Such implementation certainly undermine the beauty of the language and brings suspicion to it’s usability, but it seems that it was unavoiadble. Even today, in all practical Lisp implementations, numbers are implemented as a special type of data.
234 Woodwards & Jenkins,/ Atoms and lists/, 1961., p. 52. 235 Numbers were originally implemented in LISP I as lists of atoms, and this proved too slow for all but the simplest computations. A reasonably efficient implementation of numbers as atoms in S-expressions was made in LISP 1.5 McCarthy, History of Lisp, 1981., p. 180.