# Proposed Electronic Calculator/Chapter 4

​4. Arithmetical Considerations.
(i) Minor cycles.–It is intended to divide the information in the storages up into units, probably of 32 digits or thereabouts. Such a storage will be appropriate for carrying a single real number as a binary decimal or for carrying a single instruction. Each sub-storage of this kind is called a minor cycle or word. The longer storages of length about 1000 digits are called major cycles. It will be assumed for definiteness that the length of the minor cycle is 32 and that of the major 1024, although these need not yet be fixed.
​(ii) Use of the binary scale.—The binary scale seems particularly well suited for electronic computation because of its simplicity and the fact that valve equipment can very easily produce and distinguish two sizes of pulse. Apart from the input and output the binary scale will be used throughout in the calculator.
(iii) Requirements for an arithmetical code.—Besides providing a sequence of digits the statement of the value of a real number has to do several other things. All included, (probably), we must:
(a) State the digits themselves, or in other words we must specify an integer in binary form.
(b) We must specify the position of the decimal point.
(c) We must specify the sign.
(d) It would be desirable to give limits of accuracy.
(e) It would be desirable to have some reference describing the significance of the number. This reference might at the same time distinguish between minor cycles which contain numbers and those which contain orders or other information.
None of these except for the first could be said to be absolutely indispensable, but, for instance, it would certainly be inconvenient to manage without a sign reference. The digit requirements for these various purposes are roughly:
(a) 9 decimal digits, i.e. 30 binary,
(b) 9 digits,
(c) 1 digit,
(d) 10 digits,
(e) very flexible.
(iv) A possible arithmetical code.—It is convenient to put the digits into one minor cycle and the fussy bits into another. This may perhaps be qualified as far as the sign digit is concerned: by a trick it can be made part of the normal digit series, essentially in the same way as we regard an initial series of figures 9 as indicating a negative number in normal computing. Let us now specify the code without further beating about the bush. We will use two minor cycles whose digits will be called i$_1$ . . . i$_{32}$, j$_1$ . . . j$_{32}$. Of these j$_{24}$ . . . j$_{32}$ are available for identification purposes, and the remaining digits make the following statement about the number ξ.
There exist rational numbers β, γ and an integer m such that ​$|\xi -2^{m}\beta |<\gamma$
$\beta =\sum _{s=1}^{31}s^{s-1}i_{s}-2^{31}i_{32}$
$m=\sum _{t=1}^{9}2^{t-1}j_{t}-512$
$\gamma =\sum _{u=10}^{17}2^{u+m-n}j_{u}$
$n=\sum _{v=18}^{23}2^{v-18}j_{v}$
This code allows us to specify numbers from ones which are smaller than 10$^{−70}$ to ones which are larger than 10$^{86}$, mentioning a value with sufficient figures that a difference of 1 in the last place corresponds to from 2.5 to 5 parts in 10$^{10}$. An error can be described smaller than a unit in the last place or as large as 30,000 times the quantity itself (or by more if this quantity has its first few ‘significant’ digits zero).
(v) The operations of CA.- The division of the storage into minor cycles is only of value so long as we can conveniently divide the operations to be done into unit operations to be performed on whole minor cycles. When we wish to do more elaborate types of process in which the digits get individual treatment we may find this form of division rather awkward, but we shall still be able to carry these processes out in some roundabout way provided the CA operations are sufficiently inclusive. A list is given below of the operations which will be included. Actually this account is distinctly simplified, and an accurate picture can only be obtained by reading § 12. The account is however quite adequate for an understanding of the main problems involved. The list is certainly theoretically adequate, i.e. given time and instruction tables any required operation can be carried out. The operations are:
(1) Transfers of material between different temporary storages, and between temporary storages and dynamic storage.
(2) Transfers of material from the DS to cards and from cards to DS.
​(3) The various arithmetical operations, addition, subtraction, and multiplication (division being omitted), also ‘short multiplication’ by numbers less than 16, which will be much quicker than long multiplication.
(4) To perform the various logical operations digit by digit. It will be sufficient to be able to do ‘and’, ‘or’, ‘not’, ‘if and only if’, ‘never’ (in symbols A & B, A v B, ∼ A, A = B, F). In other words we arrange to do the processes corresponding to x y, x + y + xy, 1 + x, 1 + (x + y)$^2$, 0 digit by digit, modulo 2, where x and y are two corresponding digits from two particular TS (actually TS 9 and TS 10).