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(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 i1 . . . i32, j1 . . . j32. Of these j24 . . . j32 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
