5.0 PRINCIPLES GOVERNING THE ARITHMETICAL OPERATIONS #

5.1 #

Let us now consider certain functions of the first specific part: The central arithmetical part CA.

The element in the sense of 4.3, the vacuum tube used as a current valve or gate, is an all-or- none device, or at least it approximates one: According to whether the grid bias is above or below cut-off, it will pass current or not. It is true that it needs definite potentials on all its electrodes in order to maintain either state, but there are combinations of vacuum tubes which have perfect equilibria: Several states in each of which the combination can exist indefinitely, without any outside support, while appropriate outside stimuli (electric pulses) will transfer it from one equilibrium into another. These are the so called trigger circuits, the basic one having two equilibria and containing two triodes or one pentode. The trigger circuits with more than two equilibria are disproportionately more involved.

Thus, whether the tubes are used as gates or as triggers, the all-or-none, two equilibrium, arrangements are the simplest ones. Since these tube arrangements are to handle numbers by means of their digits, it is natural to use a system of arithmetic in which the digits are also two valued. This suggests the use of the binary system.

The analogs of human neurons, discussed in 4.2–4.3 are equally all-or-none elements. It will appear that they are quite useful for all preliminary, orienting, considerations of vacuum tube systems (cf. {6.1, 6.2}). It is therefore satisfactory that here too the natural arithmetical system to handle is the binary one.

5.2 #

A consistent use of the binary system is also likely to simplify the operations of multiplication and division considerably. Specifically it does away with the decimal multiplication table, or with the alternative double procedure of building up the multiples of each multiplier or quotient digit by additions first, and then combining these ( according to positional value) by a second sequence of additions or subtractions. In other words: Binary arithmetics has a simpler and more one-piece logical structure than any other, particularly than the decimal one.

It must be remembered, of course, that the numerical material which is directly in human use, is likely to have to be expressed in the decimal system. Hence, the notations used in R should be decimal. But it is nevertheless preferable to use strictly binary procedures in CA, and also in whatever numerical material may enter into the central control CC. Hence M should store binary material only.

This necessitates incorporating decimal-binary and binary-decimal conversion facilities into I and O. Since these conversions require a good deal of arithmetical manipulating, it is most economical to use CA, and hence for coordinating purposes also CC, in connection with I and O. The use of CA implies, however, that all arithmetics used in both conversions must be strictly binary. For details, cf. {11.4}.

5.3 #

At this point there arises another question of principle. In all existing devices where the element is not a vacuum tube the reaction time of the element is sufficiently long to make a certain telescoping of the steps involved in addition, subtraction, and still more in multiplication and division, desirable. To take a specific case consider binary multiplication. A reasonable precision for many differential equation problems is given by carrying 8 significant decimal digits, that is by keeping the relative rounding-off errors below $10^{-8}$ . This corresponds to $10^{-27}$ in the binary system, that is to carrying 27 significant binary digits. Hence a multiplication consists of pairing each one of 27 multiplicand digits with each one of 27 multiplier digits, and forming product digits 0 and 1 accordingly, and then positioning and combining them. These are essentially $27^{2} = 729$ steps, and the operations of collecting and combining may about double their number. So 1000–1500 steps are essentially right.

It is natural to observe that in the decimal system a considerably smaller number of steps obtains: $8^{2} = 64$ steps, possibly doubled, that is about 100 steps. However, this low number is purchased at the price of using a multiplication table or otherwise increasing or complicating the equipment. At this price the procedure can be shortened by more direct binary artifices, too, which will be considered presently. For this reason it seems not necessary to discuss the decimal procedure separately.

5.4 #

As pointed out before, 1000–1500 successive steps per multiplication would make any non vacuum tube device unacceptably slow. All such devices, excepting some of the latest special relays, have reaction times of more than 10 milliseconds, and these newest relays (which may have reaction times down to 5 milliseconds) have not been in use very long. This would give an extreme minimum of 10–15 seconds per (8 decimal digit) multiplication, whereas this time is 10 seconds for fast modern desk computing machines, and 6 seconds for the standard IBM multipliers. (For the significance of these durations, as well as of those of possible vacuum tube devices, when applied to typical problems, cf. {}.)

The logical procedure to avoid these long durations, consists of telescoping operations, that is of carrying out simultaneously as many as possible. The complexities of carrying prevent even such simple operations as addition or subtraction to be carried out at once. In division the calculation of a digit cannot even begin unless all digits to its left are already known. Nevertheless considerable simultaneisations are possible: In addition or subtraction all pairs of corresponding digits can be combined at once, all first carry digits can be applied together in the next step, etc. In multiplication all the partial products of the form (multiplicand) × (multiplier digit) can be formed and positioned simultaneously—in the binary system such a partial product is zero or the multiplicand, hence this is only a matter of positioning. In both addition and multiplication the above mentioned accelerated forms of addition and subtraction can be used. Also, in multiplication the partial products can be summed up quickly by adding the first pair together simultaneously with the second pair, the third pair, etc.; then adding the first pair of pair sums together simultaneously with the second one, the third one, etc.; and so on until all terms are collected. (Since $27 ≤ 2^{5}$ , this allows to collect 27 partial sums—assuming a 27 binary digit multiplier—in 5 addition times. This scheme is due to H.Aiken.)

Such accelerating, telescoping procedures are being used in all existing devices. (The use of the decimal system, with or without further telescoping artifices is also of this type, as pointed out at the end of 5.3. It is actually somewhat less efficient than purely dyadic procedures. The arguments of 5.1–5.2 speak against considering it here.) However, they save time only at exactly the rate at which they multiply the necessary equipment, that is the number of elements in the device: Clearly if a duration is halved by systematically carrying out two additions at once, double adding equipment will be required (even assuming that it can be used without disproportionate control facilities and fully efficiently), etc.

This way of gaining time by increasing equipment is fully justified in non vacuum tube element devices, where gaining time is of the essence, and extensive engineering experience is available regarding the handling of involved devices containing many elements. A really all-purpose automatic digital computing system constructed along these lines must, according to all available experience, contain over 10,000 elements.

5.5 #

For a vacuum tube element device on the other hand, it would seem that the opposite procedure holds more promise.

As pointed out in 4.3, the reaction time of a not too complicated vacuum tube device can be made as short as one microsecond. Now at this rate even the unmanipulated duration of the multiplication, obtained in 5.3 is acceptable: 1000–1500 reaction times amount to 1–1.5 milliseconds, and this is so much faster than any conceivable non vacuum tube device, that it actually produces a serious problem of keeping the device balanced, that is to keep the necessarily human supervision beyond its input and output ends in step with its operations. (For details of this cf. {}.)

Regarding other arithmetical operations this can be said: Addition and subtraction are clearly much faster than multiplication. On a basis of 27 binary digits (cf. 5.3), and taking carrying into consideration, each should take at most twice 27 steps, that is about 30–50 steps or reaction times. This amounts to .03–.05 milliseconds. Division takes, in this scheme where shortcuts and telescoping have not been attempted in multiplying and the binary system is being used, about the same number of steps as multiplication. (cf. {7.7, 8.3}). Square rooting is usually, and in this scheme too, not essentially longer than dividing.

5.6 #

Accelerating these arithmetical operations does therefore not seem necessary—at least not until we have become thoroughly and practically familiar with the use of very high speed devices of this kind, and also properly understood and started to exploit the entirely new possibilities for numerical treatment of complicated problems which they open up. Furthermore it seems questionable whether the method of acceleration by telescoping processes at the price of multiplying the number of elements required would in this situation achieve its purpose at all: The more complicated the vacuum tube equipment—that is, the greater the number of elements required—the wider the tolerances must be. Consequently any increase in this direction will also necessitate working with longer reaction times than the above mentioned one of one microsecond. The precise quantitative effects of this factor are hard to estimate in a general way—but they are certainly much more important for vacuum tube elements than for telegraph relay ones.

Thus it seems worthwhile to consider the following viewpoint: The device should be as simple as possible, that is, contain as few elements as possible. This can be achieved by never performing two operations simultaneously, if this would cause a significant increase in the number of elements required. The result will be that the device will work more reliably and the vacuum tubes can be driven to shorter reaction times than otherwise.

5.7 #

The point to which the application of this principle can be profitably pushed will, of course, depend on the actual physical characteristics of the available vacuum tube elements. It may be, that the optimum is not at a 100% application of this principle and that some compromise will be found to be optimal. However, this will always depend on the momentary state of the vacuum tube technique, clearly the faster the tubes are which will function reliably in this situation, the stronger the case is for uncompromising application of this principle. It would seem that already with the present technical possibilities the optimum is rather nearly at this uncompromising solution.

It is also worth emphasizing that up to now all thinking about high speed digital computing devices has tended in the opposite direction: Towards acceleration by telescoping processes at the price of multiplying the number of elements required. It would therefore seem to be more instructive to try to think out as completely as possible the opposite viewpoint: That of absolutely refraining from the procedure mentioned above, that is of carrying out consistently the principle formulated in 5.6.

We will therefore proceed in this direction.