designers increases the capacity of the digital computer at no sacrifice
in accuracy or reliability. The analog, unfortunately, cannot make use
of many of these solid-state devices. Again, the bugaboo of accuracy is
the reason; let’s look further into the problem.
The most accurate and reliable analog computers are mechanical in
nature. We can cut gears and turn shafts and wheels to great accuracy
and operate them in controlled temperature and humidity. Paradoxically,
this is because mechanical components are nearer to digital
presentations than are electrical switches, magnets, and electronic
components. A gear can have a finite number of teeth; when we deal with
electrons flowing through a wire we leave the discrete and enter the
continuous world. A tiny change in voltage or current, or magnetic flux,
compounded several hundred times in a complex computer, can change the
final result appreciably if the errors are cumulative, that is, if they
are allowed to pile up. This is what happens in the analog computer
using electrical and electronic components instead of precisely machined
cams and gears.
The digital device, on the other hand, is not so penalized. Though it
uses electronic switches, these can be so set that even an appreciable
variation in current or voltage or resistance will not affect the proper
operation of the switch. We can design a transistor switch, for example,
to close when the current applied exceeds a certain threshold. We do not
have to concern ourselves if this excess current is large or small; the
switch will be on, no more and no less. Or it will be completely off.
Just as there is no such thing as being a little bit dead, there is no
such thing as a partly off digital switch. So our digital computer can
make use of the more advanced electronic components to become more
complex, or smaller, or both. The analog must sacrifice its already
marginal accuracy if it uses more electronics. The argument here is
simplified, of course; there are electronic analog machines in
operation. However, the problem of the “drift” of electronic devices is
inherent and a limiting factor on the performance of the analog.
These, then, are some of the advantages the digital computer has over
its analog relative. It is more flexible in general—though there are
_some_ digital machines that are more specialized than _some_ analog
types; it is more accurate and apparently will remain so; and it is more
amenable to miniaturization and further complexity because its designer
can use less than perfect parts and produce a perfect result.
In the disadvantage department the digital machine’s only drawback seems
to be its childish way of solving problems. About all it knows how to do
is to add 1 and 1 and come up with 2. To multiply, it performs
repetitive additions, and solving a difficult equation becomes a
fantastically complex problem when compared with the instantaneous
solution possible in the analog machine. The digital computer redeems
itself by performing its multitudinous additions at fabulous speeds.
Because it must be fed digits in its input, the digital machine is not
economically feasible in many applications that will probably be
reserved for the analog. A digital clock or thermometer for household
use would be an interesting gimmick, but hardly worth the extra trouble
and expense necessary to produce. Even here, though, first glances may
be wrong and in some cases it may prove worth while to convert analog
inputs to digital with the reverse conversion at the output end. One
example of this is the airborne digital computer which has taken over
many jobs earlier done by analog devices.
There is another reason for the digital machines ubiquitousness, a
reason it does not seem proper to list as merely a relative advantage
over the analog. We have described the analog computer used as an aid to
psychological testing procedures, and its ability to handle a
multiplicity of problems at once. This perhaps tends to obscure the fact
that the digital machine by its very on-off, yes-no nature is ideally
suited to the solving of problems in logic. If it achieves superiority
in mathematics in spite of its seemingly moronic handling of numbers, it
succeeds in logic because of this very feature.
While it might seem more appropriate that music be composed by analogy,
or that a chess-playing machine would likely be an analog computer, we
find the digital machine in these roles. The reason may be explained by
our own brains, composed of billions of neurons, each capable only of
being on or off. While many philosophers build a strong case for the
yes-no-maybe approach with its large areas of gray, the discipline of
formal logic admits to only two states, those that can so conveniently
be represented in the digital computer’s flip-flop or magnetic cores.
The digital computer, then, is not merely a counting machine, but a
decision-maker as well. It can decide whether something should be added,
subtracted, or ignored. Its logical manipulations can by clever
circuitry be extended from AND to OR, NOT, and NOR. It thus can solve
not only arithmetic, but also the problems of logic concerning foxes,
goats, and cabbages, or cannibals and missionaries that give us human
beings so much trouble when we encounter them.
The fact that the digital computer is just such a rigorously logical and
unbending machine poses problems for it in certain of its dealings with
its human masters. Language ideally should be logical in its structure.
In general it probably is, but man is so perverse that he has warped and
twisted his communications to the point that a computer sticking
strictly to book logic will hit snags almost as soon as it starts to
translate human talk into other human talk, or into a logical machine
command or answer.
For instance, we have many words with multiple meanings which give rise
to confusion unless we are schooled in subtleties. There are stories,
some of them apocryphal but nonetheless pointing up the problem, of
terms like “water goat” cropping up in an English-to-Russian
translation. Investigation proved that the more meaningful term would
have been “hydraulic ram.” In another interesting experiment, the
expression, “the spirit is willing but the flesh is weak” was machine
translated into Russian, and then that result in turn re-translated back
into English much in the manner of the party game of “Telephone” in
which an original message is whispered from one person to another and
finally back to the originator. In this instance, the final version was,
“The vodka is strong, but the meat is rotten.”
It is a fine distinction here as to who is wrong, the computer or man
and his irrational languages. Chances are that in the long run true
logic will prevail, and instead of us confusing the computer it will
manage instead to organize our grammar into the more efficient tool it
should be. With proper programming, the computer may even be able to
retain sufficient humor and nuance to make talk interesting and colorful
as well as utilitarian.
We can see that the digital machine with its flexibility, accuracy, and
powerful logical capability is the fair-haired one of the computer
family. Starting with _a_ for abacus, digital computer applications run
through practically the entire alphabet. Its take-over in the banking
field was practically overnight; it excels as a tool for design and
engineering, including the design and engineering of other computers.
Aviation relies heavily on digital computers already, from the sale of
tickets to the control of air traffic.
Gaming theory is important not only to the Saturday night poker-player
and the Las Vegas casino operator, but to military men and
industrialists as well. Manufacturing plants rely more and more on
digital techniques for controls. Language translation, mentioned lightly
above, is a prime need at least until we all begin speaking Esperanto,
Io, or Computerese. Taxation, always with us, may at least be more
smoothly handled when the computers take over. Insurance, the
arrangement of music, spaceflight guidance, and education are random
fields already dependent more or less on the digital computer. We will
not take the time here to go thoroughly into all the jobs for which the
computer has applied for work and been hired; that will be taken up in
later chapters. But from even a quick glance the scope of the digital
machine already should be obvious. This is why it is usually a safe
assumption that the word computer today refers to the digital type.
_Hybrid Computers_
We have talked of the analog and the digital; there remains a further
classification that should be covered. It is the result of a marriage of
our two basic types, a result naturally hybrid. The analog-digital
computer is third in order of importance, but important nonetheless.
[Illustration:
_Minneapolis-Honeywell_
Nerve center of Philadelphia Electric Company’s digital
computer-directed automatic economic dispatch system is this console
from which power directors operate and supervise loading of
generating units at minimum incremental cost.
]
Necessity, as always, mothered the invention of the analog-digital
machine. We have talked of the relative merits of the two types; the
analog is much faster on a complex problem such as solving simultaneous
equations. The digital machine is far more accurate. As an example, the
Psychological Matrix Rotator described earlier could solve its twelve
equations practically instantaneously. A digital machine might take
seconds—a terribly long time by computer standards. If we want an
accurate high-speed differential analyzer, we must combine an analog
with a digital computer.
Because the two are hardly of the same species, this breeding is not an
easy thing. But by careful study, designers effected the desired mating.
The hybrid is not actually a new type of computer, but two different
types tied together and made compatible by suitable converters.
The composite consists of a high-speed general-purpose digital computer,
an electronic analog computer, an analog-to-digital converter, a
digital-to-analog and a suitable control for these two converters. The
converters are called “transducers” and have the ability of changing the
continuous analog signal into discrete pulses of energy, or vice versa.
Sometimes called digital differential analyzers, the hybrid computers
feature the ease of programming of the analog, plus its speed, and the
accuracy and much broader range of the digital machine. Bendix among
others produced such machines several years ago. The National Bureau of
Standards recently began development of what it calls an analog-digital
differential analyzer which it expects to be from ten to a hundred times
more accurate than earlier hybrid computers. The NBS analyzer will be
useful in missile and aircraft design work.
Despite its apparent usefulness as a compromise and happy medium between
the two types, the hybrid would seem to have as limited a future as any
hybrid does. Pure digital techniques may be developed that will be more
efficient than the stopgap combination, and the analog-digital will fall
by the wayside along the computer trail.
_Summary_
Historically, the digital computer was first on the scene. The analog
came along, and for a time was the more popular for a variety of
reasons. One of these was the naïve, cumbersome mode of operation the
digital computer is bound to; another its early lack of speed. Both
these drawbacks have been largely eliminated by advances in electronics,
and apparently this is only the beginning. In a few years the technology
has progressed from standard-size vacuum tubes through miniature tubes
and the shrinking of other components, to semiconductors and other
tinier devices, and now we have something called integrated circuitry,
with molecular electronics on the horizon. These new methods promise
computer elements approaching the size of the neurons in our own brains,
yet with far faster speed of operation.
Such advances help the digital computer more than the analog, barring
some unexpected breakthrough in the accuracy problem of the latter.
Digital building blocks become ever smaller, faster, cheaper, and more
reliable. Computers that fit in the palm of the hand are on the market,
and are already bulky by comparison with those in the laboratory. The
analog-digital hybrid most likely will not be new life for the analog,
but an assimilating of its better qualities by the digital.
------------------------------------------------------------------------
“‘_What’s one and one and one and one and one and one and one and one
and one and one?_’
‘_I don’t know,’ said Alice. ‘I lost count._’
‘_She can’t do Addition,’ the Red Queen interrupted._”
—Lewis Carroll
5: The Binary Boolean Bit
In this world full of “bigness,” in which astronomical numbers apply not
only to the speed of light and the distance to stars but to our national
debt as well, it is refreshing to recall that some lucky tribes have a
mathematical system that goes, “One—two—plenty!” Such an uncluttered
life at times seems highly desirable, and we can only envy those who
lump all numbers from three to billions as simply “plenty.”
Instead we are faced today with about as many different number systems
as there are numbers, having come a long way from the dawn of counting
when an even simpler method than “one—two—plenty” prevailed. Man being
basically self-centered, he first thought in terms of “me,” or one. Two
was not a concept, but two “ones”; likewise, three “ones” and so on.
Pebbles were handy, and to represent the ten animals slain during the
winter, a cave man could make ten scratches on the wall or string out
that many stones.
It is said that the ancient cabbies in Rome had a taximeter that dropped
pebbles one by one onto a plate as the wheels turned the requisite
number of revolutions. This plate of stones was presented to the
passenger at the end of his ride—perhaps where we get the word “fare”!
Prices have risen so much that it would take quite a bag of pebbles in
the taximeter today.
Using units in this manner to express a sum is called the unitary
system. It is the concept that gives rise to the “if all the dollars
spent in this country since such and such a time were laid end to end—”
analogies. Put to practice, this might indeed have a salutary effect,
but long ago man learned that it was not practical to stick to a
one-for-one representation.
How long it was before we stumbled onto the fact that we had a “handy”
counting system attached to our wrists is not positively known, but we
eventually adopted the decimal system. In some places the jump from one
to ten was not made completely. The Pueblo Indians, for instance, double
up one fist each time a sum of five is reached. Thus the doubled fist
and two fingers on the other hand signifies seven. In the
mathematician’s language, this is a modulo-5 system. The decimal system
is modulo-10; in other words we start over each time after reaching 10.
Besides the word digit in our vocabulary to tie fingers and numbers, the
Roman numerals V and X are graphic representations of one hand with
thumb widespread, and two hands crossed, respectively. A point worth
remembering is that the decimal system was chosen arbitrarily because we
happen to have ten digits. There is no divine arithmetical significance
in the number 10; in fact mathematicians would prefer 12, since it can
be divided more ways.
The ancient Mayans, feeling that if 10 were ten times as good as 1, then
surely 20 would be twice the improvement of the decimal system. So they
pulled off their boots and added toes to fingers for a modulo-20 number
system. Their word for 20, then, is the same as that for “the whole man”
for very good reason. Other races adopted even larger base systems, the
base of 60 being an example.
If we look to natural reasons for the development of number systems, we
might decide that the binary, or two-valued system, did not attain much
prominence in naïve civilizations because there are so few one-legged,
two-toed animals! Only when man built himself a machine uniquely suited
to two-valued mathematics did the binary system come into its own.
Numbers are merely conventions, rigorous conventions to be sure with no
semantic vagueness. God did not ordain that we use the decimal system,
as evidenced in the large number of other systems that work just fine.
Some abacuses use the biquinary system, and there are septal, octal, and
sexagesimal systems. We can even express numbers in an ABC or XYZ
notation. So a broad choice was available for the computer designer when
he began to look about for the most efficient system for his new
machine.
Considering only the question of a radix, or base, which will permit the
fewest elements to represent the desired numbers, mathematicians can
show us that a base of not 10, or 12, or any other whole number is most
efficient, but the fraction 2.71828. The ideal model is not found in
man, then, since man does not seem to have 2.71828 of anything. However,
the strange-looking number does happen to be the base of the system of
natural logarithms.
Now a system of mathematics based on 2.71828 might make the most
efficient use of the components of the computer, but it would play hob
with other factors, including the men who must work with such a weird
set of numbers. As is often done, a compromise was made between ideal
and practical choices. Since the computer with the most potential seems
to be the electronic computer, and since its operation hinges on the
opening and closing of simple or sophisticated switches, a two-valued
mathematical system, the binary system, was chosen. It wasn’t far from
the ideal 2.71828, and there was another even more powerful reason for
the choice. Logic is based on a yes-no, true-false system. Here, then,
was the best of all possible number systems: the lowly, apparently
far-from-sophisticated binary notation. As one writer exclaimed sadly, a
concept which had been hailed as a monument to monotheism ended up in
the bowels of a robot!
_The Binary System_
It is believed from ancient writings that the Chinese were aware of the
binary or two-valued system of numbers as early as 3000 B.C. However,
this fact was lost as the years progressed, and Leibnitz thought that he
had discovered binary himself almost 5,000 years later. In an odd twist,
Leibnitz apprised his friend Grimaldi, the Jesuit president of the
Tribunal of Mathematics in China, of the religious significance of
binary 1 and 0 as an argument against Buddhism!
A legend in India also contains indications of the power of the binary
system. The inventor of the game of chess was promised any award he
wanted for this service to the king. The inventor asked simply that the
king place a grain of wheat on the first square of the board, two on the
second, and then four, eight, and so on in ascending powers of two until
the sixty-four squares of the board were covered. Although the king
thought his subject a fool, this amount of wheat would have covered the
entire earth to a depth of about an inch!
We are perhaps more familiar with the binary system than we realize.
Morse code, with its dots and dashes, for example, is a two-valued
system. And the power of a system with a base of two is evident when we
realize that given a single one-pound weight and sufficient two-pound
weights we can weigh _any_ whole-numbered amounts.
At first glance, however, binary numbers seem a hopeless conglomeration
of ones and zeros. This is so only because we have become conditioned to
the decimal system, which was even more hopeless to us as youngsters. We
may have forgotten, with the contempt of familiarity, that our number
system is built on the idea of powers. In grade school we learned that
starting at the right we had units, tens, hundreds, thousands, and so
on. In the decimal number 111, for example, we mean 1 times 10^2, plus 1
times 10^1, plus 1. We have handled so many numbers so many times we
have usually forgotten just what we are doing, and how.
The binary system uses only two numbers: 1 and 0. So it is five times as
simple as the decimal system. It uses powers of two rather than ten,
again far simpler. Let’s take the binary number 111 and break it down
just as we do a decimal number. Starting at the left, we have 1 times
2^2, plus 1 times 2^1, plus 1. This adds up to 7, and there is our
answer.
The decimal system is positional; this is what made it so much more
effective in the simple expression of large numbers than the Roman
numeral system. Binary is positional too, and for larger numbers we
continue moving toward the left, increasing our power of two each time.
Thus 1111 is 2^3 plus 2^2 plus 2^1 plus 1.
[Illustration:
_System Development Corp._
A computer teaching machine answering a question about the binary
system.
]
We are familiar with decimal numbers like 101. This means 1 hundred, no
tens, and 1 unit. Likewise in binary notation 101 means one 4, no 2’s,
and one 1. For all its seeming complexity, then, the binary system is
actually simpler than the “easy” decimal one we are more familiar with.
But despite its simplicity, the binary system is far from being inferior
to the decimal system. You can prove this by doing some counting on your
fingers.
Normally we count, or tally, by bending down a finger for each new unit
we want to record. With both hands, then, we can add up only ten units,
a quite limited range. We can add a bit of sophistication, and assign a
different number to each finger; thus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Now, believe it or not, we can tally up to 55 with our hands! As each
unit is counted, we raise and lower the correct finger in turn. On
reaching 10, we leave that finger—thumb, actually—depressed, and start
over with 1. On reaching 9, we leave it depressed, and so on. We have
increased the capacity of our counting machine by 5-1/2 times without
even taking off our shoes. The mathematician, by the way, would say we
have a capability of not 55 but 56 numbers, since all fingers up would
signify 0, which can be called a number. Thus our two hands represent to
the mathematician a modulo-56 counter.
This would seem to vanquish the lowly binary system for good, but let’s
do a bit more counting. This time we will assign each finger a number
corresponding to the powers of 2 we use in reading our binary numbers.
Thus we assign the numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512.
How many units can we count now? Not 10, or 55, but a good bit better
than that. Using binary notation, our ten digits can now record a total
of 1,023 units. True, it will take a bit of dexterity, but by bending
and straightening fingers to make the proper sums, when you finally have
all fingers down you will have counted 1,023, or 1,024 if you are a
mathematical purist.
Once convinced that the binary method does have its merits, it may be a
little easier to pursue a mastery of representing numbers in binary
notation, difficult as it may seem at the outset. The usual way to
convert is to remember, or list, the powers of 2, and start at the left
side with the largest power that can be divided into the decimal number
we want to convert. Suppose we want to change the number 500 into
binary. First we make a chart of the positions:
Power of 2 8 7 6 5 4 3 2 1 0
─────────────────────────────────────────────────────────────
Decimal Number 256 128 64 32 16 8 4 2 1
─────────────────────────────────────────────────────────────
Binary Number 1 1 1 1 1 0 1 0 0