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# 2025-04-01 - On Binary Digits And Human Habits by Frederik Pohl
When an astronomer wants to know where the planet Neptune is going to
be on July 4th, 2753 A.D., he can if he wishes spend the rest of his
life working out sums on paper with pencil. Given good health and
fast reflexes, he may live long enough to come up with the answer.
But he is more likely to employ the services of an electronic
computer, which--once properly programmed and set in motion--will
click out the answer in a matter of hours. Meanwhile the astronomer
can catch up on his gin rummy or, alternatively, start thinking about
the next problem he wants to set the computer. It isn't only
astronomers, of course, who let the electrons do their arithmetic.
More and more, in industry, financial institutions, organs of
government and nearly every area of life, computers are regularly
used to supply quick answers to hard questions.
A big problem in facilitating this use, and one which costs
computer-users many millions of dollars each year, is that computers
are mostly adapted to a diet of binary numbers. The familiar decimal
digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 which we use every day upset
their digestions. They prefer simple binary digits, 0 and 1, no more.
With them the computers can represent any finite quantity quite as
unambiguously as we can with five times as many digits in the decimal
scheme; what's more, they can "write" their two digits electronically
by following such simple rules as, "A current flowing through here
means 1, no current flowing through here means 0."
In practice, many computers are now equipped with automatic
translators which, before anything else happens, convert the decimal
information they are fed into the binary arithmetic they can digest.
A few, even--clumsy brutes they are!--actually work directly with
decimal numbers.
But intrinsically binary arithmetic has substantial advantages over
decimal... once it is mastered! It is only because it has not been
entirely easy to master it that we have been required to take the
additional, otherwise unnecessary step of conversion.
The principal difficulty in binary arithmetic is in the appearance of
the binary numbers themselves. They are homely, awkward and strange.
They look like a string of stutters by an electric typewriter with a
slipping key; and they pronounce only with difficulty. For example,
the figure 11110101000 defies quick recognition by most humans,
although most digital computers know it to be the sum of 2^10 plus
2^9 plus 2^8 plus 2^7 plus 2^5 plus 2^3--i.e., in decimal notation,
1960.
To cope with this problem some workers have devised their own
conventions of writing and pronouncing such numbers. A system in use
at the Bell Telephone Laboratories would set off the above figure in
groups of three digits:
11,110,101,000
and would then pronounce each group of three (or less) separately as
its decimal equivalent. The first binary group, 11, is the equivalent
of the decimal 3; the second, 110, of the decimal 6; the third, 101,
of the decimal 5. (000 is zero in any notation.) The above would then
be read, "Three, six, five, zero."
Another suggestion, made by the writer, is to set off binary digits
in groups of five and pronounce them according to the "dits" and
"dahs" of Morse code, "dit" standing for 1 and "dah" for zero. Thus
the date 1960 would be written:
1,11101,01000
and pronounced, "Dit, didididahdit, dahdidahdahdah."
Obviously both of these proposals offer some advantages over the
employment of no special system at all, i.e., writing the number as
one unit and pronouncing it, "One one one one oh one oh one oh oh
oh." Yet it seems, if only on heuristic principles, that conventions
devised especially for binary notation might offer attractions. Such
a convention would probably prove somewhat more difficult to learn
than those, like the above, which employ features derived from other
vocabularies. It might be so devised, however, that it could provide
economy and a lessening of ambiguity.
One such convention has already been proposed by Joshua Stern of the
National Bureau of Standards, who would set off binary numbers in
groups of four--
111,1010,1000
and gives names to selected quantities, so that binary 10 would
become "ap", 100 "bru", 1000 "cid", 1,0000 "dag" and, finally,
1,0000,0000 "hi". The only other names used in this system would be
"one" for 1 and "zero" for 0, as in decimal notation. In this way the
binary equivalent of 1960 would be read as, "bruaponehi, cidapdag,
cid."
It will be seen that the Stern proposal has a built-in mnemonic aid,
in that the new names are arranged alphabetically. "Ap" contains one
zero, "bru" two zeroes, "cid" three zeroes and so on.
Such a proposal provides prospects of additions and refinements which
could well approach those features of convenience some thousands of
years of working with decimal notation have given us. Yet it may
nevertheless be profitable to explore some of the other ways in which
a suitable naming system can be constructed.
As a starting point, let us elect to write our binary numbers in
double groups of three, set off by a comma after (more accurately,
before) each pair of groups. Our binary representation of the decimal
year 1960 then becomes:
11,110,101,000
and we may proceed to a consideration of how to pronounce it. Taking
one semi-group of three digits as the unit "root" of the number-word,
we find that there are eight possible "roots" to be pronounced:
Binary
------
000
001
010
011
100
101
110
111
The full double group of six digits represents 8x8 or 64 possible
cases. By assigning a pronunciation to each of the eight roots, and
by affixing what other sounds may prove necessary as aids in
pronunciation, we should then be able to construct a sixty-four word
vocabulary with which we can pronounce any finite binary number.
The problem of pronunciation does not, of course, limit itself to the
case of one worker reading results aloud to another. It has been
suggested that we all, in some degree, subvocalize as we read. To see
the word "cat" is to hear the sound of the word "cat" in the mind,
and when the mind is not able instantly to produce an appropriate
sound reading falters. (This point is one which probably needs no
belaboring for any person who has ever attempted to learn a foreign
language out of books.) Reading is, indeed, accompanied often by a
faint mutter of the larynx which can be detected and amplified to
recognizable speech.
Our first suggestion for pronunciation is that there is no need to
assign word equivalents to each digit in binary notation, whether the
words be "one" and "zero", "dit" and "dah" or any other sounds. We
may if we choose regard each digit as a letter, and pronounce the
word they construct as a unit in its own right--as, indeed, the
written word "cat" is pronounced as a unit and not as "see, ay, tee."
Perhaps the adoption of vowel sounds for the digit 0 (if only because
0 looks like a vowel) and consonant sounds for 1 will give us a
starting point in this attempt.
A useful consonant would be one which takes different sounds as it
is moved forward and backward in the mouth: "t" and "d" are one such
group in English. We can then assign one of these values for the
single 1, one for a double 1, etc.
Let us assign to the single 1 the sound "t" and to the double 1 the
sound "d." Postponing for a moment the consideration of the triple 1,
and representing the as yet unassigned vowel sounds of 0 by the
neutral "uh," we can pronounce the first seven basic binary groups as
follows:
Binary Pronunciation
------ -------------
000 (uh)
001 (uh) t
010 (uh) t (uh)
011 (uh) d
100 t (uh)
101 t (uh) t
110 d (uh)
The eighth case, 111, can be represented in various ways but, by and
large, it seems reasonable to represent it by the sound "tee."
We now have phonemic representation of each of the eight possible
cases, as follows:
Decimal Written binary Pronounced binary
------- -------------- -----------------
0 000 uh
1 001 ut
2 010 uttuh
3 011 ud
4 100 tuh
5 101 tut
6 110 duh
7 111 tee
We can then continue to count, by compounding groups,
ut-uh (001 000), ut-ut (001 001), ut-uttuh (001 010), etc. In this
way the binary equivalent of the date 1960 might be read as
"Ud-duh, tut-uh."
Conceivably such a system, spoken clearly and listened to with
attention, might serve in some applications, but merely by applying
similar principles in assigning variable vowel sounds to the digit 0
we can much improve it. One such series of sounds which suggests
itself is the set belonging to the letter O itself: the single o as
in "hot" the double o as in "cool," and, for the triple o, by the
same substitution as in the case of "tee," the sound of the letter
itself: "oh." Our first few groups then become:
Decimal Written binary Pronounced binary
------- -------------- -----------------
0 000 Oh
1 001 Oot
2 010 Ahtah
3 011 Odd
4 100 Too
5 101 Tot
6 110 Dah
7 111 Tee
followed by:
8 001 000 Oot-oh
9 001 001 Oot-oot
10 001 010 Oot-ahtah
et cetera. Pronounceability has been somewhat improved, although we
are still conscious of some lacks. The phonemic distinction between
"ahtah" and, say, "odd-dah," the pronunciation of 011 110 is not
great enough to be entirely satisfactory. At any rate, on the above
principles our binary equivalent of 1960 is now pronounced,
"odd-dah, tot-oh."
At this point we may introduce some new considerations. We have
proceeded thus far on mainly logical grounds, but it is difficult to
support the thesis that logic is the only, or even the principal,
basis for constructing any sort of language. Irregularities and
exceptions may in themselves be good things, as promoting recognition
and lessening ambiguity. It may suffice to have only an approximate
correlation between the symbols and the sounds they represent, as,
indeed, it does in the written English language.
As the author feels an arbitrary choice of supplemental sounds will
enhance recognition, he has taken certain personal liberties in their
selection. The consonant L is added to "oh," making it "ohl"; all
root-sounds beginning with a vowel are given a consonant prefix when
they occur alone or as the second part of a compound word; certain
additional sounds are added at the end of the same roots when they
occur as the first part of a compound word; "dah" becomes the
diphthong "dye"; etc. The final list of pronunciations, then,
becomes:
Binary quantity Pronunciation when Pronunciation when alone
in first group or in second group
--------------- ------------------ ------------------------
000 ohly pohl
001 ooty poot
010 ahtah pahtah
011 oddy pod
100 too too
101 totter tot
110 dye dye
111 teeter tee
Decimal 1960 is now in its binary conversion pronounced,
"Oddy-dye, totter-pohl."
We may yet make one additional amendment to that pronunciation,
however. The conventions of reading decimal notation aloud provide a
further convenience for reading large numbers or for stating
approximations. In a number such as:
1,864,191,933,507
we customarily read "trillions," "billions," "millions," etc.,
despite the fact that there is no specific symbol calling for these
words: We read them because we determine, by counting the number of
three-digit groups, what the order of magnitude of the entire
quantity is. The spoken phrase "two million" is a way of saying what
we sometimes write as 2x10^6. In the above number we might say that
it amounts to "nearly two trillion" (in America, at least), which
would be equivalent to "nearly 2x10^12"
A similar convention for binary notation might well be merely to
announce the number of groups (i.e., double groups of six digits) yet
to come. An inconveniently long number might be improved in this way,
a number such as 101 001, 111 011, 001 010, 000 100 being read as
"Totter-oot three groups, teeter-odd, ooty-pahtah, ohly-too." By the
phrase "one group" we would understand that the quantity of the
entire number is approximately the product of the number spoken
before it times 2^6 or 64 (1,000 000). "Two groups" would indicate
that the previous number was to be multiplied by 2^12, "three groups"
by 2^18, etc. Or, in binary notation:
One group: x 10^(1,000 000^1)
Two groups: x 10^(1,000 000^10)
Three groups: x 10^(1,000 000^11)
Four groups: x 10^(1,000 000^100)
And so on, so that as we say, in round decimal numbers: "Oh, about
three million," we might say in round binary numbers, "Oh, about
ooty-poot three groups."
It may be considered that there is an impropriety in using a term
borrowed from decimal notation to indicate binary magnitudes. Such an
impropriety would not be entirely without precedent--the word
"thousand," for example, etymologically related to "dozen," being an
apparent decimal borrowing from a duodecimal system. In any event, as
logic and propriety are not our chief considerations, we may reflect
that the group-numbering will occur only when, sandwiched between
binary terms, there will be small chance for confusion, and elect to
retain it. With this final emendation our spoken term for the binary
equivalent of 1960 becomes, "Oddy-dye one group, totter-pohl."
Let us now consider that we have achieved a satisfactory pronouncing
system for binary numbers and take up the question of whether similar
principles can lead us to a more compact and recognizable method of
graphically representing these quantities. The numbing impact on the
senses of even a fairly large number expressed in binary terms is
well known. Although conventions for setting off groups and stating
approximations, as outlined above, may be helpful, there would appear
to be intrinsic opportunities for error in writing precise
measurements, for example, in binary notation: One of the most common
writing errors in numbering arises from transposing digits, and as
binary numbers have in general about three times as many digits as
their decimal equivalents, they can be assumed to furnish three times
as many opportunities for error.
We have previously chosen to write binary numbers in double groups of
three digits each. As each group represents eight possible cases we
would, in the manner of the Bell workers, represent each group by its
decimal equivalent, so that decimal 1960, which we have given as
binary 011 110, 101 000, could be written in some such fashion as
B36,50. Yet again we may hope to find advantages in a uniquely
designed system for binary numbers.
The author has experimented with a system which has little in common
with notations intended for human use but some resemblance to records
prepared for, or by, machines. For example, a rather rudimentary
"reading" machine intended to grade school examinations or similar
marked papers does so by taking note not of the symbol written but of
its position on the paper--check marks, blacked-in squares, etc. An
abacus, too, denotes quantities by in effect recording the position
of the space between the beads at the top of the wire and the beads
at the bottom.
Indeed, one such system might be designed after the model of the
abacus, requiring the use of paper pre-printed with a design like
this:
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
Each vertical column of eight dots can represent one three-digit
group. To represent the binary equivalent of 1960, one would circle
the fourth dot in the first column, the seventh dot in the second,
the sixth dot in the third and the first dot in the fourth. The first
dot in each column would represent 000, the second 001 and so on.
If we permit the use of preprinted paper, a more compact design might
be a series of drawings of two squares, one above the other, thus:
__ __ __ __ __
| | | | | | | | | |
`--' `--' `--' `--' `--'
__ __ __ __ __
| | | | | | | | | |
`--' `--' `--' `--' `--'
Each pair of two squares is made up of eight lines. By assigning to
each side of a square a value from 000 through 111, a simple check
mark or dot would show the value for that group:
010 (2)
+-------+
| |
001 (1) | | 011 (3)
| |
+-------+
100 (4)
110 (6)
+-------+
| |
101 (5) | | 111 (7)
| |
+-------+
000 (0)
A vertical pair of squares in which the left-hand side of the lower
square was checked would then show that the value 101, or 5, was
recorded for that group.
Some readers will have remarked that the orderly system of
representing the groups from 000 to 111 has been abandoned, the 000
group coming last in this representation. The reason for this is that
we have a perfectly adequate symbol for a zero quantity already--that
is, 0, which is unambiguous in numbering to almost all bases. (The
sole exception is the monadic system, to the base 1. As this does not
permit positional notation it does not require any zero.) Using the 0
symbol here does not, of course, fit in with our plan of checking off
lines on squares, but this is itself only a way-station in the
attempt to find a suitable notation which will not require the use of
preprinted paper.
We might find such a notation by drawing the squares ourselves as
needed, or at least drawing such parts of them as are required. For
001 we would have to show only the first line of the first square.
For 010 we need two lines, the left-hand side and the top. For 011 we
required three lines. At this point we begin to find the drawing of
lines laborious and reflect that, as it is only the last line drawn
which conveys the necessary information, we may be able to find a way
of omitting the earlier ones. Can we do this?
We can if, for example, we explode the square and give it a clockwise
indicated motion by means of arrowheads:
A------->
| |
| |
| |
<-------V
Each arrow has a unique meaning. A is always 001; < is always 100. In
order to distinguish the arrows of the first and second squares, we
can run a short line through the arrows of the lower square; thus V+
is 111, not 011. Having come this far, we discover that we are still
carrying excess baggage; the shaft of the arrow is superfluous, the
head alone will give us all the information we need. We are then able
to draw up this table:
Decimal Binary, digital Binary, graphic
------- --------------- ---------------
0 000 0
1 001 A
2 010 >
3 011 V
4 100 <
5 101 A+
6 110 >+
7 111 V+
and our binary equivalent for 1960 may be written as V>+0.
The writer does not suggest that in this discussion all problems have
been solved and binary notation has been given all the flexibility
and compactness of the decimal system. Even if this were so, the
decimal system possesses many useful devices which have no parallel
here--the distinction between cardinal and ordinal numbers, accepted
conventions for pronouncing fractions, etc. It seems probable,
however, that many nuisances found in conversion between binary and
decimal systems can be alleviated by the application of principles
such as these, and that, with relatively little difficulty,
additional gains can be made--for example, by adapting existing
computers to print and scan binary numbers in a graphic
representation similar to the above. Indeed, it further seems only a
short step to the spoken dictation to the computer of binary numbers
and instructions, and spoken answers in return if desired; but as the
author wishes not to confuse his present contribution with his prior
publications in the field of science fiction he will defer such
prospects to the examination of more academic minds.
Decimal Binary, digital Binary, graphic Pronounced
------- --------------- --------------- ----------
0 000 0 pohl
1 001 A poot
2 010 > pahtah
3 011 V pod
4 100 < too
5 101 A+ tot
6 110 >+ dye
7 111 V+ tee
8 001 000 AO ooty-pohl
9 001 001 AA ooty-poot
10 001 010 AV ooty-pahtah
11 001 011 AV ooty-pod
12 001 100 A< ooty-too
13 001 101 AA+ ooty-tot
14 001 110 A>+ ooty-dye
15 001 111 AV+ ooty-tee
16 010 000 >0 ahtah-pohl
17 010 001 >A ahtah-poot
18 010 010 >> ahtah-pahtah
19 010 011 >V ahtah-pod
20 010 100 >< ahtah-too
30 011 110 V>+ oddy-dye
40 101 000 A+0 totter-pohl
50 110 010 >+> dye-pahtah
100 001,100 100 A,<< poot one group too-too
1,000 001 111,101 000 AV+,A+0 ooty-tee one group
totter-pohl
... ... ... ...
Decimal: 1,000,000
Binary, digital: 011,110 100,001 001,000 000
Binary, graphic: V,>+<,AA,00
Pronounced: pod three group dye-too ooty-poot ohly-pohl
OR
"Oh, about pod three groups."
From:
On Binary Digits And Human Habits by Frederik Pohl (1962)
See also:
How To Count On Your Fingers by Frederik Pohl (1956)
BCR Bytewords
RFC 1751 S/KEY, Another Human Readable Encoding (1994)
PGP Word List (1995)
* * *
The number 31337 could be represented as:
111,101 001,101 001
And spoken as:
tee two group totter-poot totter-poot
Below is an AWK script to convert between decimal and Pohl code.
pohlcode.awk
Extra credit ideas:
* Floating point numbers
* Equally amusing hand signs for a signed version
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