💾 Archived View for spam.works › mirrors › textfiles › programming › endian.dc captured on 2023-07-22 at 20:43:47.
⬅️ Previous capture (2023-06-16)
-=-=-=-=-=-=-
Article 4959 of comp.misc:
Path: ucdavis!ucbvax!pasteur!ames!mailrus!iuvax!bobmon
From: bobmon@iuvax.cs.indiana.edu (RAMontante)
Newsgroups: comp.misc
Subject: Re: "big endian" and "little endian" - first usage for computers
Summary: Danny Cohen's article
Message-ID: <16070@iuvax.cs.indiana.edu>
Date: 31 Dec 88 06:22:06 GMT
Reply-To: bobmon@iuvax.UUCP (RAMontante)
Organization: malkaryotic
Lines: 837
Since this appears to be a fresh topic for lotsa people, I'll repost
Danny Cohen's article in which he (originated or) popularized the terms
Big-Endian and Little-Endian. It's pretty big; I don't know who typed
it all in (maybe John Gilmore, info at the end), but I'm glad they did.
I don't know about the 1 April 1980 date shown, but I've seen the IEEE
version and it has a similar date.
IEN 137 Danny Cohen
U S C/I S I
1 April 1980
ON HOLY WARS AND A PLEA FOR PEACE
INTRODUCTION
This is an attempt to stop a war. I hope it is not too late and that
somehow, magically perhaps, peace will prevail again.
The latecomers into the arena believe that the issue is: "What is the
proper byte order in messages?".
The root of the conflict lies much deeper than that. It is the question
of which bit should travel first, the bit from the little end of the
word, or the bit from the big end of the word? The followers of the
former approach are called the Little-Endians, and the followers of the
latter are called the Big-Endians. The details of the holy war between
the Little-Endians and the Big-Endians are documented in [6] and
described, in brief, in the Appendix. I recommend that you read it at
this point. [I have inserted it -- gnu]
13
A P P E N D I X
Some notes on Swift's Gulliver's Travels:
Gulliver finds out that there is a law, proclaimed by the grandfather of
the present ruler, requiring all citizens of Lilliput to break their
eggs only at the little ends. Of course, all those citizens who broke
their eggs at the big ends were angered by the proclamation. Civil war
broke out between the Little-Endians and the Big-Endians, resulting in
the Big-Endians taking refuge on a nearby island, the kingdom of
Blefuscu.
Using Gulliver's unquestioning point of view, Swift satirizes religious
wars. For 11,000 Lilliputian rebels to die over a controversy as
trivial as at which end eggs have to be broken seems not only cruel but
also absurd, since Gulliver is sufficiently gullible to believe in the
significance of the egg question. The controversy is important
ethically and politically for the Lilliputians. The reader may think
the issue is silly, but he should consider what Swift is making fun of
the actual causes of religious- or holy-wars.
In political terms, Lilliput represents England and Blefuscu France.
The religious controversy over egg-breaking parallels the struggle
between the Protestant Church of England and the Catholic Church of
France, possibly referring to some differences about what the Sacraments
really mean. More specifically, the quarrel about egg-breaking may
allude to the different ways that the Anglican and Catholic Churches
distribute communion, bread and wine for the Anglican, but bread alone
for the Catholic. The French and English struggled over more mundane
questions as well, but in this part of Gulliver's Travels, Swift points
up the symbolic difference between the churches to ridicule any
religious war.
For ease of reference please note that Lilliput and Little-Endians
both start with an "L", and that both Blefuscu and Big-Endians start
with a "B". This is handy while reading this note.]
[End of appendix -- gnu]
The above question arises from the serialization process which is
performed on messages in order to send them through communication media.
If the communication unit is a message - these problems have no meaning.
If the units are computer "words" then one may ask in which order these
words are sent, what is their size, but not in which order the elements
of these words are sent, since they are sent virtually "at-once". If
the unit of transmission is an 8-bit byte, similar questions about bytes
are meaningful, but not the order of the elementary particles which
constitute these bytes.
If the units of communication are bits, the "atoms" ("quarks"?) of
computation, then the only meaningful question is the order in which
bits are sent.
Obviously, this is actually the case for serial transmission. Most
modern communication is based on a single stream of information
("bit-stream"). Hence, bits, rather than bytes or words, are the units
of information which are actually transmitted over the communication
channels such as wires and satellite connections.
Even though a great deal of effort, in both hardware and software, is
dedicated to giving the appearance of byte or word communication, the
basic fact remains: bits are communicated.
Computer memory may be viewed as a linear sequence of bits, divided into
bytes, words, pages and so on. Each unit is a subunit of the next
level. This is, obviously, a hierarchical organization.
2
If the order is consistent, then such a sequence may be communicated
successfully while both parties maintain their freedom to treat the bits
as a set of groups of any arbitrary size. One party may treat a message
as a "page", another as so many "words", or so many "bytes" or so many
bits. If a consistent bit order is used, the "chunk-size" is of no
consequence.
If an inconsistent bit order is used, the chunk size must be understood
and agreed upon by all parties. We will demonstrate some popular but
inconsistent orders later.
In a consistent order, the bit-order, the byte-order, the word-order,
the page-order, and all the other higher level orders are all the same.
Hence, when considering a serial bit-stream, along a communication line
for example, the "chunk" size which the originator of that stream has in
mind is not important.
There are two possible consistent orders. One is starting with the
narrow end of each word (aka "LSB") as the Little-Endians do, or
starting with the wide end (aka "MSB") as their rivals, the Big-Endians,
do.
In this note we usually use the following sample numbers: a "word" is a
32-bit quantity and is designated by a "W", and a "byte" is an 8-bit
quantity which is designated by a "C" (for "Character", not to be
confused with "B" for "Bit)".
MEMORY ORDER
The first word in memory is designated as W0, by both regimes.
Unfortunately, the harmony goes no further.
The Little-Endians assign B0 to the LSB of the words and B31 is the MSB.
The Big-Endians do just the opposite, B0 is the MSB and B31 is the LSB.
By the way, if mathematicians had their way, every sequence would be
numbered from ZERO up, not from ONE, as is traditionally done. If so,
the first item would be called the "zeroth"....
Since most computers are not built by mathematicians, it is no wonder
that some computers designate bits from B1 to B32, in either the
Little-Endians' or the Big-Endians' order. These people probably would
like to number their words from W1 up, just to be consistent.
Back to the main theme. We would like to illustrate the hierarchically
consistent order graphically, but first we have to decide about the
order in which computer words are written on paper. Do they go from
left to right, or from right to left?
3
The English language, like most modern languages, suggests that we lay
these computer words on paper from left to right, like this:
|---word0---|---word1---|---word2---|....
In order to be consistent, B0 should be to the left of B31. If the
bytes in a word are designated as C0 through C3 then C0 is also to the
left of C3. Hence we get:
|---word0---|---word1---|---word2---|....
|C0,C1,C2,C3|C0,C1,C2,C3|C0,C1,C2,C3|.....
|B0......B31|B0......B31|B0......B31|......
If we also use the traditional convention, as introduced by our
numbering system, the wide-end is on the left and the narrow-end is on
the right.
Hence, the above is a perfectly consistent view of the world as depicted
by the Big-Endians. Significance consistently decreases as the item
numbers (address) increases.
Many computers share with the Big-Endians this view about order. In
many of their diagrams the registers are connected such that when the
word W(n) is shifted right, its LSB moves into the MSB of word W(n+1).
English text strings are stored in the same order, with the first
character in C0 of W0, the next in C1 of W0, and so on.
This order is very consistent with itself and with the English language.
On the other hand, the Little-Endians have their view, which is
different but also self-consistent.
They believe that one should start with the narrow end of every word,
and that low addresses are of lower order than high addresses.
Therefore they put their words on paper as if they were written in
Hebrew, like this:
...|---word2---|---word1---|---word0---|
When they add the bit order and the byte order they get:
...|---word2---|---word1---|---word0---|
....|C3,C2,C1,C0|C3,C2,C1,C0|C3,C2,C1,C0|
.....|B31......B0|B31......B0|B31......B0|
In this regime, when word W(n) is shifted right, its LSB moves into the
MSB of word W(n-1).
4
English text strings are stored in the same order, with the first
character in C0 of W0, the next in C1 of W0, and so on.
This order is very consistent with itself, with the Hebrew language, and
(more importantly) with mathematics, because significance increases with
increasing item numbers (address).
It has the disadvantage that English character streams appear to be
written backwards; this is only an aesthetic problem but, admittedly, it
looks funny, especially to speakers of English.
In order to avoid receiving strange comments about this orders the
Little-Endians pretend that they are Chinese, and write the bytes, not
right-to-left but top-to-bottom, like:
C0: "J"
C1: "O"
C2: "H"
C3: "N"
..etc..
Note that there is absolutely no specific significance whatsoever to the
notion of "left" and "right" in bit order in a computer memory. One
could think about it as "up" and "down" for example, or mirror it by
systematically interchanging all the "left"s and "right"s. However,
this notion stems from the concept that computer words represent
numbers, and from the old mathematical tradition that the wide-end of a
number (aka the MSB) is called "left" and the narrow-end of a number is
called "right".
This mathematical convention is the point of reference for the notion of
"left" and "right".
It is easy to determine whether any given computer system was designed
by Little-Endians or by Big-Endians. This is done by watching the way
the registers are connected for the "COMBINED-SHIFT" operation and for
multiple-precision arithmetic like integer products; also by watching
how these quantities are stored in memory; and obviously also by the
order in which bytes are stored within words. Don't let the B0-to-B31
direction fool you!! Most computers were designed by Big-Endians, who
under the threat of criminal prosecution pretended to be Little-Endians,
rather than seeking exile in Blefuscu. They did it by using the
B0-to-B31 convention of the Little-Endians, while keeping the
Big-Endians' conventions for bytes and words.
The PDP10 and the 360, for example, were designed by Big-Endians: their
bit order, byte-order, word-order and page-order are the same. The same
order also applies to long (multi-word) character strings and to
multiple precision numbers.
5
Next, let's consider the new M68000 microprocessor. Its way of storing
a 32-bit number, xy, a 16-bit number, z, and the string "JOHN" in its
16-bit words is shown below (S = sign bit, M = MSB, L = LSB):
SMxxxxxxx yyyyyyyyL SMzzzzzzL "J" "O" "H" "N"
|--word0--|--word1--|--word2--|--word3--|--word4--|....
|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|.....
|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|......
The M68000 always has on the left (i.e., LOWER byte- or word-address)
the wide-end of numbers in any of the various sizes which it may use: 4
(BCD), 8, 16 or 32 bits.
Hence, the M68000 is a consistent Big-Endian, except for its bit
designation, which is used to camouflage its true identity. Remember:
the Big-Endians were the outlaws.
Let's look next at the PDP11 order, since this is the first computer to
claim to be a Little-Endian. Let's again look at the way data is stored
in memory:
"N" "H" "O" "J" SMzzzzzzL SMyyyyyyL SMxxxxxxL
....|--word4--|--word3--|--word2--|--word1--|--word0--|
.....|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|
......|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|
The PDP11 does not have an instruction to move 32-bit numbers. Its
multiplication products are 32-bit quantities created only in the
registers, and may be stored in memory in any way. Therefore, the
32-bit quantity, xy, was not shown in the above diagram.
Hence, the above order is a Little-Endians' consistent order. The PDP11
always stores on the left (i.e., HIGHER bit- or byte-address) the
wide-end of numbers of any of the sizes which it may use: 8 or 16 bits.
However, due to some infiltration from the other camp, the registers of
this Little-Endian's marvel are treated in the Big-Endians' way: a
double length operand (32-bit) is placed with its MSB in the lower
address register and the LSB in the higher address register. Hence,
when depicted on paper, the registers have to be put from left to right,
with the wide end of numbers in the LOWER-address register. This
affects the integer multiplication and division, the combined-shifts and
more. Admittedly, Blefuscu scores on this one.
Later, floating-point hardware was introduced for the PDP11/45.
Floating-point numbers are represented by either 32- or 64-bit
quantities, which are 2 or 4 PDP11 words. The wide end is the one with
the sign bit(s), the exponent and the MSB of the fraction. The narrow
end is the one with the LSB of the fraction. On paper these formats are
clearly shown with the wide end on the left and the narrow on the right,
according to the centuries old mathematical conventions. On page 12-3
6
of the PDP11/45 processor handbook, [3], there is a cute graphical
demonstration of this order, with the word "FRACTION" split over all the
2 or the 4 words which are used to store it.
However, due to some oversights in the security screening process, the
Blefuscuians took over, again. They assigned, as they always do, the
wide end to the LOWer addresses in memory, and the narrow to the HIGHer
addresses.
Let "xy" and "abcd" be 32- and 64-bit floating-point numbers,
respectively. Let's look how these numbers are stored in memory:
ddddddddL ccccccccc bbbbbbbbb SMaaaaaaa yyyyyyyyL SMxxxxxxx
....|--word5--|--word4--|--word3--|--word2--|--word1--|--word0--|
.....|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|
......|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|
Well, Blefuscu scores many points for this. The above reference in [3]
does not even try to camouflage it by any Chinese notation.
Encouraged by this success, as minor as it is, the Blefuscuians tried to
pull another fast one. This time it was on the VAX, the sacred machine
which all the Little-Endians worship.
Let's look at the VAX order. Again, we look at the way the above data
(with xy being a 32-bit integer) is stored in memory:
"N" "H" "O" "J" SMzzzzzzL SMxxxxxxx yyyyyyyyL
...ng2-------|-------long1-------|-------long0-------|
....|--word4--|--word3--|--word2--|--word1--|--word0--|
.....|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|
......|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|
What a beautifully consistent Little-Endians' order this is !!!
So, what about the infiltrators? Did they completely fail in carrying
out their mission? Since the integer arithmetic was closely guarded
they attacked the floating point and the double-floating which were
already known to be easy prey.
7
Let's look, again, at the way the above data is stored, except that now
the 32-bit quantity xy is a floating point number: now this data is
organized in memory in the following Blefuscuian way:
"N" "H" "O" "J" SMzzzzzzL yyyyyyyyL SMxxxxxxx
...ng2-------|-------long1-------|-------long0-------|
....|--word4--|--word3--|--word2--|--word1--|--word0--|
.....|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|-C1-|-C0-|
......|B15....B0|B15....B0|B15....B0|B15....B0|B15....B0|
Blefuscu scores again. The VAX is found guilty, however with the
explanation that it tries to be compatible with the PDP11.
Having found themselves there, the VAXians found a way around this
unaesthetic appearance: the VAX literature (e.g., p. 10 of [4])
describes this order by using the Chinese top-to-bottom notation, rather
than an embarrassing left-to-right or right-to-left one. This page is a
marvel. One has to admire the skillful way in which some quantities are
shown in columns 8-bit wide, some in 16 and other in 32, all in order to
avoid the egg-on-the-face problem.....
By the way, some engineering-type people complain about the "Chinese"
(vertical) notation because usually the top (aka "up") of the diagrams
corresponds to "low"-memory (low addresses). However, anyone who was
brought up by computer scientists, rather than by botanists, knows that
trees grow downward, having their roots at the top of the page and their
leaves down below. Computer scientists seldom remember which way "up"
really is (see 2.3 of [5], pp. 305-309).
Having scored so easily in the floating point department, the
Blefuscuians moved to new territories: Packed-Decimal. The VAX is also
capable of using 4-bit-chunk decimal arithmetic, which is similar to the
well known BCD format.
The Big-Endians struck again, and without any resistance got their way.
The decimal number 12345678 is stored in the VAX memory in this order:
7 8 5 6 3 4 1 2
...|-------long0-------|
....|--word1--|--word0--|
.....|-C1-|-C0-|-C1-|-C0-|
......|B15....B0|B15....B0|
This ugliness cannot be hidden even by the standard Chinese trick.
8
SUMMARY (of the Memory-Order section)
To the best of my knowledge only the Big-Endians of Blefuscu have built
systems with a consistent order which works across chunk-boundaries,
registers, instructions and memories. I failed to find a
Little-Endians' system which is totally consistent.
TRANSMISSION ORDER
In either of the consistent orders the first bit (B0) of the first byte
(C0) of the first word (W0) is sent first, then the rest of the bits of
this byte, then (in the same order) the rest of the bytes of this word,
and so on.
Such a sequence of 8 32-bit words, for example, may be viewed as either
4 long-words, 8 words, 32 bytes or 256 bits.
For example, some people treat the ARPA-internet-datagrams as a sequence
of 16-bit words whereas others treat them as either 8-bit byte streams
or sequences of 32-bit words. This has never been a source of
confusion, because the Big-Endians' consistent order has been assumed.
There are many ways to devise inconsistent orders. The two most popular
ones are the following and its mirror image. Under this order the first
bit to be sent is the LEAST significant bit (B0) of the MOST significant
byte (C0) of the first word, followed by the rest of the bits of this
byte, then the same right-to-left bit order inside the left-to-right
byte order.
Figure 1 shows the transmission order for the 4 orders which were
discussed above, the 2 consistent and the 2 inconsistent ones.
Those who use such an inconsistent order (or any other), and only those,
have to be concerned with the famous byte-order problem. If they can
pretend that their communication medium is really a byte-oriented link
then this inconsistency can be safely hidden under the rug.
A few years ago 8-bit microprocessors appeared and changed drastically
the way we do business. A few years later a wide variety of 8-bit
communication hardware (e.g., Z80-SIO and 2652) followed, all of which
operate in the Little-Endians' order.
9
Now a wave of 16-bit microprocessors has arrived. It is not
inconceivable that 16-bit communication hardware will become a reality
relatively soon.
Since the 16-bit communication gear will be provided by the same folks
who brought us the 8-bit communication gear, it is safe to expect these
two modes to be compatible with each other.
The only way to achieve this is by using the consistent Little-Endians
order, since all the existing gear is already in Little-Endians order.
We have already observed that the Little-Endians do not have consistent
memory orders for intra-computer organization.
IF the 16-bit communication link could be made to operate in any order,
consistent or not, which would give it the appearance of being a byte-
oriented link, THEN the Big-Endians could push (ask? hope? pray?) for an
order which transmits the bytes in left-to-right (i.e., wide-end first)
and use that as a basis for transmitting all quantities (except BCD) in
the more convenient Big-Endians format, with the most significant
portions leading the least significant, maintaining compatibility
between 16- and 32-bit communication, and more.
However, this is a big "IF".
Wouldn't it be nice if we could encapsulate the byte-communication and
forget all about the idiosyncrasies of the past, introduced by RS232 and
TELEX, of sending the narrow-end first?
I believe that it would be nice, but nice things do not necessarily
occur, especially if there is so much silicon against them.
Hence, our choice now is between (1) Big-Endians' computer-convenience
and (2) future compatibility between communication gear of different
chunk size.
I believe that this is the question, and we should address it as such.
Short term convenience considerations are in favor of the former, and
the long term ones are in favor of the latter.
Since the war between the Little-Endians and the Big-Endians is
imminent, let's count who is in whose camp.
The founders of the Little-Endians party are RS232 and TELEX, who stated
that the narrow-end is sent first. So do the HDLC and the SDLC
protocols, the Z80-SIO, Signetics-2652, Intel-8251, Motorola-6850 and
all the rest of the existing communication devices. In addition to
these protocols and chips the PDP11s and the VAXes have already pledged
their allegiance to this camp, and deserve to be on this roster.
10
The HDLC protocol is a full fledged member of this camp because it sends
all of its fields with the narrow end first, as is specifically defined
in Table 1/X.25 (Frame formats) in section 2.2.1 of Recommendation X.25
(see [2]). A close examination of this table reveals that the bit order
of transmission is always 1-to-8. Always, except the FCS (checksum)
field, which is the only 16-bit quantity in the byte-oriented protocol.
The FCS is sent in the 16-to-1 order. How did the Blefuscuians manage
to pull off such a fiasco?! The answer is beyond me. Anyway, anyone
who designates bits as 1-to-8 (instead of 0-to-7) must be gullible to
such tricks.
The Big-Endians have the PDP10's, 370's, ALTO's and Dorado's...
An interesting creature is the ARPANet-IMP. The documentation of its
standard host interface (aka "LH/DH") states that "The high order bit of
each word is transmitted first" (p. 4-4 of [1]), hence, it is a
Big-Endian. This is very convenient, and causes no confusion between
diagrams which are either 32- (e.g., on p. 3-25) and 16-bit wide (e.g.,
on p. 5-14).
However, the IMP's Very Distant Host (VDH) interface is a Little-Endian.
The same document ([1], again, p. F-18), states that the data "must
consist of an even number of 8-bit bytes. Further, considering each pair
of bytes as a 16-bit word, the less significant (right) byte is sent
first".
In order to make this even more clear, p. F-23 states "All bytes (data
bytes too) are transmitted least significant (rightmost) bit first".
Hence, both camps may claim to have this schizophrenic double-agent in
their camp.
Note that the Lilliputians' camp includes all the who's-who of the
communication world, unlike the Blefuscuians' camp which is very much
oriented toward the computing world.
Both camps have already adopted the slogan "We'd rather fight than
switch!".
I believe they mean it.
11
SUMMARY (of the Transmission-Order section)
There are two camps each with its own language. These languages are as
compatible with each other as any Semitic and Latin languages are.
All Big-Endians can talk to each other with relative ease.
So can all the Little-Endians, even though there are some differences
among the dialects used by different tribes.
There is no middle ground. Only one end can go first.
CONCLUSION
Each camp tries to convert the other. Like all the religious wars of
the past, logic is not the decisive tool. Power is. This holy war is
not the first one, and probably will not be the last one either.
The "Be reasonable, do it my way" approach does not work. Neither does
the Esperanto approach of "let's all switch to yet a new language".
Our communication world may split according to the language used. A
certain book (which is NOT mentioned in the references list) has an
interesting story about a similar phenomenon, the Tower of Babel.
Little-Endians are Little-Endians and Big-Endians are Big-Endians and
never the twain shall meet.
We would like to see some Gulliver standing up between the two islands,
forcing a unified communication regime on all of us. I do hope that my
way will be chosen, but I believe that, after all, which way is chosen
does not make too much difference. It is more important to agree upon
an order than which order is agreed upon.
How about tossing a coin ???
12
time time
| |
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
<-MSB---------------LSB- -MSB---------------LSB->
order (1) | | order (2)
time time
| |
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
/ | | \
<-MSB---------------LSB- -MSB---------------LSB->
order (3) | | order (4)
Figure 1: Possible orders, consistent: (1)+(2), inconsistent: (3)+(4).
14
R E F E R E N C E S
[1] Bolt Beranek & Newman.
Report No. 1822: Interface Message Processor.
Technical Report, BB&N, May, 1978.
[2] CCITT.
Orange Book. Volume VIII.2: Public Data Networks.
International Telecommunication Union, Geneva, 1977.
[3] DEC.
PDP11 04/05/10/35/40/45 processor handbook.
Digital Equipment Corp., 1975.
[4] DEC.
VAX11 - Architecture Handbook.
Digital Equipment Corp., 1979.
[5] Knuth, D. E.
The Art of Computer Programming. Volume I: Fundamental
Algorithms.
Addison-Wesley, 1968.
[6] Swift, Jonathan.
Gulliver's Travel.
Unknown publisher, 1726.
15
OTHER SLIGHTLY RELATED TOPICS (IF AT ALL)
not necessarily for inclusion in this note
Who's on first? Zero or One ??
People start counting from the number ONE. The very word FIRST is
abbreviated into the symbol "1st" which indicates ONE, but this is a
very modern notation. The older notions do not necessarily support this
relationship.
In English and French - the word "first" is not derived from the word
"one" but from an old word for "prince" (which means "foremost").
Similarly, the English word "second" is not derived from the number
"two" but from an old word which means "to follow". Obviously there is
an close relation between "third" and "three", "fourth" and "four" and
so on.
Similarly, in Hebrew, for example, the word "first" is derived from the
word "head", meaning "the foremost", but not specifically No. 1. The
Hebrew word for "second" is specifically derived from the word "two".
The same for three, four and all the other numbers.
However, people have,for a very long time, counted from the number One,
not from Zero. As a matter of fact, the inclusion of Zero as a
full-fledged member of the set of all numbers is a relatively modern
concept.
Zero is one of the most important numbers mathematically. It has many
important properties, such as being a multiple of any integer.
A nice mathematical theorem states that for any basis, b, the first b^N
(b to the Nth power) positive integers are represented by exactly N
digits (leading zeros included). This is true if and only if the count
starts with Zero (hence, 0 through b^N-1), not with One (for 1 through
b^N).
This theorem is the basis of computer memory addressing. Typically, 2^N
cells are addressed by an N-bit addressing scheme. Starting the count
from One, rather than Zero, would cause either the loss of one memory
cell, or an additional address line. Since either price is too
expensive, computer engineers agree to use the mathematical notation of
starting with Zero. Good for them!
The designers of the 1401 were probably ashamed to have address-0 and
hid it from the users, pretending that the memory started at address-1.
16
This is probably the reason that all memories start at address-0, even
those of systems which count bits from B1 up.
Communication engineers, like most "normal" people, start counting from
the number One. They never suffer by having to lose a memory cell, for
example. Therefore, they are happily counting 1-to-8, and not 0-to-7 as
computer people learn to do.
ORDER OF NUMBERS.
In English, we write numbers in Big-Endians' left-to-right order. I
believe that this is because we SAY numbers in the Big-Endians' order,
and because we WRITE English in Left-to-right order.
Mathematically there is a lot to be said for the Little-Endians' order.
Serial comparators and dividers prefer the former. Serial adders and
multipliers prefer the latter order.
When was the common Big-Endians order adopted by most modern languages?
In the Bible, numbers are described in words (like "seven") not by
digits (like "7") which were "invented" nearly a thousand years after
the Bible was written. In the old Hebrew Bible many numbers are
expressed in the Little-Endians order (like "Seven and Twenty and
Hundred") but many are in the Big-Endians order as well.
Whenever the Bible is translated into English the contemporary English
order is used. For example, the above number appears in that order in
the Hebrew source of The Book of Esther (1:1). In the King James
Version it is (in English) "Hundred and Seven and Twenty". In the
modern Revised American Standard Version of the Bible this number is
simply "One Hundred and Twenty-Seven".
INTEGERS vs. FRACTIONS
Computer designers treat fix-point multiplication in one of two ways, as
an integer-multiplication or as a fractional-multiplication.
The reason is that when two 16-bit numbers, for example, are multiplied,
the result is a 31-bit number in a 32-bit field. Integers are right
justified; fractions are left justified. The entire difference is only
a single 1-bit shift. As small as it is, this is an important
difference.
Hence, computers are wired differently for these kinds of
multiplications. The addition/subtraction operation is the same for
either integer/fraction operation.
17
If the LSB is B0 then the value of a number is SIGMA<B(i)*[(2)^i]>,
for i=0,15, in the above example. This is, obviously, an integer.
If the MSB is B0 then the value of a number is SIGMA<B(i)*[(1/2)^i]>,
for i=0,15. This is, obviously, a fraction.
Hence, after multiplication the Integerites would typically keep B0-B15,
the LSH (Least Significant Half), and discard the MSH, after verifying
that there is no overflow into it. The Fractionites would also keep
B0-B15, which is the MSH, and discard the LSH.
One could expect Integerites to be Little-Endians, and Fractionites to
be Big-Endians. I do not believe that the world is that consistent.
SWIFT's POINT
It may be interesting to notice that the point which Jonathan Swift
tried to convey in Gulliver's Travels in exactly the opposite of the
point of this note.
Swift's point is that the difference between breaking the egg at the
little-end and breaking it at the big-end is trivial. Therefore, he
suggests, that everyone does it in his own preferred way.
We agree that the difference between sending eggs with the little- or
the big-end first is trivial, but we insist that everyone must do it in
the same way, to avoid anarchy. Since the difference is trivial we may
choose either way, but a decision must be made.