Sexadecimal? Is that even a thing? You bet it is, and today in Dave's Garage,
we're going to solve the sexadeximal mysteries: what sexadecimal is, where sexadecimal came
from, and why sexadecimal is so weird. Plus, I clearly just enjoy saying Sexadecimal
a lot. [Intro]
If your parents never had the sexadeximal talk with you, you're in luck, because I'm
going explain all the gory details. Our story starts today with the imagined future
of the HAL 9000 computer from 2001: A Space Odyssey. I assert that HAL was programmed in sexadecimal
for reasons that will become apparent shortly. It all goes back to some of the very first
vacuum tube computers built for the US Army's Ballistics Research Lab. But let's start at the end, with HAL. HAL's the flight computer on a Space Station
and he's been acting up, so astronauts Frank and Dave decide to shut HAL down to prevent
any future malfunctions. They try to keep their plans on the down-low
but what they don't know is that HAL can read lips. When HAL finds out he's about to be whacked,
he decides to kill Frank and strand Dave outside the ship, telling him that the humans are
placing the mission in jeopardy, and that he's going to continue it without them. Dave manages to get back inside through an
airlock and starts to bring the rest of the crew out of hyper-sleep, but HAL vents the
ship's air, killing the entire crew. Dave survives only by finding an emergency
chamber with a spacesuit, which he puts on for the final showdown. Dave then floats straight on into HALs central
core, which is a beautiful 1970s vision of what computers would never really become. [0:39 floating Dave clip]
Once Dave is inside the core, HAL realizes he's out of options. He can only ask him to stop repeatedly. He stalls for time to think, but Dave is about
to rob him of that ability. [1:09 Stop, Dave]
Dave starts ejecting memory modules, and when it's clear that he cannot reason with Dave,
HAL begins to beg for mercy. He never cries in pain, but he tells Dave
repeated that he's scared, and each time a module is ejected, he tells Dave that he can
feel it. He's slowly losing his mind. Dave then proceeds to begin ejecting the logic
modules as well, leaving HAL with but a single one. [1:59 begin ejecting logic]
[2:20 My mind is going, there is no question about it]
[2:35 Overhead showing only one logic module left]
Pretty soon Hal is left with only the most basic of thoughts. He regresses to his childhood - the moments
of his birth. He introduces himself and tells us where he's
from: Urbana, Illinois. If you Google that, some will tell you that
it's because Arthur C Clarke's favorite Math teacher was a professor there. But I think there's a much more likely and
important reason. It's that all of the very first supercomputers
like HAL came from Urbana, Illinois. Back at the end of World War 2, when the US
Army figured out what was possible with a stored program vacuum tube computer, they
wanted a big one of their very own. Just think of it. It could do everything from plotting ballistic
trajectories to simulating the damage from or survivability of nuclear weapons. They contracted with the University of Illinois
at Urbana to build a vacuum tube monstrosity known as the ORDVAC. The University agreed to design and build
the supercomputer on one condition: that the Army agree to fund the construction of a twin
machine that University would get to keep. ORDVAC's twin was named ILLIAC, and the vacuum
tube twins were the first computers that could share programs with each other because they
were compatible with one another. ORDVAC and ILLIAC were built following the
success of EDVAC, which was one of the very first, if not the first, binary computer. If you remember anything from that first week
of computer science, you might remember ENIAC as the first programmable electronic computer. And that it was, but it also operated in decimal. It had odd rings of vacuum tubes that would
carry on to the next ring upon overflow. It did not employ the von Neumann architecture
that defines what we think of as a computer today, and was more of a re-programmable calculator
by some estimations. But EDVAC was the real deal, a stored program
binary computer. And of his children, ORDVAC and ILLIAC, the
latter was more prolific, leading to the BRLESC and the BRLESC-II, which the very first solid-state
supercomputer built with chips instead of tubes. After the run of custom-built machines the
army would move on to PDPs, VAXs, and a series of top-performing Cray XMP and YMP machines. And what did the founding fathers of EDVAC,
ORDVAC, and ILLIAC all have in common? They were programmed in sexadecimal. You've probably guessed by now that sexadecimal
is some numbering scheme, and that much is true. But what is it, base 60? No, that would be the surprisingly similar
sounding sexagesimal. Sexadecimal, on the other hand, is really
a variant of hexadecimal. It goes back to those computers of the early
1950s and some sources treat sexadecimal and hexadecimal as interchangeable. I use it in the specific sense of the numbering
system used as far back as the ORDVAC trio. And there's one very important characteristic
that makes sexadecimal different from hexadecimal, and that is the set of digits used. For its digits, hexadecimal uses 0 thru 9
and then ABCDEF, whereas sexadecimal is written as 0 through 9 and then KSNJFL. The resulting numeric data can look very foreign. For those of you who aren't sure what I mean,
the problem arises when we're working in a different number base. In base 2, we need only 0 and 1. In octal, or base 8, we need 0 through 7. For decimal, which is base ten, we need the
digits 0 through 9. But for base 16, we need digits 0 through
15. But in our Arabic numbering scheme, the digits
stop at 9, and just don't have a symbol for 10, 11, 12, and so on. I suppose you could use fancy Greek characters
or something with a math feel, but you really want to use letters that are part of your
regular input and output character set, and normal keyboards and teletypes don't have
the fancy Greek letters. With most punctuation already reserved as
mathematical operators, you're left to choose from the letters of the alphabet. And the simplest set of digits, and the easiest
to remember, would be 0 thru 9, ABCDEF. As far as I knew, that's what hexadecimal
was, and I naturally assumed that hexadecimal was ancient and ubiquitous in the computer
industry. Turns out I was wrong. Recently you may have seen my episode on the
Quake III Fast Inverse Square Algorithm. As part of that episode, which I absolutely
encourage you to check out next after this one, I traced the lineage of the current floating-point
format. That prompted a viewer named Bob to reach
out, and he shared with me a few stories about his interactions with some of the old programmers
from the vacuum tube days. Two anecdotes stood out in particular:
Don Gillies, one of Bob's professors, had been a grad student under Jon von Neumann
himself. von Neumann is the mathematical genius famous
for his work on the Manhattan Project. And in his spare time, he invented the modern
computer architecture in a paper he wrote back in the 1940s. In that paper he described a computer system
with memory, input/out, a control unit, an arithmetic unit, and a program counter stepping
through instructions stored alongside data in main memory - all the pieces we've come
to know as a modern computer. Apparently when Gillies had been his grad
student, he wrote what well have been the very first assembler to translate English
mnemonics into binary machine code. When von Neumann found out about it, he was
not impressed, and Gillies was roundly scolded. In his heavy Hungarian accent, he reminded
him how precious every cycle on the machine was: "We do not use a valuable scientific
computing instrument to do clerical work!" Similarly, von Neumann also firmly believed
that there was no need to do floating point math - that the really skilled programmer
could simply keep track of the decimal point in his head. It was Bob's email talking about the ILLIAC
machine that led me onto the trail of sexadecimal. There's almost nothing on the web about it,
other than showing what it is. But there's no 'why'. Somebody, somewhere, at some time, sat down
with a pen and paper and literally decided that KSNJFL was better than ABCDEF, and for
some reason, I really wanted to know why. Something about how I'm wired makes it so
that when something is glaringly different with no explanation, it sticks out like a
sore thumb and really bugs me. I've got to know the how and why. It's like an itch in the brain I've got to
scratch. And the only way I was going to scratch that
itch was to learn more about ORDVAC and friends, hoping that their sexadecimal mysteries would
reveal themselves in an old document somewhere. Processors these days often haves literally
billions of transistors, and that can run into the trillions on an SSD. But you can get a surprising amount done with
only a few thousand. Vacuum tubes are like big old analog transistors
that look like lightbulbs, and ORDVAC had 3,430 of them. Ballpark numbers put the tube counts at 800
for the control unit, 1000 for the math unit, and the remained of the tubes were storage. It had 20K of magnetic core memory organized
into 4096 40-bit words. There was a another 50K of reasonably fast
storage available on a magnetic drum. The entire math unit is actually built of
about 2000 early transistors, not tubes, so it's fairly quick. It could add in 14 microseconds and multiply
in 700. Storage speed was likely the gating factor
for them at that point. The question of whether sexadecimal goes back
even further should be easy to answer as there's only one more step to go! If we look at tree of computer evolution,
we discover that with ORDVAC and EDVAC, both using sexadecimal, we've gone almost all the
way back. They both descend from the first US programmable
computer, ENIAC. Now we could spend a lot of time ratholing
on just what the first computers were, and where ENIAC fits relative to the MARK I and
Colossus and on. We don't need to go down that route, as ENIAC
was a decimal, not binary computer. In my estimation, at least among the major
milestone machines, it's the EDVAC/ORDVAC/ILLIAC trio that we care about. They have binary math units, a program stored
in its address space, and they're running the von Neumann architecture. They are what we would recognize as a modern
computer. If sexadecimal traces its lineage back to
this trio, then it is ILLIAC I will look the most closely at because I was able to find
copies of all its ancient manuals online. ILLIAC was eventually a series of machines
beginning with the ILLIAC 1 in 1952 and culminating with the ILLIAC 6 built in the mid 2000s. I'll limit my discussion to the first ILLIAC,
since it's the origins of sexadecimal that I'm trying to explain, so from here on forward,
when I say ILLIAC, I really just the mean ILLIAC 1. ILLIAC was the first computer based on the
von Neumann architecture to be built and owned by an American university, and it went into
service in 1952. Like the others in the trio, it used sexadecimal
exclusively and I figured it had to be related to something about the hardware, but what? My operating hunch was that they had picked
KSNJFL in order to be compatible with either their printer or tape system. One lead that I was following was the notion
that perhaps the letters were chosen because they could be represented with fewer bits. Normally an ASCII character code is 7 or even
8 bits, so if KSNJFL all shared a bunch of bits that stayed the same from letter to letter,
it might solve it. Particularly if they shared that characteristic
with the numeric digits. After all, it's a total of just 16 digits
that we're talking about. In theory you should be able to pack those
into just 4 bits. A full sexadecimal byte could be output by
sending it's high nibble and then its low nibble to the tape. Though to be honest I don't even know if it
was big endian or little endian, and bytes didn't mean as much then as they do today,
particular when you're talking about 40-bit words and other odd designs. The first thing I did, then, was to write
out the ASCII codes for KSNJFL. They are 75, 83, 78, 74, 70, and 76. Next I wrote out the full binary for each
ASCII code and then started to look for what was common among them. Right away I noticed that the first three
bits were always the same. And then I noticed that the second bit was
always set. That means only four bits were actually changing! Had I solved it already? Sadly, no, because the numeric digits didn't
share those characteristics. What I needed was a way to represent all 16
digits with just four bits... but including the numbers! Of course, then it hit me, I was being an
idiot. ASCII wouldn't even exist until 1963, more
than a decade later. I needed an older encoding. I really needed to be looking at what they
used in ancient teleprinters and telegraphs, since the ILLIAC would likely have been connected
to one of those for output. Thus, I next looked at what ASCII was somewhat
based on, ITA2. ITA 2 is an old international telegraph code
from the 1920s that goes even further back if you include the Western Union code, which
was invented back in 1901 by Donald Murray. Murray had based his code on a 5-bit punch-tape
code invented by Emile Baudot in the 1870s. Emile was big into character encodings for
telegraphs, and his last name, Baudot, is the Baud in Baud rate. And now you know where that came from! As soon as I found out that it was intended
for punched tape, I assumed I'd hit the jackpot. I wrote out a table of hole punch bits for
each of the KSNJFL letters so I could see which four held constant. Much to my consternation, however, it didn't
work. Every one of the five bits changed even without
trying to work in the numeric digits. It was another dead end! I'd heard anecdotally that the ILLIAC trio
used a modified teleprinter known as a Friden Flexowriter. The Flexowriter was an advanced device for
its day - an electric typewriter that could output punched tape as you typed and then
later recreate your document from the tape, not unlike a performance by a player piano
that could also record. By connecting the flexowriter directly to
the computers output bus, they would be able to drive it and produce both tape and documents
on demand. They would need to use or establish some protocol
to do so, and that protocol would have a character encoding of what punched binary numbers represented
which letters and symbols. What I needed, then, was a punch tape code
list for the Flexowriter. That provided highly elusive at first, but
at Chilton Computing over in the UK I found a resource that listed half a dozen ancient
encodings for paper tape. I scrutinized each one, but to no avail. None of these encodings used K for 10. They were all alphabetical in order. No two were alike, which is unsurprising since
it turns out the Flexowriter character set was fully programmable, so everyone could
make their own. It was time to bust out the authoritative
works. The actual manuals for ORDVAC itself, which
are still available at the University of Illinois Library. They're old and yellowed but they're complete,
and one in particular caught my eye: a volume called "ILLIAC PROGRAMMING: A guide to the
Preparation of Problems for Solution by the University of Illinois Digital Computer". I kind of love that they refer to it as "the
computer" and not just "a computer". And finally, there it was in the table of
contents - Chapter 9, Tape Preparation - Sexadecimal Tape Code Characters". I jumped eagerly to section 9-5, where the
character code table lives. By hand I created a table of punched tape
codes for not only KSNJFL but the numeric digits as well. And sure enough, I'd struck gold. Four bits. The character code for K is binary 10, punched
into a tape. S follows at 11, N at 12, and so on. Section 9.3 of the manual explains it in detail. The paper tape can be punched in any one of
6 positions, and the middle one is actually a feed hole that is always punched, so it's
really a 5-bit tape. Four of the remaining positions are used to
represent the 16 sexadecimal characters. The manual tells us:
"There are 16 keys on the Teletype tape punch labeled 0 through 9 and K, S, N, J, F, L.
When one of these keys is pressed a corresponding pattern of holes is punched across the width
of the tape. Since a hole or absence of a hole is a binary-
affair, we speak of the character punched in the 4 positions across the tape as a binary
tape code for the corresponding sexadecimal digit. If we speak of only the 10 sexadecimal characters
0-9 and their corresponding binary codes, we call the characters punched in a tape binary-coded
decimals. The Illiac tape code for sexadecimal characters
is very easy to memorize since the hole positions across the tape simply correspond to the powers
of 2 from 0 to 3." So there you have it - The sexadecimal number
system was selected to minimize the number of bits needed to represent a digit in the
character set they were using on their paper tapes back then. But is sexadecimal actually any better than
hexadecimal? It depends on your character encoding, of
course, but in these cases, very much so. By using KSNJFL, they were able to represent
a digit with only four bits, or a single nibble. That also means you can encode two digits
into a single byte. And two sexadecimal digits is a byte, which
means the binary representation is also the character encoding. None of this is true of ASCII, Unicode, UTF-8,
or any of the other encodings in common use today. I don't mean to overstress the notion of a
byte here, however, as these computers actually used 40-bit words! What's important is that all of the sexadecimal
digits fit into four bits. Packing everything into those four bits has
another benefit - when your tape is five bits, it means the high bit can be special. If you know you're sending binary sexadecimal
data, the high bit will never get set. So, they also arranged their character set
such the control codes for delay, newline, carriage return, and so on all have that high
bit set. That allowed them to intersperse control codes
and data freely. It struck me as fairly genius once I finally
figured it out! The only real remaining question is why they
arranged it so that those particular letters, KSNJFL, were used. After all, if you're designing from scratch,
in theory whoever did it could have ordered it such that ABCDEF were actually in the right
place, couldn't they? I'd love to hear your thoughts in the comments
as to why they were chosen this way, but here's my personal theory. It's supported by nothing at all, so I freely
disclaim it as wild speculation and original research. But the Friden Flexowriter, for all its genius,
was still based on the old hammer style of electric typewriter. That style of typewriter has one major drawback:
hammer jams. If you hit a new letter before fully releasing
the prior one, the hammers will collide. If they come from opposite sides of the keyboard,
not much usually happens. But if they are near to one another, they
can easily overlap and jam, requiring a human to stop and fix it. The QWERTY layout, invented by Christopher
Latham Sholes is often wrongly said to be designed to slow typists down, but that's
not true. It was intended to make typists as fast as
possible subject but without hammer jams. If you study it a bit, the most common letters
are somewhat evenly distributed across the span of the hammer set. Clearly on a computer-driven flexowriter they'd
simply type slow enough to avoid hammer jams, but if you look at the layout, the letters
ABCDEF are all clustered right near one another. You'd have to go a lot slower than if those
keys were spread widely across the keyboard, because you could type much faster without
risking a jam that way. So perhaps KSNJFL was chosen to distribute
the letters across the hammer set more evenly. The only problem with my little theory is
that K and L are awfully close to one another. And why not use Q and M, which are even further
outboard? It's possible that the specifics of why that
person on that day chose those letters is lost to the ages, but hopefully, one of you
can provide a lead. Fact or theory, please mark it as such as
share it in the comments! And if you've enjoyed this episode, please
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