Hexadecimal mechanical calculator from the 1970s
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- čas přidán 16. 12. 2022
- The Hexadder, made by Hexco Inc in the early 1970s. Sent to me by viewer Tom Carlson. Thanks Tom!
This is episode 66 of my video series about old calculating devices.
Original instructions PDF: cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=hexadderinstructions.pdf
Maya numeral table by Wikimedia user Neuromancer2K4, CC-BY-SA-3.0
Roman numeral and Chinese counting rod tables from Wikipedia.
HTML color codes from htmlcolorcodes.com/
End song inspired by "Hotter than a Molotov" by The Coup.
Chris Staecker webarea: cstaecker.fairfield.edu/~cstaecker
#hexadecimal #base
I just love the philosophical bit about what 27 is or does. Somehow it summarizes a lot of what people (and math teachers) don't get about mathematics.
The cubes rearranging themselves is very powerful at conveying how math likes to look at things under so many different perspectives all at once.
It reminds me of that painting “this is not a pipe” what is it? It’s a _picture_ of a pipe.
I am so jealous. I have an Addiator, but as a programmer I would love to have one in hexadecimal too. I had no idea they existed!
Shouldn't be too hard to 3D print addiators in any base. I've never seen models online but it would be a cool project. I can imagine writing some python in blender that could output STLs for any base you like.
@@ChrisStaecker This would be so cool!
@@ChrisStaecker Well, there goes my weekend.
I’m interested if you come up with anything!
@@ChrisStaecker probably not python in blended, but openscad with parametric design.
~~o r i g i n a l~~
Yeah it literally looks like a bootleg addiator
*O R I G I N A L*
The animation for "what 27 is like, what it can do" at 9:37 is really great. In 8 seconds it shows 27 = 3 * 9 = 9 * 3 = 3 * 3 * 3 in a fun way. Reminded me a bit of the educational series "The Mechanical Universe" or "Project Mathematics!". Thanks for putting these videos together!
All my button-up shirts have pockets that are three times as long as normal. Finally an adding machine for me!
The hex adder sounds like a magical snake from a DnD campaign.
I may try to incorporate it in my next game.
Gotta get a 1d16
This was amazing thank you, subscribed.
I am interested in 'Turing Tumble' type mechanical computers. So this (and many of the computers in the end montage) fill(s) tons of gaps in my need for computing real-world problems in real time using only mechanical computers.
Feels like I learned a year of study in minutes.
Love the picture from the Margarita Philosophica! Good video, this.
Your demonstration of counting in binary was the clearest portrayal that I have ever seen. Binary numbers tend to look cryptic but understanding how they are 'built' makes them easy to understand. As always, an excellent and informative video! Thanks Chris!
Thanks Tom! Thanks Chris!
Amazing production. Well done.
This video is gold
what a fantastic video, you not only get to see a cool mechanical calculator, but get a great explanation of numbers and base-systems.
I still use hexadecimal at work nearly every day. It's the natural representation for a pointer. I guess I rewired my brain for that in the 80's when it was still more pliable than it is today.
Man that got real philosophical towards the end, glad I stay for the whole thing!
I only recently discovered your channel but I've loved all these videos about these old devices, history, and your takes on it all. Thank you for sharing all this great information!
Every time I see your videos I learn something new about math! Thanks Chris!
Another great video! You exactly described my confusion as a young kid about different bases. I found it not at all clear that another base would have the same prime numbers or 6 would still divide 12.
Excellent! I guess the sliding up is a version of 15s compliment? Sorry if you did mention that! Fantastic on all counts (no pun intended!) Your '27' story is fantastic and a great bit of storytelling too! Thanks so much for making this video!
Thanks! The sliding up isn’t really any kind of complement- it’s more of a direct subtraction. Sliding down makes it add, and sliding up is just the opposite which makes it subtract. (Typical subtraction by complements would move the mechanism in the same direction as adding, but suppress or modify the carry to make the details work out.)
When I was a kid, I had one of those! I'm pretty sure that mine was base-10, and that my dad (who was a businessman) brought it back from a trade show and gave it to me. It had the bar to clear it. This would have been c. 1970.
As a kid in the '60s, I was fascinated by the mechanical calculating machines (some did more than add / subtract) in my dad's office, and when electronic calculators became affordable, I started a collection.
Probably you had the adductor- very easy to get nowadays on eBay!
Been learning about early computer history-including mechanical computing-lately and stumbling upon this video was wonderful surprise. So glad you went into different base systems too. Our relationship to and development of mathematics is just so weird and amazing.
This would have come in very handy when I was in high school, back in the stone age, as I learned to program in hex on punched paper tape.
That's a cool piece to have!
My grandpa, a civil engineer, had this. Clearly remember it and playing with it. Even as a kid I thought it was nifty.
I also love this series!
I had a hand-held decimal adding machine that worked almost exactly like this back in the 80's. I lost the stylus for it, but a toothpick worked well enough as a replacement.
this is really cool
Very cool gadget! I have a similar calculator (decimal, about 6-8 digits wide, not by Hexco) that my father gave me back in the 1960s(?).
Wow that is so damn cool!
359°59’60” when the computer tries to convert 359.999997° to dms. The extraneous 9s is due to an imperfect base conversion. If I add the angles up on paper they equal 360°. I use Excel to do map check calculations, conversions on certain angles it gets hung up. So I throw a little extra .01 on the end of my dms entry of 890101.01 (89°01’01”). If my polygon closes perfectly I get a divide zero error so I put in a slight error in that case too, must please the binary squirrels inside the computer.
Sweet item, mint! I've usually seen this style in decimal with subtraction just run by reverse rules.
I wonder if they changed the clear mechanism on the Hexadder from what the Adductor had, so the Hexadder can be cleared without picking it up. I imagine that when doing a bunch of sequential math problems being able to clear the Hexadder without putting down the stylus, could be very handy. The Adductor's clear mechanism is nice as it keeps it small, but the Hexadder is already so long that it doesn't matter if it has the stylus clear.
Fact: the Chinese abacus can also compute hex if it has 7 beads on each column.
1 bottom bead = 1
1 top bead = 5
2 top beads = A
2 top + 1 bottom = B
2 top + 2 bottom = C
2 top + 3 bottom = D
2 top + 4 bottom = E
2 top + 5 bottom = F
Fuck yeah a new video. I love these old calculators i never knew they had ones for HEX now find one for OCT
A hexadecimal Troncet adder for programmers? This is the kind of thing I can only imagine existing as a product in a very very specific historical moment (a few years in the 1970s).
I had one! Cool!
Addiator also made a calculator for the pre-decimal(pre 1971) british pound with 2 digits of base 12(pence), 2 digits of base 20(shillings), and 3 digits of base 10(pounds).
i recently bought an addiator for the pre-decimal British currency. It works great with £sd, But I can't find a Hexco or IBM addiator that works with hexadecimal numbers like the one you have here. They can't all have been thrown away. Someone must have one at the back of a dusty drawer. Please get searching folks!
My mom had one of these long calculators...she would take it to the grocery store and add up the groceries!
I love this video. It made me want to buy one of those hex adders.
Just a small note: at 08:57 you said the number 27 in base 2 is 11001.
That is not right ... The correct base2 representation of the number 27 is 11011
so want one of these...
NIce! Did you know "one zero" in any base expresses the value of that base? So 10 in base pi is pi.
Big fan
That last bit of you questioning the number 27 broke my brain a bit...I've never really thought of the relationship between numbers and what they represent. This had me wonder if there were a life form that uses a different number base than 10, how would that impact their math theorems? Would their preference of a different base lead to some total unique theories, or would math theories from different base civilizations ultimately converge?
Note to self...careful with Chris's videos when it's late at night.
The choice of base doesn't effect the theorems at all. (Except the very few theorems which are specifically about base 10 expressions of numbers.) The base only determines what the number looks like when we write it, but not what its properties are. A number is even or odd, prime or not prime, etc- these are properties of the number itself and don't change depending on the base.
For this reason, facts which are specifically about base 10 descriptions of numbers are generally regarded by mathematicians as unserious.
Other-base mathematical cultures would eventually realize the practical importance of binary for machine computation, so a preference for binary in technical contexts is probably universal.
every time you make a video I want the thing you're presenting. I actually found myself the tool for measuring area by perimeter that you showed. it's neat but I absolutely don't need it lol
True, but I do not regret at all searching for a Monroe L-Series. I use it in place of computer-based methods frequently.
Base 12 has a lot of advantages (whole number 3/4, 2/3, 1/2, 1/3, 1/4, 1/6 fractions) and has been ysed (we have individual names for the first twelve numbers.
Wonder why it lost out to base 10? Just can't finger math out sometimes.
I often make my quizzes out of 12 points so I can easily deduct half, third, or fourth credit. Students don't like it though! They get 7/12, but then need to convert to a percentage to find out their "real grade".
Soon as the vlog started .... I seemed to recall having/seeing something like this. Then the initials showed up. CDP .... those were my dad's initials.
Ironic
"where does he get all those toys" - the Joker - 🤩
Tom!
The word “numeral” helps when talking about notations for numbers. “Numeric” “numeral” … well, I suppose that’s where it ends.
Hexco also still makes very nice stainless steel rulers which were nice to have in the days of columnar RPG. It was a badge of a System/38 or AS/400 programmer to have one of these.
Yes- this was the only other old school hexco product I could find. Any others? Possible for a video some day.
Hex adder sounds like a snake that puts a spell on people.
I did not realize how much I was missing your calculating machine descriptions and especially the flights of mathematics they often inspire! Happy holidays to you and yours. And a question: why only 6 digits? I suppose the byte was possibly less important a thing when this was made, but nowadays I am so used to seeing things in every increasing multiples of 2 that 6 hex digits seems very odd!
I agree 6 digits is a bit weird- maybe just copying the addiator? In decimal, 6 digits gets you "up to a million", which seems natural I guess. To me the hex adder is very long and skinny- making it slightly wider to 8 digits would've been fine.
Possibly copying the Addiator model? Hexadecimal is commonly pronounced as hex for short which is technically 6? 6 hex digits will get you to 16,777,215 in decimal which is probably high enough for what you'd use it for?
I can do Hex in my head!
I remember as a kid learning that clocks were based on the sexagesimal system from ancient Mesopotamia. That broke my brain when my teacher said that there were other base systems… I was like… numbers aren’t real 😱
Of course the real weird is L.S.D. calcuilators. Not the drug, but pounds shillings and pence. You have different bases for different digits. Base 4 for farthings, base 10 and base 2 for pence, base 12 for shillings and base 10 for pounds.
Great video. My Granddad had one of these adders, except it used binary. I think it was called the Binosaur. It was wide instead of long, and I think the carry method was easier. If I ever find it, I'll take a pic.
Never heard of this- I’m interested!
@@ChrisStaecker Sorry; I just made it up. Would be cool though.
Damned, I wish I had one of those...
Isn't there *anyone* producing and selling those things anymore?
I don't know of any modern instrument based on the troncet (addiator-type) design.
You might be interested to know there were some computers that weren't binary! Some Computers do have shades of gray! Or at least they used to. They stopped making them because binary computers were less money for more power.
I think there were only ever ternary computers though, nothing higher (other than analog computers), so not too many shades of grey
yeah and they might be bringing back analog computers according to a veritasium video (because they can apparently do the multiplications used in neural networks more efficiently than digital computers)
Yes- I've heard of ternary machines in the USSR. Wild!
They even made base 5 machines too (It was called Biquinary), where they used a few bits to encode a group of 5, and then used another one one the left to determine which range of 5 it was in. Quite interesting to read about
@@lapatatadelplato6520 Interesting- the standard chinese abacus (suanpan) is also called biquinary because each digit has a base-5 piece, and a base-2 piece.
The music is too quiet, we can still hear you. Please increase the soundtrack volume another 40%
What would happen if you added, say, 1 and 1FF? How does it handle the second 1 being carried over?
Perceptive question! The mechanism can't handle double-carries. The Hex Adder handles this exactly like the addiator- see my addiator video around 3:10: czcams.com/video/2mv45XP48bQ/video.html
I shot about 30 seconds for this video explaining it, but I ended up cutting it.
Hey Chris. Any plans on doing a Curta Calculator video?
Way out of my price range! Unless you know a guy...
@@ChrisStaecker If I can make another suggestion what about a Line of Chords ruler. My favourite workshop tool.
@@ChrisStaecker I don't know a guy but I'm sure one of your hardcore fans can make a plan.
@@brendtkieffer7095 I've never heard of this but i will have to look into it. I've been planning a video about the sector rule for a long time, which looks similar. Thanks for the tip!
@@ChrisStaecker It's out of my price range too, but I'd kick in $20. If enough people did, you could have a Curta.
Negative bases are weird and interesting.
Trinary computers are (marginally) a thing. What kind of base would work best with them? 9? 27?
Ive wondered about this too, It almost depends on what you define your "tryte" as, the trinary version of the byte.
I got b27 to work and used 0-9 and letters A-Z to, almost work, at one point I think i used greek numerals. I was working in balanced ternary though
jeez i still have trouble with times tables.....
1:49 What are those symbols between 4 and 6?
I'm not really sure- that image comes from here: commons.wikimedia.org/wiki/File:Ms.Thott.290.2%C2%BA_150v.jpg
You can see elsewhere on the page the numeral for 5 being treated strangely. I guess these are just two variations on the glyph? I can't find more info.
Okay thank you very much. What happens when you carry over to an F on the machine? Thanks in advance.
@@YamamotoTV2021 A "double carry" is a bit tricky on a machine like this- I didn't get into it in this video, but I discuss it in detail in my addiator video. See around 3:08 here: czcams.com/video/2mv45XP48bQ/video.html
I'm with you, 27 is a name for this thing |||||||||||||||||||||||||||, but you don't need to know that to do math. But I feel like, by now, SOMEONE should've figured that out. So yeah, nothing wrong with saying it is a row of marks.
"If we ever encounter like aliens with different anatomy frorn us they might count in terms of ... sixes and twelves."
you just described americans
Смейтесь не смейтесь, но десятичной системы такая линейка была у моего деда. Отечественная.
If you want to discover a 'new kind of math', check out negative bases lol. Yes they are a thing.
There are also even weirder bases: for example complex bases or rational bases.
@@pdote Base _i_ is a weird one for sure. But yeah, all positive and negative integers can be represented as a positive integer in base -x. Meaning all the positive and negative numbers are represented twice, if you include the negative numbers in that base as well. Kind of disproves the idea of 'bigger infinities', at least infinities that are bigger in a countable way.
I truly believed base -1+i was the future at one point. Since used only zero and one I encoded it with hex digits. Had a hard time getting division to work. I eventually left that project for geometric algebra. It was a weird experience graphing things in 4 dimensions.
Balanced ternary is gorgeous if you get s chance to see it.
Well yeah there's more, there's always more. 28.
We probably would not discover anything by looking at other bases simply because when talking about properties of numbers in mathematics, we don't use any bases at all - just the abstract operations and decompositions that exist. For example, primes are primes in any base.
Yes- what a number looks like in a specific base is not often very helpful in mathematical settings. I can't think of many examples, but one favorite is that the Cantor Set is easy to describe in terms of base-3 numbers, and very complicated to describe in base-10. But 99% of the time it doesn't matter.
@@ChrisStaecker Bases of numbers are useful of course, but they are something more like coordinates - in fact you can think of the carrying operation by constructing an algebra of polynomials with natural number coefficients. The base ends up being something you stick in the variable to evaluate the number.
It’s the Davie 504 reference for me lol
Never heard of him! I guess it's a pretty obvious gag though- I've heard something similar from Adam Neely.
should have called it the hex deck
Computers don't HAVE to be in binary. Humans have already made some non-binary computers. But they're the simplest to make, so they were the first; and their complexity (or lack thereof) makes them incredibly reliable, so momentum has just kind of carried us through to today.
There are people who use a base 12 system
Yeah this guy 2:24
Now I want to make some base 12 Rugen’s Bones.
As someone working construction in the US of A, I love 12! So divisible!
niche is pronounced like "neesh"
I wonder what that thing is worth today. It's in Fucking mint condition.
$25
if there was an allignment chart of different number bases, Base π would be chaotic evil.
And then you got the madlad Babylonians with their Base 60 system.
"Most human beings throughout history have thought of numbers as organized into groupings of five and ten" is probably the funniest thing you've said in all of your vids combined. Then proceeding to give a highly selective set of examples is the funniest thing you've presented (at least funnier than the monkey).
And regarding your comment about number of fingers for counting, I recently started practicing counting in base 12 with my fingers and it's easier than expected even though not common where I was born and wasted. Different cultures have different bases they count with their hands.
Like you said in the finale the only reason we're used to base 10 is for cultural reasons, has nothing to do with human affinity to counting in groups of 5 and 10 because there are stronger examples for 3 12 and 16. We use base 10 because it's easier to machine-calculate positionally, and positional systems only started being useful when 0 was invented. base 10 is cheaper on an abacus than base 12, and that's why the French Revolution happened.
Do you disagree that most humans have used base 10? This certainly must be true if we count person-by-person since modern humans are far more populous than in the past. But even if not, is there any real argument? Obviously I chose my examples to support my point, but which am I missing?
I'm aware of historic sexagesimal systems (which are still based on 10s, together with 6s), and some duodecimal systems for counting certain things. But base-10 counting has arisen independently in world cultures many times, right? (I'm not an anthropologist) I can't possibly believe that the choice of base 10 is to facilitate machine calculation- this is a concern that developed long after cultures had already decided how to group their numbers.
@@ChrisStaecker Nevermind that it's futile to come up with based statistics of how common different systems were throughout pre-modern history and decide which system was "the most" used, and for many reasons. (which alone makes the initial claim unfoundable)
If we go by historical evidences alone, disregard lesser-known systems, and only include those that are *at least somewhat* used to this day without special aids:
Binary is used all the time, certainly if you ever ordered in pints.
Base-3 is one of the more natural and intuitive groupings a human can perceive, also a tsp is a third of a tbsp.
Base-6 whenever you roll a dice (can't say "two dies" without base-6)
Base-7 for weeks (can't say "couple of weeks" without base-7)
Base-12 for time and calendar ("couple of months", AM/PM time system)
Base-24 again for time ("couple of days")
Base-60 yet again for time ("half a minute" is base-60)
Base-360 for degrees of angles (you could argue it's also pi but i think it's fair to say most often when someone thinks of quarter of a circle, or a right angle, they're thinking 90 degrees rather than half-pi but speaking of which base-pi is not uncommon for this use)
I completely disagree with your claim that base-60 and base-100 and other bases which are multiples of 10 are duplicates of a small base by x10ⁿ. Regardless of notation, which as you said in the finale of the video is mere representation of an idea, the different base systems are used for their meaning whereby the base-n stands for a single whole (or, drumroll, "100 per *cent*").
Anything less than that whole is part of it - a tenth, a quarter, a third. While any multiples of that whole are counted as multiple wholes.
You'd have to do a whole lot of convincing to probably anyone who uses seconds-minutes-hours system that a minute *means* "ten six seconds" or "six ten seconds", rather than it means "sixty seconds". If you said "half a minute is three ten seconds" it might be perceived as funny in some circles, but only because it's contrary to the common meaning of a minute. Therefore the 60 in a minute is the whole, and that's why it's base-60. In the same way "two minutes" don't mean "twelve ten seconds", rather they're "two sixty seconds".
So I didn't bother you with a list that includes base-100, 1000, but to claim these bases are irrelevant would be very contrary to the way numbers are used by people (and mathematicians) for millennias.
And if I already mentioned base-1000, then you can probably guess what I think about base-1024.
And yes I know you can represent all of these with "x10ⁿ" but like you said numbers are an idea beyond the representation of them, reducing every multiple of 5 to base-10 is falling for the exact fallacy you warned against.
@@pubcollize I think I disagree, though what you say is interesting. I think I believe in true duodecimal systems, but I can't really imagine sexigesimal as a non-subdivided true base-60 system. The Babylonian symbols for writing sexagesimal numbers are clearly based on 6x10. It seems to me that a pure base-60 system with 60 different symbols per digit would be totally unusable for people. Our use of base 60 in time and angle measurements is only workable because we write those numbers in a more understandable base. (Though I agree that people don't really think of 60 minutes as composed of 6 groupings of 10 minutes.)
Anyway thanks for the comments-
@@ChrisStaecker Some bases divide better than others. For example, 1/7 in base 10 is a pain. It has a nice cyclical pattern, but it's incredibly cumbersome to use. Some bases are better at this than others. For example, even though in base 16 there are fewer nonrecurring reciprocals of integers, their recurring nature is easier to use than that of 1/7 in base 10. (All the reciprocals of the integers below 20 have single or doubly recurring hexadecimal expansions, which is really good.) The best base that I have seen for this effect is base 6. For all the reciprocals of integers below 20, 9 of them terminate in base 6. 8 of them have easy to use recurring expansions(where it recurs with a "width" of 2 or fewer digits). The only terrible ones are 13, 17, and 19. So, I would say that in all honesty, from a pure ease of use standpoint, base 6 makes more sense. Oddly enough, very few cultures use base 6 though.
@@ChrisStaecker If we only count base systems based on the number of unique symbols they have then the roman nonpositional system would be base-7, and most duodecimal uses we're familiar with would be base-10. With all due respect it doesn't make much sense. Sorry if I'm misrepresenting what you meant.
Either way it again falls to the same problem of giving too much weight to the glyphs than to the ideas.
My first exposure to different bases was courtesy of Schoolhouse Rock. Blew my biscuiting mind.
czcams.com/video/pqGyUvZP0Zg/video.html
Also, thank you for the kind letter and nomograph!
That's a pretty trippy animation!
Yes! Me too! I loved that one. It was the trippiest psychedelic my 5 yr old brain could have. So cool.