Binary number system from 1617 (Napier's Location Numerals) Review / HowTo

Oh, he was so close. He figured out the main idea, but didn't know how good it was. I guess he was amazed by how well the subtraction worked. No borrowing, just a couple of prep steps on both numbers and the actual operation is really simple.

[@jetison333]

extending it is basically just preborrowing all of the numbers though.

[@awesoman12]

The ampersand was actually part of the alphabet after Z, so I guess he used it as it was just another letter

[@thinkbolt]

500 years from now, all mathematicians will be shouting "CURTA!"

[@michaelbauers8800]

Maybe Curta! Can become a meme like Sparta! But a better meme

[@OldManBOMBIN]

Professor Chris! You disappeared from my subscriptions and feed sometime around the Cuuurrta series! I had to go out of my way to track you back down when I realized I hadn't seen any number gadgets recently. I've missed your weird, polymonotone voice and interesting inflections. I mean that as a compliment, though I know it doesn't read like one.

[@michaelwoodhams7866]

This reminds me in some ways of fibinary notation. In binary, your digits represent the numbers 1, 2, 4, 8, 16 etc. In fibinary, they represent 1, 2, 3, 5, 8, 13, 21 etc. (Fibonacci). Like Napier's notation, numbers have non-unique representations, and to do addition or subtraction you need intermediate steps of expansion and contraction. The rule in fibinary is that anywhere in the binary sequence you can substitute 011 for 100 or vice versa. It isn't too hard to figure out how to add and subtract. Essentially you write the numbers one above the other, and for adding you are allowed to move a 1 in one number to the same location in the other number were there is a 0. For subtraction, you are allowed to substitute 1 to 0 in both numbers where the 1s occur in the same location. Then use the 011100 substitution as needed to shift things around until you are able to completely zero one of the numbers. I know of no sensible multiplication algorithm, and expect none exists. I invented fibinary notation (including the name) and then though to myself 'this is obvious enough that I bet I'm not the first to have come up with it', and web search revealed that indeed I was not (including the name). So far as I know, fibinary notation is purely recreational mathematics. I know of no practical application. Edit: Just thinking on it now, I do have a multiplication algorithm, but it is a real pain. If I want A x B, where A and B are in fibinary, I calculate A+A = 2A using the addition algorithm, then 2A+A = 3A, then 2A+3A=5A etc, keeping all my intermediate results, Then when I have as many of those results as B has digits, I select the intermediate results corresponding to the 1s in the fibinary representation of B, and use the addition algorithm to add them all together.

[@p-j-y-d]

7:03 "What we have here really is a great idea being weighed down by some bad ideas." And this is a great video being elevated by some great insights. Thank you.

[@Jimorian]

Somewhere there's an alien civilization using this number system.

[@ChrisStaecker]

hope they never have to count past 2^46

[@DanDart]

It's okay, they just have a lot of letters in their alphabet, and never have used numbers nearly that large.

[@thezipcreator]

@@ChrisStaecker what if they used a logographic writing system (like chinese)? that way they already have thousands of symbols

[@michaelbauers8800]

Perhaps those space lizards some people talk about use this.

[@Meophist]

I do like the compactness of the resultant numbers, you can represent very large numbers in a very small amount of space using this. Not to mention, this allows you to turn words into numbers and vice versa. I think the simplified version in alphabetical order can be considered the canonical representation for a number, similar to 10 is for ten, even though you can write 010, 10.0, or 1000%, and they'd all mean "ten".

[@stevecummins324]

Yes re Canonical form aka normal form And it can be a very useful technique. There are other ways such can be used but Eg 1 is also sqrt[2]/sqrt[2] Great for multiplying by 1 thus not changing value. And shifting awkward square roots out of denominator of a fraction.

[@tqnism]

It clicks better if you compare it with greek or hebrew or russian numeral systems. Those are strange animals for us, but were used by many people for long time.

[@ChrisStaecker]

I think that roman numerals is a more likely inspiration, which uses a different symbol for powers of 10 (plus the 5s, but whatever). Roman numerals also include subtractions when you write them on the left, but if you're going to use base 2 this doesn't apply anyway. So Napier's concept is the simplest and most natural base 2 version of roman numerals that you could do. The basic idea of using letter symbols as numerals is common to all of these, but it's maybe the least important thing about Napier's system.

[@Vallee152]

1:08 the ampersand was originally considered to be a letter. It was at the end of the alphabet. When singing the alphabet, instead of saying "x, y, and z" at the end, they said "x, y, z, and per and." It was originally just called "and," but then "and per and" was misinterpreted as "ampersand."

[@Eidolon2003]

and per se and

[@p-j-y-d]

Curiously, although the concept of base 2 did not exist in ancient Egypt, their method of multiplication is essentially the same as our method of long multiplication after converting the multiplier and the multiplicand to binary.

[@GroundThing]

The ampersand was used because up until the 1800s, the ampersand was considered the 27th (or 26th without J) letter. I don't know if it is true that this is the case, but the timelines match up, so my belief is that what made it fall out of fashion as a letter was the alphabet song, which albeit does actually attempt to include the ampersand ([...] X, Y -and- Z), but presumably its swapped position with "Z" and the fact that "and" is also just a natural word to go there made people learning the alphabet only think of the 26 letters.

[@mildlycornfield]

If I was an ARG creator, I'd be perking up at this so much

[@noucuratlex]

This is a great video. I learned about numbers, I learned history of math, I got some good life advice. A great video.

[@shoofle]

I kinda love the subtraction tho. He went off with that

[@stevecummins324]

It can be argued that standard maths has plural representation. The most usual representation gets called the normal form Sometimes it's very useful to know a values alternative form. Eg when multiplying both sides of an equation by alternative forms of "1". 1 is the normal form of (2^½/2^½), and more generally (a/a) Can even use alternative forms of 1 to perform unit conversion. Just treat the units akin to undeclared algebraic variables. Multiply by a suitable form of "1" and unwanted units are cancelled out. 1[mile]=1.6[km] -->1=1.6[km]/[mile] 5[mile]×1 ---> 5[mile]×1.6[km]/[mile] ---> 5×1.6 [km]

[@retrogiftsuk4812]

Great video as always. A couple of other problems with the system that I can see. 1) ambiguity. With so many different characters it will be easy to misread them, particularly when some Greek letters are similar to the regular alphabet and nearly everything was handwritten back then. 2) comparison. Because there's multiple ways of writing the same number it's not obvious which is larger, or any sense of scale. If I sell apples and have "accdf" in stock and an order for "bbbeee" comes in, do I have enough? And if I need "deefhh" and they come in packs of "ceggh" roughly how many will I need? Both of these problems are easy in base10 or binary because the length (and leading digit in base10) tell you the relative size of the numbers which is often all you need to know (or is at least a good starting point, and is useful for checking your answer)

[@Terigena]

For people still using Roman counting boards, this must have seemed a lot easier than those heretical Mohammadan numbers, even easier than Roman numerals. And compared to Roman numerals it allows you deal with much larger numbers. Common operations such as doubling or halving are also very easy. And if you can't remember the order of all the letters, the board functions as a cheat sheet. It is easy to critise Napier for not inventing binary as we know it, but he may have been writing for a different audience. I bet he also realised that this would provide a cool way to encode text. You can use it as a hash function for words.

[@argonwheatbelly637]

Exchequer boards are fun. We used something similar to it in kindergarten. Now, try using calculi on a Salamis tablet! Way cool.

[@pierQRzt180]

The last sentence is golden

[@herzogsbuick]

i love reviews of something i've never learned before! :-D

[@joshuasims5421]

His system looks like a clever iteration on Greek and Hebrew style numerals. Both systems assign letters to numbers 1-9, 10-90 and then 100-900, and then add together values. So alpha was one, beta was two, etc. I suspect Napier had this in mind when he created his system, but realized the power of binary letter values.

[@musicalBurr]

Wonderful video! Thanks as always.

[@krishnachoubey8648]

7:09 He actually did have a precedent. Pingala(2nd-3rd BCE) of India described the permutations of _Laghu(Short)_ and _Guru(Long)_ Syllables in music in a binary notation very similar to the modern one and gave rules to generate them and a formula to find the total permutations of a sequence with length n as [1 + (1+2+4+ . . . + 2^n)]=2^(n+1) (Please read the wiki; I can't type further :| )

[@ChrisStaecker]

Yes there were a few historical precedents, but as far as I could tell none of them were known to Napier. Thanks for this reference-

[@felixecho]

What does a binary abacus look like?

[@txikitofandango]

That's not a real napier binary counter location numeral that's paper and markers you got at office depot

[@maxmn5821]

That‘s just a neat disguise. Look at the match box. There is a time machine in the attic.

[@markhughes7927]

9:05 Napier’s system seems - from this showing - to relate to the principles of biblical gematria in that it may combine identical values for additions with anagrams some of which are loaded with significant meaning.

[@Rainbow_Oracle]

Yeah modern binary is definitely a lot more elegant and refined. But the thing about using letters as stand-in for numeric values means that you can encode information into text. Vedic scholars are the ones that are most notable for doing such things, using the values to code mathematical information -- or other information -- into memorable sentences, phrases, and aphorisms, but Greek and Hebrew scholars were up to similar tricks hiding information in text. That's why textual analysis of the bible and the torah were such big things, because there was the superficial information and the deeper information. Same thing with even the texts of ancient Greek philosophers. That's where most of the allure of using letters for numbers comes from, even if it's initially more limiting for pure maths purposes.

[@aotmr1604]

This reminds me of an obscure algebra called "evolution algebras"; it isn't one, but we can treat a...f are basis elements e_1...e_6 in a commutative algebra, and there is a bijection between e_1+e_1 and e_2, e_2+e_2 and e_3, etc. so the canonical form of a number has each basis element represented no more than once. Now, what happens if this algebra is noncommutative, we disallow arbitrary summand swaps, so that e_1+e_2 represents the same number e_2+e_1? If we add these distinct representations of three together we get e_1+e_2+e_2+e_1 = e_1+e_3+e_1; which represents six; in a commutative algebra this would reduce to e_2+e_3.

[@JefiKnight]

It I grew up from childhood with this as my number system, it might just be super awesome. Maybe. I wonder if you would add and subtract just by visualizing.

[@TXDude]

Hmmm.... One supposes that if Napier had a modern roll of adding machine (cash register) paper, he could have limited his scope of letters to a single a, then repeat as necessary!

[@JR13751]

If we were using base 2/8/16 instead of decimal, Napier's board would be good idea for multiplication/division.

[@argonwheatbelly637]

What about a video on the Exchequer Table or the Tablet of Salamis?

[@Katachresis2000]

I'm surprised Napier didn't use lower case and upper case letters when going to larger numbers (rather than to the Greek alphabet).

[@argonwheatbelly637]

Location Arithmetic was quite lovely when it hit in the early 17th century. Only 407 years ago. +/-

[@psiryan]

Napier's system seems like it might have excellent application as a tool for cryptography.

[@JaePlay]

how to use 1s and 0s but with extra steps.

[@leduyquang753]

The sound of marker against paper sure isn't very pleasant.

[@ChrisStaecker]

Really? I considered cutting it out (voice was all recorded separately) but I liked having it in there.

[@yogimew]

@@ChrisStaecker You could have written them on a whiteboard.

[@BrianTRice77]

Yeah I would appreciate at least a remix of the audio to subdue that marker sound on paper. I can’t think while listening to it!

[@mesientogut6701]

Fuck this guy, the sounds of reality existing are always pleasant

[@_D_P_]

I quite like it. Its like rhythmic white noise.

[@billsimons4113]

I love your progress in motion videos, but I can't be the only one squirming under the marker swishing sounds? Like nails on a chalkboard, but not quite as bad.

[@hendrikd2113]

Will there be a video about the 'promptuary'? It would be a good click-bait opportunity! Napier invented logs, binary counting, and prompting, kinda like the technological progression from slide rulers to electronic calculators to using ChatGPT.

[@ChrisStaecker]

maybe some day- i mentioned it in the previous video

[@maxmn5821]

There is an interesting video from Welch Labs about almost-logs by a different inventor. A table of powers of 1.0001, basically. We do all inventors more justice when we try to be specific about what step they accomplished - and from where.

[@jeremydavis3631]

Hey, the sum at the end of the video is wrong: you used the letter j! 😛

[@Vaaaaadim]

good catch

[@ChrisStaecker]

Napiers mind would be blown.

[@fxkr8091]

badge

[@ar_xiv]

bitwise enums! kinda. you usually aren't adding them but OR-ing them

[@DanDart]

That's for nominal numbers but you can't do the extension thing nor the simplification step, but sure, I see where you're coming from.

[@ar_xiv]

@@DanDart yeah it's a loose connection. also they become meaningless above 2x the highest defined entry (which is kinda like the needing more symbols thing....)

[@jamiehardt3061]

What he needed to do was each letter was a prime number and you multiply them, then you can get up to some real nonsense.

[@MattMcIrvin]

It's kind of like ancient Greek numerals if the Greeks only had two fingers.

[@seabiscuitthechallenger6899]

👍

[@michaelbauers8800]

So basically, non positional numbering system like Roman numerals, but with base 2. Like if the Romans had only had two fingers total :)

[@pierQRzt180]

There are over 9000 ways to write the number 17. I see what you did there.

[@ChrisStaecker]

Did you see this? oeis.org/A023359

[@DanDart]

I wrote a Haskell program to figure out the value of words which are in alphabetical order. Mwahaha.

[@michaelbauers8800]

One day, maybe when I retire, I will sit down and learn Haskell. Friend of mine really likes it.

[@StrayCatHomestead]

Who's this? I got a new phone

[@agranero6]

Why the hell you have a Russian match box? I happy you said " *whole* numbers have unique representations", as real numbers don't.

[@maxmn5821]

It could be a Soviet one since the череповецкая спичечная фабрика started production in 1973. I‘m not sure if the picture on the box ever changed and can give us a clue. Those boxes were great to store small electronic components (some of which were notoriously difficult to get).

[@agranero6]

@@maxmn5821 When I was a boy I stored my electronic components on match boxes too.

[@ChrisStaecker]

@maxmn5821 I got it new in Kazakhstan in 2002, so probably not Soviet era, but maybe.

[@Michael-Sanford]

Most new ideas look stupit until you prove it.

[@StevenSiew2]

It is just ducking binary numbers using algebra with single character variables!!!!

[@joehopfield]

Binary's place in the development of digital tech is overrated. There have been other schemes... (mechanical and electronic). Compiler and OS classes at UCLA in the late 70s revolved around Knuth's MIX, a hybrid binary-decimal "machine" that prevented punning.

[@michaelbauers8800]

How can you say binary is overrated when it's used in probably billions of CPUs? My question would be more like "Wasn't it obvious?". Given that quite long ago, someone like Zuse was building electromechanical computers with relays.

[@Pallethands]

Make no BONES about it, it's not great

[@michaelbauers8800]

I see what you did there

[@hughobyrne2588]

Unique representation? Really? Didn't you know, 1 = 0.11111...?

[@ChrisStaecker]

I made sure to say "whole numbers"

[@Keldor314]

It's still a unique representation for any number with a finite number of non-zero digits. And location numerals only work these numbers anyway, since otherwise you would need an infinite alphabet, which would be inconvenient to teach at schools.

[@stevecummins324]

​@@Keldor314 " sqrt[2] over sqrt[2] " has exact same value as 1.

[@timetraveller6643]

TOPIC SUGGESTION: If you like Napier's number system, You'll LOVE Zeckendorf Fibonacci representation. 144 89 55 34 21 13 8 5 3 2 1 --- 101000101 = 80

[@DanDart]

I thought it was interesting to find out which numbers divided their Zeckendorffian format and which did not.