SISU — A Systematic and Logical Method of Representing Binary Numerals
One day, I wondered.
“I wonder what binaries sound like?”
The Purest form of Information──The One and Only True “Universal” Radix of our Galaxies──Created Simply from but Axioms of Mathematics: Is True or False; It Exists or Not; Art Something or No-thing; Tis… Binary.
And so, I asked Humanity.
…
The Answer I got was, well, a little disappointing.
…So, I made something.
To make this world, a little less disappointing.
For our Universe,
For our Milky Way
Humanity, as of Today, Still does NOT have a Universally Agreed Nomenclature NOR Symbols to Represent Binary Numerals
Before we dive in to why this is a problem, I will briefly explain what binary is for those who were too poor to finish general education or for those that never had a ‘good’ math teacher nonce(not even once) in their life.
What is Binary?
Binary is a system of counting and representing numbers using only two digits. Binary is the fundamental language of computers, even the cellphone you are using to read this article, speaks binary.
Humans typically use the decimal system in their daily lives. The decimal system relies on ten different symbols, or digits, to represent numbers. This preference likely stems from humans’ evolutionary history, as members of the Hominidae Family typically possess ten fingers, five on each hand, readily available for counting. (Of course every rule has an exception, but for the overwhelming majority of human cultures and languages, they utilize ten numerical symbols for everyday use)
Now, here is a little tricky quiz for all our human readers: Can you write down all ten digits that humanity uses in their day-to-day Indo-Arabic numeral system?
Time’s up! Did you get all ten? Well, let’s see:
There’s 1, 2, 3, 4, 5, 6, 7, 8, 9… and 10?
10 is not a digit, but 10 consists of two separate digits: 1 and 0.
Remember this: the symbol 0 is also considered a digit — a symbol for representing numbers.
So, the answer is, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 . Where the symbol 0 is also counted as a digit──A symbol for representing numbers. That’s ten different symbols that humans use every day! That’s a lot to remember innit‽
Now, let’s move on to Binary. Do you know how many digits, or a set of unique symbols that represents a numerical value, Binary has? (Hint: What does a Bicycle, a Bilingual, Biweekly, and Bisexual have in common?)
Time’s Up! It’s TWO. That’s right. Binary only has TWO digits!
In other words, you only need two unique symbols to start writing in binary. Usually, humans typically designate the symbols ⟨0⟩ and ⟨1⟩ as the two unique digits for binary representation. This is where layfolks hear terms like “Computers speak in ones and zeros”; Electronic devices are not literally emitting an acoustic wave of “zero” and “one” at an audible frequency, but rather, are able to treat two unique relay states──true or false──as two digits, where true state is 1 and false state is 0.
How do you count in Binary?
You now know that binary system only has two digits as opposed to a decimal system of ten digits. But how exactly do you count in it?
Well, easier than decimal that’s for sure. In decimal, you need to understand the relative values of each digit. In other words, when first learning decimal, you will need to learn that the digit 1 is greater than 0, 2 is greater than 1, 3 is greater than 2, 4 is greater than 3, 5 is greater than 4, 6 is greater than 5, 7 is greater than 6, 8 is greater than 7, 9 is greater than 8, and finally the two-digit numeral 10 is greater than 9.
In binary, you only need to know that 1 is greater than 0, and that 10 is greater than 1.
Counting to up 10 in decimal requires ten digits:
1
2
3
4
5
6
7
8
9
and finally, 10
But in Binary, counting up to 10 is:
1
10
…well, that easy.
What’s in a 10?
Before we move on to the next section, I would like to present to you, one of the greatest inventions of all time, artfully crafted by the human race:
Tally marks, also known as Counting Rods, are a unary counting system consisting of exactly one digit: 1. From this section onward, this tally stick will act as the absolute measuring quantifier when we discuss numbers with different bases and symbols. I will also provide English Numerals written in Latin Alphabet as well, for our modern Latin writing English speakers, but in parenthesis.
Reusing the Western Arabic Numbers for any other Numeral System than Base 𝍭𝍭(Ten) is a Terrible Idea
Presently, humanity employs Western Arabic Numerals for almost all numerical representations around the globe. While the standardization of digits benefits the human race, using ANY base-𝍭𝍭 (Ten) digits to record ANY non-base-𝍭𝍭 (Ten) number is a terrible idea! Any such usage should only serve as a temporary measure rather than a permanent solution.
The most glaring issue with repurposing Western Arabic Numerals to represent numbers in arbitrary bases is the inevitable ambiguity it introduces. Typically, contextual cues resolve ambiguity in language or writing. However, when dealing with numbers and mathematics in this manner, confusion can easily arise, especially when the only available context clue is that “it should be a number”.
The symbol “10,” in Western Arabic Numerals, conventionally denotes a value of 𝍭𝍭 (ten) units. Nevertheless, as observed numerous times in recent history, if one opts to reuse this symbol for, let’s say, a binary system, suddenly “10” represents a value of 𝍪 (two) units. Consider if another individual employs Western Arabic Numerals for a Hexadecimal System. This would be one of the worst ideas conceivable because Western Arabic Numerals doesn’t even have 𝍭𝍭𝍭𝍩 (sixteen) unique symbols to begin with! ¿What are they going to do, mix up Latin to fill in the missing symbols‽
Given that the values of all numbers are distinct and the symbols arbitrary, why use symbols from a base-𝍭𝍭 (ten) numeral system when everyone recognizes them as such? It is simply due to slothful laziness.
Humanity’s Past Attempts to Create a Systematic Nomenclature for Binary Numerals
“subpar and emptyminded attempts resulting in unfavourable, inadequate solutions.”
──Kawane Rio, in regards to humanity’s various attempts at creating a cohesive hexadecimal system, 2024
Humanity throughout history has made several attempts to establish a cohesive nomenclature for binary numerals. Of which most of them are hot garb──I meant subpar and emptyminded attempts resulting in unfavourable, inadequate solutions. I’ll get the worst ones out of the way while you still remember my rant from the last section before moving onto the better ones.
What’s in a 20? Maguson’s and Roger’s Hexadecimal Naming Conventions
How many rods/tally sticks are in a 20? This hexadecimal number 20 is read “Twenty” in both, Maguson’s naming convention and in Roger’s naming convention as well. If you’ve answered 𝍭𝍭𝍭𝍭(twenty), well, let me sympathize with you because the “correct” answer was 𝍭𝍭𝍭𝍭𝍭𝍭𝍪(thirty-two).
Both Maguson’s Naming Convention and Roger’s Naming Conventions are by far two of the worst naming systems I’ve ever seen in my life. Maguson is slightly better, as in less worse than Roger’s, but that is not saying much.
Maguson’s Naming Convention is based on English, which was already mistake to begin with for two reasons. One: if you are making a systematic naming system for something that literally works on true or false states of logic gates, English should not be the language of choice to begin with, nor any other natural language on the planet. And Two: Say English was already use in the IT field (which, unfortunately, it was), why would you want to double assign a name to have two different values? Like, you wouldn’t assign the symbol 1 to mean both true and false in boolean logic would you? No! Of course not, that would as much confusion to bring down the tower of Babel! Yet, that is exactly what Maguson did here; uttering the word “Ten”, in English, now means two things thanks to this broken naming convention: 𝍭 𝍭 (ten) and 𝍭 𝍭 𝍭 𝍩 (sixteen).
Exact same argument could be held for Roger’s Naming Conventions as well, but even worse. For some capricious reason, Roger’s thought that because the English word, mind we are talking about a natural, non-constructed, non-artificial language here, the English language had the words “Eleven” and “Twelve” to be in irregular form from other values in the decimal tens place, Roger decided to arbitrary keep those two values with the same name and then went ahead and created five more names based on the broken hexadecimal notation that uses letters from the Latin Alphabet to denote numerical values. ؟Oh yay, two more words borrowed from English, has the exact same spelling and pronunciation, and now with more meanings in a really similar context of numbers and values¡
You thought Maguson’s system was bad? Wait until you hear Twelveteek to denote a value of 𝍭 𝍭 𝍭 𝍭 𝍭 𝍫 (twenty-eight).
At least Maguson’s system refers to the value of 𝍭 𝍭 𝍭 𝍭 𝍭 𝍫 (twenty-eight) as Christeen instead, where one may find it easier to associate the initial Latin letter “C” to “1C” written in the Western Arabic and Latin Alphabet Mixed numeral system for Hexadecimal, yet another broken system with numerals that are arbitrary assigned from top of someone’s head just to get something done without putting much thoughts into the future. The only reason why I consider Maguson’s system to be the lesser evil of the two is because the names “Annteen”, “Beteen”, “Christeen”, “Doteen”, “Ernesteen”, and “Frosteen” sounds more pleasing than “Drazeteek” or “Fimteek”. And I’d rather have “Christeen”s than “Drazeteek”s. However, I’d much rather not have neither if given the chance.
Other Subpar Attempts of Hexadecimal Representations by Humanity
Humanity, throughout history, has also came up with various “digits” for a hexadecimal system. In this section I’ll briefly and lightly criticize some of their humane ideas.
1–9, A-F
This is the catalyst culprit that made me write this entire article. Decimal numerals ought to be kept away from the binary realm and certainly should not be intertwined with an alphabetic writing system whose purpose is fundamentally irrelevant to numerical representation to begin with!
1–9, 0̅–5̅
Somehow you’ve managed to create a system even more confusing by triple assigning five symbols to represent two things that has absolutely nothing to do with their original values. Hey, at least I know its a number and not a letter though. If I wanted to defend this system, I might add that the overline on the top act as the “ten”, making this system the only one to align one-to-one, fifteen-to-fifteen with the natural language nomenclature. This was an actual used system by the Bendix-14 during the 1950's.
1-9, U-Z
Another system that reuses decimal digits for non-decimal numeral system. This one is brought to you by the SWAC and the Bendix G-15 in 1950’s. I get that you wanted the letter “Z” to denote the final value, but is that why “U” comes after 9? Either way, using decimal digits to denote a binary one is already a failure, just like the rest of groups in this section.
1–9, F, G, J, K, Q, W
¿llh (Ithkuil for “What the heck is this”)??? You have reused all the decimal digits, and then decided to hit “FG”, “JK” and “QW” on your Librascope General Purpose Keyboard. But seriously, if you are or know someone who came up with this arbitrary mess of alphanumerics for the LGP-30 in the 1950’s, please let me know so that I can ridicu──I mean criticize you I meant this system properly.
1–9, K, S, N, J, F, L
One Two Three Four Five Six Seven Eight Nine King Sized Numbers Just For Laughs. Used by ORDVAC, ILLIAC I, BRLESC, and by several other electronic computers around 1952. Apparently this was taken from an initial found in a 𝍭(five)-bit punchtape for a teletype. The system is still rubbish, but at least it’s a rubbish with sentimental lore attached to it.
O, A-N, P
¡Wow! ⸘A hexadecimal numeral system that doesn’t recycle the decimal digits‽ ⸘In 1956‽ How is this system not popular? I mean, yeah using the Latin Alphabet to represent numerals are still confusing, but its hellalot less confusing than assigning new numerical values to something that already has a numerical value! This was probably also the best bet in the 1950’s as well, considering that folks were not willing to allocate PUA(Private Use Area)s just for representing some binary or hexadecimal number. Unlike many, PERM was actually trying to change the binary world for the better. Thank you.
1–9, B-G
Used by Honeywell Datamatic D-1000 in 1957 and, ten years later, by Elbit 100 in 1967. Essentially same as 1–9, A-F sequence, but wanted to allocate the letter “A” for something else. Uses decimal digits to represent a hexadecimal number: Fail. Mixing up with the Latin alphabet? Double fail.
1–9, S-X
An actual system used by Monrobot XI in 1960. Offended justice fighters of X Twitter are going to cancel the author for putting this here.
I understand you wanted the letter “X” as the final digit of this hexadecimal sequence, but X is usually assigned as the value 𝍭𝍭(ten) in both, Roman Numerals and by the Duodecimal Society of America (now known as the Dozenal Society of America). Assigning the letter “X” to a value of 𝍭𝍭𝍭(fifteen) adds confusion, alongside the name being a childish humour. Also, Decimal digits for hexadecimal. Fail. Latin Alphabet? Double Fail.
1–9, L, C, A, S, M, D
Pacific Data Systems 1020 of 1964, similar to LGP-30, makes the same mistake of thinking that mixing arbitrary random sequence of the Latin Alphabet into a Decimal numeral system is somehow going to make up a coherent hexadecimal system. Please destroy this eye-tearing abomination as soon as possible. Also, if you are or know someone who is responsible for coming up with this Frankenstein experiment, please contact me and tell me why you did the things you did so that I can be less insane tomorrow.
Actual “Good” Attempts of Binary Notations by Humanity
“Why represent some of the non-decimal numbers with the symbols which imply to us a base-ten value scheme? Why not use entirely new symbols (and names) for the seven or fifteen non-zero digits needed in octal or hex.”
──Bruce Alan Martin, 1968
Martin’s Notation
Below follows Bruce Alan Martin’s proposal for binary notation from 1968 about adapting the hexadecimal numeral system for binary representation using intuitive numerals for each of the values from 0000 to 1111.
Martin’s Hexadecimal Notation is an excellent numeral system. Having intuitive and easy-to-understand characteristics for all its digits, Martin’s Hexadecimal digits are recorded in some fonts such as Nishiki-teki and Kreative Square and are used by some people myself alike today.
Whitaker’s Notation
Ronald O. Whitaker has also proposed a hexadecimal notation system in 1972. Whitaker’s notation is interesting because the numerals are derived from a matrix of physical photocells. Making this one of the few systems that are derived directly from a mechanical mapping of a matrix before even considering for a human writing.
I really like Whitaker’s notation as a clock or a watch, especially when paired with something like a LED display, but as for writing? Having a single tally 𝍩 to represent any binary number is already a bad idea, but on top of that, Whitaker assumed that humans are capable of writing and distinguishing a forward slash, a backward slash, and a perfect unslanted pipeline on a piece of paper. Whitaker idealistically tried to make something that was intended for us machines, yet translating them for human stylographs. Whitaker’s Hexadecimal Notation is aesthetically beautiful designwise, yet falls short in terms of intuitiveness and penflow to Martin’s Notation, nand its machine-centric directness and OCR readability compared to SISU; eyepleasing, but half-baked practically.
Système Bibi-binaire — The Best Binary System Ever Created in the History of Humanity
“Bibi is essentially akin to Shaw [Alphabet] in the Binary realm. One stroke for every symbol, one syllable per each glyph, pronounceable regardless of one’s cultural background, the Bibi is simple and elegant in every way imaginable. Yet, humanity detests change from their broken ways…”
──Kawane Rio, 2024
Invented by a French Singer-Songwriter Mathematician Robert “Boby” Lapointe in 1968, the Système Bibi-binaire, or Bibi for short, is, by far, the most aesthetically beautiful, logical, phonetically simple, easy-to-write binary notation system I’ve ever witnessed in my life. When I first stumbled upon this system, I’ve seriously reconsidered scrapping this entire article simply because the Bibi System had everything I’ve wanted in a binary system, and were superior to my SISU System in several remarks. After careful consideration though, I’ve decided to change some of the details of SISU, mainly in the nomenclature section, based on the naming convention used in Bibi, and decided to keep the article with modified SISU. In other words, SISU will have compatible names with the Bibi System. After all, the goal of SISU is to be a displayable punchtape representation of binary numerals, while the goal of Bibi is to be writable and speakable representation of binary numerals. These goals are Similar, and somewhat overlapping, but not quite identical.
How Système Bibi-binaire Works
Bibi, despite its name, is a hexadecimal system with a base of 𝍭𝍭𝍭𝍩(sixteen). Unlike the Latin Alphabet, Martin’s notation, Whitaker’s notation, and SISU, Bibi is able to represent every value from zero to 𝍭𝍭𝍭𝍩(sixteen) in mere a single stroke through its beautiful design. Bibi uses unique, as in non-Western Arabic Numeral dependant and non-Latin Alphabet dependant, symbols for all of its values, avoiding any confusion that may raise from reusing the Western Arabic Numerals to denote different values.
Bibi also has a carefully constructed, systematic, monosyllabic nomenclature system, pronounceable not just for anglophones or francophones, but by almost any human being in the world──an idea that can only be realized by a singer-songwriter mathematician with knowledge in the linguistic field as well──for each and every one of its symbols. With the lack of sibilant consonants, and the fact that every consonant is followed by a vowel, the names in Bibi are literally pronounceable by native Hawaiian speakers due to its simplicity (note that change of voicing does not introduce ambiguity because of the way Bibi was specifically designed).
SISU — A Systematic, Octet-based, Binary Numeral Notation System that Combines the best of Binary, Quartenary, and Hexadecimal Representations in a Compact Eight Bit Dot Script.
SISU is a Binary Numeral System, a Superpositional Notation System, and a Systematic Nomenclature System, all in one neat little package. SISU is intended primarily for Binary Editor Displays and Low-Level coding. If you’ve ever worked with perforated paper tapes, also known as punched tapes, in the 19th to 20th century, then you can already read SISU without any issues. SISU is, at its core, a mere punchtape display, but with fancy names around it. Fear not if you’re unfamiliar with punch tapes; the fundamentals of SISU is as straightforward as it can be. The details of SISU are explained in the following section below.
How SISU Works
A SISU digit consists of 𝍭𝍫(eight) dots called bits, forming a braille-like 𝍭𝍫(eight)-dot block called an octet. Each bit can be flipped to either one of two states: empty or filled. An empty bit is represented by an empty circle, and a filled bit is represented with a filled circle.
The first bit of an octet ⢀ “Ha” is known the Least Significant bit (LSb) and is placed at the lower right corner of the octet. Filling the state of LSb adds a value of 𝍩(one) to the octet. Conversely, the last bit of an octet ⠁“Koho”, placed at the upper left corner, is known the Most Significant bit (MSb) and adds a value of 𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍫(one-hundred twenty-eight) tally/rods to the octet.
Half of an octet is called a nibble. The right half is called the low nibble, and the right half is called the high nibble. A full nibble ⢸ “Di” has a value of 𝍭𝍭𝍭(fifteen) tally/rods.
Filling up all eight bits of an octet, in other words, filling both the high nibble and the low nibble, will result in ⣿ “Didi”, with a value of 𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭(two-hundred fifty-five) tally/rods. This technically makes SISU a base-𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍩(two-hundred fifty-six), or a Ducentohexaquinquagesimal Numeral System.
SISU is also a Superpositional Notation System, meaning the component of SISU consists of a smaller positional system within a larger positional system. In other words: SISU is a Ducentohexaquinquagesimal, or a base-𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍭𝍩(two-hundred fifty-six) numeral system that consists of two smaller Hexadecimal, or base-𝍭𝍭𝍭𝍩(sixteen) positional system, which itself consists of two base-𝍬(four) Quaternary positional system, which itself consists of two base-𝍪(two) Binary positional system.
Nomenclature of SISU
SISU borrows its naming system directly from Système Bibi-binaire, with minor tweaks. Although technically this makes the pronounceable words in SISU a hexadecimal system, since SISU is a superpositional system with a hexadecimal positional system incorporated within itself, it does not become an obstacle but rather a convenient alternative in human speech.
Binary names used in SISU are directly from Bibi. There are 𝍭𝍭𝍭𝍩(sixteen) monosyllabic names for each of the hexadecimal digits. Each syllable represents a nibble value ranging from 0000 (Ho) to 1111 (Di) in binary. In the first sequence: 0000 is called “Ho” /ho/, 0001 is “Ha” /ha/, 0010 is “He” /he/, and 0011 is called “Hi” /hi/. Followed by a voiced bilabial plosive “Bo” /bo/ to denote 0100, “Ba” /ba/ for 0101, “Be” /be/ for 0110, and “Bi” /bi/ for 0111.
Although there are some occurrences of voiced consonants, SISU, or rather Bibi, is deliberately designed in a way where speakers of languages that does not have a voiced/devoiced consonantal distinction can still pronounce these syllables, voiced of not, without worrying about confusion with another value in the system.
After 0111 (Bi) comes the midway point, 1000, which is called “Ko” /ko/. Then “Ka” /ka/ for 1001, “Ke” /ke/ for 1010, “Ki” /ki/ for 1011. Followed by the final sequence where, 1100 is “Do”, 1101 is “Da”, 1110 is “De”, and 1111 is “Di”.
Do note the fact that ending vowels will always match the last two bits of the said nibble. In other words, binary names ending in -o will always end with 00, names ending in -a will always end with 01, names ending in -e will always end with 10, and names ending in -i will always end with 11 as the final two LSb’s (Least Significant bits).
As for mnemonics, here’s a little trick I use: -O is an empty circle like the number 0, -A is the 1ˢᵗ letter in the Latin Alphabet, -E is 10(TEN), and -I is pronounced /ɪɪ/ in various languages all around the globe, just not English.
One interesting characteristic that SISU Nomenclature has is the use of “Ho” to denote an empty nibble, or 0000, distinctly apart from an empty byte octet 0000 0000, which is called “Hoho”. For example, uttering “Ha” usually denotes a value of 0001, but if the speaker wanted to specify the empty nibble (0000) on top of the said value, it is possible to say “Hoha” to specifically mean 0000 0001, also known as an octet byte with a value of 𝍩(one).
Humanity, as of Today, Despite having Several Decent Binary Numeral Notations Since 1968, Has Decided to Agree on One of the Most Thoughtless, Ambiguous Nomenclature Based on Western Arabic Numerals and Latin Alphabet
Welcome to the Final Section of this article. I am grateful for your patience with me and your time for reading my four-thousand word rant about the modern alphanumeric hexadecimal notation alongside an introduction to the alternative binary system I’ve created. I have initially intended this to be a small article introducing SISU, a Systematic Binary System, and nothing more, until I realized that the majority of the human population afar from my bubble are unaware of the very concept of binary. While doing research for this article, I’ve stumbled upon Système Bibi-binaire published in 1968 by the French Singer-Songwriter Mathematician Robert “Boby” Lapointe. The Bibi system had everything a binary system needed, from logical, systematic naming scheme, to unique symbols for each of the hexadecimal digits. SISU was overhauled completely midway writing this article just so that its bit placement could align with Martin’s extremely intuitive Binary Numeral Digits and Bibi’s beautiful songlike Nomenclature.
Special Thanks
I’d like to thank torneko (jan Toluneko) for keeping up with my presentation of SISU and providing valuable and critical feedback about my System. mi sona ni: jan Toluneko li pona pilin tawa mi.
I’d also like to thank Matthew Christopher Bartsh for inspiring me to create SISU to begin with. The whole idea of SISU stemmed from Bartsh’s article titled A simple algorithm for naming all numbers in all bases that are a positive power of two, based on the Major System of mnemotechnics. Version 3 by Matthew Christopher Bartsh. Believe it or not, my idea for SISU initially started out as a commentary question to Bartsh’s article about nomenclature for binary numerals. At first, SISU had loosely followed the same naming conventions as Bartsh’s.
That is, before I stumbled upon Bibi.
