Tag: number-theory

Despite their importance to modern mathematics, alternative numeric systems are a concept often left towards the end of standard education throughout the English-speaking world. This may be in part to their diminishing presence in the last 6⛂_STK years with the introduction of the metric measurement system. However, even in current day USA, we still use the imperial measurement system, with our less standard mix of binary, trinary, quaternary, dozenal, and arbitrarily large bases. Our way of describing numbers shapes our understanding of things.
In 2018, internet personality Mitchel John “jan Misali” Halley made a numeric base naming convention that focused on factor pairs that make up a given radix. The system suffers from its searing focus on the even treatment of all bases given the very limited source of affixes representing the factors. In addition, jan Misali only targeted integer and rational radices (or more accurately: natural numbers; their opposites; and their ratios). This is not an alternative, but rather a fork of this idea. This is the Misali-Dual numeric base naming convention.

You may notice words throughout are highlighted. Moving your finger or cursor over these will reveal a brief definition of the word. This is to make this article as accessible as possible.

Natural Numbers

The foundation of our understanding of numbers, the numbers by which we can count are naturally the first any system will want to address. For our purposes, natural numbers will also include zero, though this is not the only definition. The first idea I had was the use of multiple roots as seen in medical terminology, typically Greco-Roman, but not always for reasons that will become more obvious as we work our way down the list.

Radix Root DEC	Named Consesus	Abbreviation	Affix-A	Affix-B
0	Nulinary	NUL	nil/le-	zilch/i-
1	Unary	UNI	un/i-	hen/a-
2	Binary	BIN	b/i-	d/i-
3	Trinary	TRI	tert/i-	t/ri-
4	Quaternary	QUA	quandr/i-	tetr/a-
5	Pentinary	PEN	quint/i-	pent/a-
6	Seximal	SXM	sex/i-	hex/a-
7	Septinary	SPT	sept/a-	hept/a-
8	Octal	OCT	tav/o-	oct/o-
9	Ennibainal	ENN	nun/o-	enn/e-
10	Decimal	DEC	den/a-	dec/a-
11	Elenary	ELE	el/a-	maor/i-
12	Dozenal	DOZ	doz/ena-	mon/e-
13	Bakinary	BAK	chef/a-	bak/e-
14	Fortecimal	FRT	fort/e-	macr/o-
15	Hulibainal	HUL	hul/i-	guiran/i-
16	Hexcode	HEX	comp/u-	ord/ni-
17	Subinary	SUB	sub/o-	hup/o-
18	Telimal	TEL	adult/u-	tele/i-
19	Fininary	FIN	fin/o-	esch/a-
20	Vigesimal	VIG	vig/e-	ikos/i-
23	Erinary	ERI	kal/a-	er/i-
24	Daily	DAY	day- dai-	emar/i-
26	Alphacimal	ALP	alph/a-	gram/ma-
29	Lunar	LUN	lun/a-	artem/i-
36	Niftimal	NFM	nif/ti-	dict/o-
52	Stakimal	STK	pap/ie-	stak/o-
60	Hourly	HRL	hor/i-	min/o-
65	Senjobainal	SNJ	senj/o-	ger/a-
100	Centesimal	CEN	cent/e-	hect/o-
120	Cardiac	CRD	cardi/o-	hem/ato-
360	Circular	CIR	radi/o-	degr/i-
1,000	Milleceimal	MIL	mill/e-	kil/o-
5,040	Platonial	PLT	put/o-	plat/o-
1,000,000	Lardocimal	LAR	lard/o-	lip/o-

Radix Property	Suffix
Indivisible by natural numbers less than itself	-in/ary
Odd	-bainal
Even	-c/imal

There is a lot going on up here so let’s have a brief discussion of medical terminology, the second easiest language an English speaker can learn. When you want to describe something in medical terminology, you take all the items in question such as stomach and intestines, find their medical roots such as gastr/o- and enter/o-, and add them to a suffix to make the meaning such as -ology to make gastroenterology which is the study of the digestive tract and its function. In this same sense, we take the factors of smaller bases such as base-21_DEC being the product of 3_DOZ and 11_SXM. This gives us the name triseptabinal. However, numbers like 31_DEC belong to a category of numbers known as prime numbers. Before we can create naming conventions, we need to get into more basic arithmetic.

Rational Numbers

Natural numbers are a subset of all integers, which include the opposite of natural numbers or negative numbers and always includes zero. Integers are then a subset of all rational numbers, numbers that can be represented as an integer divided by a natural number where it is defined without zero. Similarly to before, roots are predominately represented by Greco-Roman affixes.

Arithmetic Operational Root	Abbreviation	Affix-A	Affix-B
Addition +	ADD	et/a-	ka/i-
Negation 0-	NEG	neg/a-	yam/i-
Inversion ⅟n	DIV	inv/e-	vot/i-
Exponentiation n	EXP	exp/a-	pow/a-
Logarithm ㏒	LOG	log/o-	fract/o-

Without arithmetic, the creation of names of prime radices purely through factorization is impossible. Instead, we conceptualize primes as the addition of one to the number that proceeds them. This means 31_DEC is simply just kaipentasexihenary, made up of roots ka/i-, pent/a-, sex/i-, hen/a-, and -in/ary or “and”, “five”, “six”, “one”, and “prime”. I didn’t want to separate the arithmetic table, so I did elect to push up discussion of all rational numbers.
Negative radices are represented as the very simple “nega” or literally negative versions of their natural numbers. Negabinary is just base-(10)_BIN.
Ratio radices are a bit odd. All conventions to this point suggests the largest factor goes last. As such, we’d expect 5/10_SPT to be votiseptapentinary. What about 1/35_DEC? Votiseptapentinary. That is the opposite of what we want. Instead, when inverting a number, it needs to terminate the same way we do with addition. Votiseptahenaquintary is base-10/12_PEN and votiseptapentunary is base-1/55_SXM.
These last three operators are not useful in the rational side of numbers.

Irrational Numbers

Irrational numbers are numbers that cannot be created as the output of field operations. Pi (generally represented as lowercase Greek π) is the result of taking the circumference of a circle and dividing it by twice its radius. This number is not a ratio of two natural numbers, which is among one reason we have given it a fun name. Similarly, the sum of the series of the inverse of all factorialized natural numbers cannot be rationalized as a ratio between any less than infinite factors. Euler’s number (generally represented as italics, lowercase Latin e) also then cannot be represented in our system yet.

Irrational Radix Root DEC	Named Consensus	Abbreviation	Affix-A	Affix-B
12√2	Equalary	EQL	equal/a-	temp/o-
Plastic Ratio ρ	Rhonary	PLA	plast/i-	pis/o-
Golden Ratio ϕ	Phinary	AUR	aur/a-	khrus/o-
Supergolden Ratio ψ	Psinary	SAU	plentin/o-	lefk/o-
Silver Ratio δS	Dhelsinary	ARG	argent/u-	argyr/o-
Euler’s number e	Eulariary	EUL	eul/e-	nat/o-
Pi π	Pinary	PIR	p/e-	p/i-

We can then create more irrational bases like natopary and powapeheneulinary. I’m not sure how they are useful, and honestly much of everything going forward makes less and less sense to me. I knew irrational radices were a thing, but I always wondered why exactly you would use them. Equalary is used in music temperament which makes sense, and psinary was used in early analog-to-digital conversion. But dhelsinary and phinary just seem to exist.

Imaginary and Complex Numbers

Imaginary numbers are this side effect of doing operations you cannot normally do, like the square-root of a negative number. Why these are useful is complicated and has a lot to do with gimbal-lock and fractals among other problems. One idea is to treat it literal, making base-i powinvanegahenunihenary which is a mouthful and not very useful. Instead, we conceptualize the idea as a whole category of what is this imaginary plane.
This is the part where the table would go if describing more than one imaginary item made sense. However, I can find little reason to create more than one row of imaginary inventory as well as an additional operation.

Imaginary i

Imaginary

IMG

im/a-

phant/ou-

Bi-operational Increment ±

ASE

ou-

eith/re-

Of note is that ± is read plus-or-minus. In cases where minus-or-plus is required, the item is negated before incrementing the operations. Is there a better name for these operations?
With both of these, we can finally make our favorite fractal, the Twindragon, into a base. -1 ± i is… Complicated question, is i further to zero than 1? They are supposedly equidistant, so we will need to make our own rules. For the sake of clarity, imaginary numbers are smaller (that is closer to zero) than negative integers, and negative integers are smaller (that is closer to zero) than natural numbers. This isn’t actually true, but rather a “we need to draw a line in the sand somewhere”. So, eithimanuyamuninulinary is base-(1) ± i and phantobinary is base-2i.

Mixed radices

For the most part, we keep the tune of the regular system. You just list it with a few notable exceptions.

Mixed Radix Roots	Named Consensus	Abbreviation	Affix-A	Affix-B
Primes ℙ	Primorial	PRI	prim/o-	prot/o-
Factorials n!	Factorial	FCT	perm/u-	fact/o-
Fibonacci n!F	Fibonorial	FBN	fib/o-	self/i-

Shorthand Representation

I do not believe bases without naming consensus should have true abbreviations. Abbreviations are useless unless they genuinely make identification of things faster. As such, to shorthand write any base, you can tabulate the name back to their roots, take the first two letters of each abbreviation. This means the base is always identifiable, similar to chemical composition notation. This does mean the eithimanegahennulinary would become the “not-much shorter” AsImNuNeUnNu, but a novice would be able to recognize it as an imaginary or complex radix without significant effort.

Name calculation

We’ve briefly covered the dynamics of calculation. Numbers are sorted in order of distance to zero, prioritizing imaginary components over negative over positive components. But how does this function as a whole? I feel the best way to describe this is programmatically.

Gather affixes
 Search radix
 If radix has Named Consensus
  Return Named Consensus
  Exit function
 Set suffix to "-in/ary"
 Select case radix type
  Case natural number
   Find factor pairs
   If number of items of factor pairs is one
    Add affix "ADD"
    With radix one less than current
     Gather affixes
     For each affix
      Add current affix
    Add affix "UNI"
    Process affixes
    Exit Function
   For each factor pair
    If total number of Named Consensus in factor pair is greater than preference
     Set preference to current total number of Named Consensus
   Delete items in factor pairs with fewer Named Consensus than preference
   Sort factor pairs on difference of members, less to great
   Select item with smallest difference
   Sort factor pair on difference of members to zero, less to great
   For each member in factor pair
    Gather affixes
    For each affix
     Add current affix
   If modulos of radix by one plus one is one
    Set suffix to "-bainal"
   Else
    Set suffix to "-c/imal"
   Process affixes
   Exit Function
  Case rational number
   Select case nonnatural rational radix type
    Case negation
     Add affix "NEG"
     With absolute value of radix
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function
    Case inversion
     Add affix "DIV"
     With denominator
      Gather affixes
      Reverse affix order
      For each affix
       Add affix
     Add affix "UNI"
     With numerator
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function
  Case Else
   Select case operand type
    Case exponentation
     Add affix "EXP"
     With exponent
      Gather affixes
      For each affix
       Add affix
     Add affix "UNI"
     With base
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function
    Case logarithm
     Add affix "LOG"
     With antilogarithm
      Gather affixes
      For each affix
       Add affix
     Add affix "NUL"
     With base
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function
    Case factorial
     Add affix "FCT"
     With elements
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function
    Case addition
     Add affix "ADD"
     For each term
      Gather affixes
      For each affix
       Add affix
      Add affix "NUL"
     Process affixes
     Exit function
    Case bioperation increment
     Add affix "ASE"
     For each term
      Gather affixes
      For each affix
       Add affix
      Add affix "NUL"
     Process affixes
     Exit function
    Case imaginary
     Add affix "IMG"
     With coefficient
      Gather affixes
      For each affix
       Add affix
     Process affixes
     Exit function

Process affixes
 For each affix
  Search affix
  If index modulus one plus one is one
   Select item-A
  Else
   Select item-B
  If item is starts vowel sound
  or item first consonant is equal to item index one less than current last consonant
   Set item index one less than current to end to ends consonant
 If suffix is "-bainal"
  Set last item to ends vowel
 If last item last consonant not equal to suffix first consonant
  Set last item to ends vowel
  Set suffix to start syllable
 Else
  Set last item to ends consonant
  Set suffix to start vowel
 Concatenate items and suffix
 Return name
 Exit function


Stray thoughts

You may have also noticed the use of subscript letter with abbreviations instead of decimal numeric reference for written numbers. It’s far clearer to say 10_DEC than 10₁₀. Because what is 10? That’s kinda the whole point of this discussion.

Conclusion

The original draft of this from nearly six months ago included a pronunciation guide that was more unhelpful that anything else. It can be summarized as “If it feels like the correct pronunciation, then that’s right” and “Here’s how I, someone from the Midwestern USA, pronounce these things” with a lot of IPA symbols.
The original creator, jan Misali, should be proud of the word they put in, it creates a more realistic and more universally understandable system that depends less on preexisting notions that create bias. Most importantly, I see this not as a replacement, but an extension to jan Misali’s proposed system. I also see this as not complete, but another stepping-stone in number theory. One thing I would like further emphasis is counting systems as used by various world cultures. Already through this did I discover that the Huli people used a mostly undocumented base-13_DOZ and the Māori people use a much more documented base-15_SXM.
In my opinion, this system is at half its reasonable capacity, being capable of well accommodating 20_STK named items. In theory, it could have as many 2A4_HEX items, but pronounceability and distinct symbolization will become a problem.
Further on that idea, it could also make use of less common Latin characters like ẞß (Eszett), Þþ (Thorn), Ƿƿ (Wynn), Ƕƕ (Hwair), Ꞷꞷ (Omega), Ꞵꞵ (Beta), Ꜫꜫ (Tresillo), Ꜧꜧ (Heng), Ƹƹ (Ayin), Ʒʒ (Ezh), and Ȝȝ (Yogh). I would like to note that it makes the most sense to use non-diacritical glyphs that exist already in the Latin alphabet with some actual use outside of linguistics. Another option is making use of a completely different writing system, but the choice of Latin was because I already use that and of the 1 languages I absolutely speak and 1 language I can speak use this writing system. The only other writing system I know is the Arabic Abjad, and of the 28 official characters, that only extends our valid name capacity by 73_HUL entries. It goes further with the extended use characters in other languages or use to accommodate international communication.
One last option is a unique script designed for this explicit purpose. This would require some effort more than what I’ve already put in here. At least for now.

 
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Bonus: JavaScript Popup Dictionary

This shockingly small piece of accessibility is not integrated into every OS or browser in popular use. On current Android devices and Apple products, selecting a section of text will enable you to get a brief definition as brought up by the dictionary engine the device is using or search the web without leaving the tab.
While searching the web is an option, an academic resource really needs a glossary or annex with definitions of words. This little script simply detects when a word of a given CSS class is rolled over. When this happens, it moves an HTML element to the position in question with prepopulated meanings and pronunciations. When the element is rolled off, the element is moved, made transparent, and depopulated for the next word to be loaded up.
The truly hardest part of this was ensuring its functionality on touch devices and ensuring the aesthetics were just right. Positioning is close enough that you can still see it, but just far enough that you aren’t robbed of the sentence context clues.
I would enjoy making this work everywhere on my website, referencing a JSON dictionary, but for the moment, it exists in this article.

Legal

Zachary Yarnot and DualVission do not hold any rights to these owners’ contents.
jan Misali is the internet person of Mitchel John Halley.
JavaScript is a trademark of Oracle Corporation.
JSON is a standard maintained by Ecma International created by Douglas Crockford.