# A simple algorithm for naming all numbers in all bases that are a positive power of two, based on the Major System of mnemotechnics.

## Introduction.

The base ten counting system has a set of spoken names and rules for pronouncing them that works well, and that we take for granted. Other bases, such as binary, or base eight, or base sixteen seem inhuman, difficult, and useless mainly because they lack an analogous system of names, so that people resort to reading the numbers out like telephone numbers and then saying what base it is to allow the listener to decipher what the number means. To remedy this deficiency in my favorite bases, which are those that are powers two, I developed a unified names and pronunciation system for all base that are a positive power of two. This allows one to use any base that is a power of two, and therefore a form of binary, in a way that is analogous to how we use base ten. In other words, I have created a spoken language for these other bases, and I optimized it for ease of learning and concision.

## Concisely pronouncing powers of two both positive and negative using a set of names generated by an easily learned algorithm.

I have created a simple algorithm that allows one to deduce how to pronounce any number in any form of binary. This is so as for people to be able to try out the system with minimal investment of time.

I used a subset of the encoding system used in the well-known (in the world of mnemotechnics) Major System of mnemotechnics to link the ten numerals of base ten to ten consonants that are easily recognized and distinguished from each other. They are as follows.

0 = s

1 = t

2 = n

3 = m

4 = r

5 = l

6 = sh

7 = k

8 = f

9 = p

Unlike in the Major System, in my system vowels also encode something. For now I use only four vowels, viz i, a, u, and o, chosen for being at or near the corners of the vowel chart of phonology, and thus as far apart as possible from each other on the chart, and therefore as unlike each other as possible in sound:

normal = i, reciprocal = a, negative = u, negative reciprocal = o

Vowels are pronounced as short, i as in hit, a as in cat, u as in foot, o as in cot.

The vowels are pronounced with a final glottal stop when at the end of a syllable, for example -2⁸ = fu is pronounced like in the Cockney English (and increasingly in standard British English) pronunciation of the first syllable of “football” (which is pronounced foo’ball) ie foo’. The t sound is replaced with a glottal stop, meaning a closure of the glottis in the throat. An example from American English of a final glottal stop is in the first syllable of “uh oh”.

If confusion with existing English words, when written longhand, is a problem, a silent h could be used in front. Thus, optionally, not “pip” but “hpip” and so on for all those that are English words. This is analogous to the silent w in “two”, which serves to distinguish it in writing from “too”, and “to”. Note that in speech there is no confusion, and likewise with “one” (sounds like “won”), “four” (sounds like “for”), “six’ (sounds like “sicks”), and “eight” (sounds like “ate”). Somehow the context and tone of voice and so on disambiguate adequately between, say, “I have one” and “I have won”. Note that if the silent h is left off for any reason, there is still no danger of misreading it as a different number, so it is optional to include the silent h. In any case, it is not necessary to write numbers longhand to be understood clearly. It just looks better. Compare “I have one.” with “I have 1.”

## The table of names.

To take the third row as an example,

2² means two to the power two, i.e. two squared, which is four. Thus, the word “ni” means four.

Only the first column needs to be understood at this stage, but for completeness, here are explanations of the other three.

2^-2 means two to the power negative two, i.e. the reciprocal of four, which is a quarter. Thus, “na” means a quarter.

-2² means minus one times 2², i.e. minus four. Thus, “nu” means minus four.

-2^-2 means minus one time 2^-2, i.e. minus a quarter. Thus, “no” means minus a quarter.

Since “no” is already an English word, optionally “hno” would be used instead, later. But for now, I want to keep things simple.

Counting from one to ten in binary can be done in a variety of ways:

Base 2: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010.

One, two, two one, four, four one, four two, four two one, eight, eight one, eight two.

Si, ti, ti si, ni, ni si, ni ti, ni ti si, mi, mi si, mi ti.

Note that there is no need to specify the base except when using Arabic numerals. Just as three score years and ten unambiguously means seventy, ni unambiguously means four, whether in base two or base four, or any other base.

Here’s the table of names (it’s infinite, so only parts of it are shown here):

2⁰ si 2⁰ sa -2⁰ su -2⁰ so

2¹ ti 2^-1 ta -2¹ tu -2^-1 to

2² ni 2^-2 na -2² nu -2^-2 no

2³ mi 2^-3 ma -2³ mu -2^-3 mo

2⁴ ri 2^-4 ra -2⁴ ru -2^-4 ro

2⁵ li Etcetera…

2⁶ shi

2⁷ ki

2⁸ fi 2^-8 = hfa -2⁸ fu -2^-8 fo

2⁹ pi Etcetera…

2¹⁰ tis

2¹¹ tit

2¹² tin

…

2⁹⁸ pif

2⁹⁹ pip 2^-99 pap -2⁹⁹ pup -2^-99 pop

2¹⁰⁰ tisis 2^-100 tasas -2¹⁰⁰ tusus -2^-100 tosos

2¹⁰¹ tist 2^-101 tast -2¹⁰¹ tust -2^-101 tost

2¹⁰² tisin 2^-102 tasan -2¹⁰² tusun -2^-102 toson

…

2⁹⁹⁸ pipif 2^-998 papaf -2⁹⁹⁸ pupuf -2^-998 popof

…

2²³⁴⁵ nimril

…

2¹²³⁴⁵⁶ tinmirlish

…

Ad infinitum.

The concision and simplicity of the system is especially marked with large numbers, but I must use small numbers to explain it properly.

This is intended to be a preliminary system, and so it is optimized for ease of learning by users of base ten, so that it can be quickly learned and tried out. If it works well, I propose longuistic and psychological research is carried out to see whether a system of names based on binary or on a number base that is a power of two (base sixteen, say) is worth it and how best to construct it.

If the names sound too alike, one might double all the names, so that fifteen which is mi ni ti si becomes “mimi nini titi sisi”. Or it might become “mim nin tit sis”. The second vowel is free to be anything, and so could match the consonant to make the names more distinct when spoken. Note that when using the numbers for solitary thinking, including thinking and counting out loud, there is no problem with mistaking one for another, solitary use should be enough to find out which base is best for each person.

Note that all the consonants are voiceless. This is to make it easier to learn and pronounce.

The way the numbers are encoded as consonants is the same as in the Major System of mnemonics, except that in my system, 1 is always t, rather than t or d, and so on. I use vowels differently, encoding being a reciprocal and/or a negative number using the vowels. Other vowels and diphthongs could be used to enlarge the table to include imaginary numbers like 2³ i, 2^-3 i, -2³ i, and -2^-3 i, ie eight i, one eighth i, minus eight i, and minus one eighth i. Yet other vowels could indicate a unit, meters for example. Thus, a large power of two of meters might be a single syllable and the same power of two of seconds, a different syllable.

## Counting higher in binary.

Counting the numbers off using the fingers of one or both hands works well to prevent me from losing count. Once I get to the end of my ten manual digits, I just start again at the beginning calling it eleven and so on. Thus, when I say, “two one,” I am holding up three fingers. When I say eight two, I am holding up ten digits. When I say ri four, I am holding up ten digits again, which confirms that I haven’t lost count, because that’s as it should be, since ri is sixteen and sixteen plus four is twenty.

One, two, two one, four, four one, four two, four two one, eight, eight one, eight two, eight two one, eight four, eight four one, eight four two, eight four two one, ri, ri one, ri two, ri two one, ri four, ri four one, ri four two, ri four two one, ri eight, ri eight one, […], ri eight four two one, li, li one, li two, […], li ri eight four two one, shi, shi one, […]

Si, ti, ti si, ni, ni si, ni ti, ni ti si, mi, mi si, mi ti, mi ti si, mi ni, mi ni si, mi ni ti, mi ni ti si, ri, ri si, ri ti, ri ti si, ri ni, ri ni si, ri ni ti, ri ni ti si, ri mi, ri mi si, […], ri mi si ti si, li, li si, li ti, […], li ri mi ni ti si, shi, shi si, […]

Base 2: 1, 10, 11, 100, 101, 110, 111, 1000, etc.

## “Binary decimals”.

Base 2: 0.011[2]

= 2^-2 + 2^-3 = na ma

Base 2: -0.00000000001

= -2^-11 = tot

Base 2: -0.00010000001

= -2^-4 + -2^-11 = ro tot

## Unconventional binary counting.

When telling the time in English, “five to twelve” is the same as “eleven fifty-five”.

Analogously ri su = 2⁴ — 2⁰ is the same as mi ni ti si = 2³ + 2² + 2¹ + 2⁰

7 = 8 -1 = mi plus su = mi su

7/8 = 1–1/8 = si plus ma = si ma

62 = 64 -2 = 2⁶ — 2¹ = shi plus tu = shi tu

63 = 64 -1 = 2⁶ — 2⁰ = shi plus su = shi su

64 1/64 = 2⁶ + 2^-6 = shi plus sha = shi sha

63 63/64 = 64–1/64 = 2⁶ — 2^-6 = shi sho

## An integrated system of counting systems based on binary, quaternary, octal, hexadecimal, and so on, being binary counting in another form, using just one set of names for all the powers of two.

One pronunciation system and regular set of names can be used for all bases that are a positive power of two.

Any base that is a power of two, such as base four, base eight, base sixteen and so on can use the same set of names. Ni is 2² = 4. Thus, in base four one might count one, two, three, one ni, one ni one, one ni two, one ni three, two ni, two ni one, two ni two, two ni three, three ni, three ni one, three ni two, three ni three, one ri, one ri one , one ri two, one ri three, one ri one ni, one ri one ni one, ….

By saying one ri one ni rather than ri ni, I am following the English “One thousand one hundred” style. In French the article is omitted: “Mille cent” (literally: “thousand hundred.” Perhaps the French would say “ri ni” while the English say “one ri one ni”. In binary there is never more than one of a power of two, so it’s not an issue with binary.

Using the shorter “French” style: One, two, three, ni, ni one, ni two, ni three, two ni, two ni one, two ni two, two ni three, three ni, three ni one, three

With English style pronunciation of numbers, one could count in base four like this:

(Imagine a subscript ‘4’ next to each number in the leftmost column. A four in square brackets means subscript four which means the number that it is subscripted to is in base 4. They are base four numbers written in Arabic numerals.)

Base 4: 1 si 2⁰

Base 4: 2 ti 2¹

Base 4: 3 ti si 2¹ + 2⁰

Base 4: 10 si ni 2⁰ * 2² = 2²

Base 4: 11 si ni si 2⁰ * 2² + 2⁰

Base 4: 12 si ni ti 2⁰ * 2² + 2¹

Base 4: 13 si ni ti si 2⁰ * 2² + 2¹ + 2⁰

Base 4: 20 ti ni 2¹ * 2²

Base 4: 21 ti ni si 2¹ * 2² + 2⁰

Base 4: 22 ti ni ti 2¹ * 2² + 2¹

Base 4: 23 ti ni ti si …and so on.

Base 4: 30 ti si ni

Base 4: 31 ti si ni si

Base 4: 32 ti si ni ti

Base 4: 33 ti si ni ti si

Base 4: 100 si ri 2¹ * 2⁴ = 2⁴

Base 4: 101 si ri si

Base 4: 102 si ri ti

Base 4: 103 si ri ti si

Base 4: 110 si ri si ni

Base 4: 111 si ri si ni si

Base 4: 1000 si shi 2¹ * 2⁶ = 2⁶

The form of the number ti si ni ti si meaning base 4: 33 is analogous to base ten’s

Base eight is often briefly taught at school, but I think it would be better to substitute base four, because the times table and addition table is so much easier for base four and when counting in base four one gets a reasonable number of digits very soon allowing students to see a bit more of the big picture.

Let us look at base sixteen now (imagine the numerals all have subscript sixteens):

Base 16: 1 One,

Base 16: 2 two,

Base 16: 3 three,

Base 16: 4 four,

Base 16: 5 five,

Base 16: 6 six,

Base 16: 7 seven,

Base 16: 8 eight,

Base 16: 9 nine,

Base 16: A ten, or “ay”

Base 16: B eleven, or “bee”

Base 16: C twelve, or “see”

Base 16: D thirteen, or “dee”

Base 16: E fourteen, or “ee”

Base 16: F fifteen or “eff”

Base 16: 10 one ri,

Base 16: 11 one ri one,

Base 16: 12 one ri two

…

Base 16: 1F one ri fifteen, or “one ri eff”

Base 16: 20 two ri,

Base 16: 21 two ri one,

…,

Base 16: 3F three ri fifteen, or “three ri eff”

Base 16: 40 four ri,

Base 16: 41 four ri one,

….

Base 16: FF fifteen ri fifteen, or “eff ri eff” = 15 * 2⁴ + 15

Base 16: 100 one fi,

Base 16: 101 one fi one,

…

Base 16: 10F one fi fifteen, or “ one fi eff”

Base 16: 110 one fi one ri,

Base 16: 111 one fi one ri one,

….

Base 16: FFF fifteen fi fifteen ri fifteen, or “eff fi eff ri eff

Base 16: 1000 one tin = 2¹²

Base 16: 1001 one tin one = 2¹² + 2⁰

Note: I am not endorsing the use of the letters A to F to stand for ten to fifteen. I just want to make things easy here by using a familiar system. I think a much better set of symbols (probably some foreign script could be borrowed from) and spoken names can be found or created, but that can come later. Likewise, my use of base ten as is not an endorsement of base ten, but only an acknowledgement of its universal use.

One pronunciation system can thus be used for all bases that are a positive power of two. This way, people can experiment with all of them at the same time, with no confusion or conflict, because the names for the powers of two never change, and people can find out which base works best for them at a later time. People are arguing about which base is best. The only way to find out is for all the candidate bases to be mastered by at least some people, so that a comparison can be made. Ideally several people would master at least two bases each.

In a way base four and base eight are the same base, both being base 2 in essence, and in a way the set of all bases that are positive powers of two are one base. This fact is obscured by the fact that 30 say means three eights in base eight, but three fours on base two. But when pronounced according my new system, three eights is perfectly clear and so is three fours, and every term has the one meaning. Note that there is no need to say which base you are using, unlike in writing. Three eights is the same number regardless of the base and likewise three fours so there is no need to specify the base. Indeed bases can be combined without confusion, as in “I need three eight(s) (of) apples, but I have only three four(s) (of) apples.” Saying “three zero base eight” and “three zero base four” makes things seem less clear than they are, and is more complicated, and harder to understand. It’s also not analogous to how we pronounce base ten, which makes it seem more different from base ten than it really is, and more difficult than it really is, and not as good as it really is. It is as if there’s an ethnocentric, for want of a better word, love of base ten, that causes people to make other bases look weirder and more difficult and less useful than they really are.

With computers getting good at translating text and even speech from one language to another, it seems that we might be able to have numbers translated automatically from one base to another, which would allow one to read numbers, or even listen to them, in any base one chooses. Being the only person thinking in a base that is a power of two would then be okay, as one could have most of ones numbers translated to and from that base when communicating with others who use another base. Using machines, we could be free to be fluent in several bases and use all of them daily.

Base ten is not very well designed. It seems to be designed to let us count using all ten digits on both hands. Using base eight, we could count on our fingers while ignoring our thumbs. Anyway, we could still count to five and ten on our hands using base eight: one, two, three, four, five, six , seven, eight, eight one, eight two. Base ten is based on ten which is not a power of two. Ten is two times five. Five is an awkward number, and therefore ten is, and therefore ten is an inconvenient choice of base. Five cannot be neatly divided by two. Doubling and halving are incredibly important operations and with a base that is a positive power of two one can repeatedly halve and/or double without the irregularity (chaos) that happens with base ten. I could go on, but that would be to digress for too long.

## Names for new numerals, especially of very large bases, e.g. base 64 and above.

One can create a pseudo-numeral by simply putting brackets around a number. Eg. 63 in base 64 might be (63) or ’63’ and it can be used exactly like a numeral.

Base 64: 10–1 = (63) i.e. 64–1 = 63

Base 64: 100 -10 = (63)0 i.e. 64² — 64 = 63 x 64 = 4032

One can pronounce (63)0 as “sixty-three sixty-four” or “sixty-three shi.

That seems to work well, but if completely new names are wanted for the new numerals of very large number bases, one can say that the vowel ‘e’ (pronounced as short e like in “bet”) indicates a numeral, and then (63) can be “shem”. Because, following the same pattern that links numbers to consonants, 6 = sh, and 3 = m. Then (63)0 is the same as shem shi. Note that shem doesn’t involve a power of two and the presence of the e tells you that. Note that shim (or hshim) is 2⁶³ = approximately 10³⁰.

(123456) In base 131072 ( base 2¹⁷) is tenmerlesh. Three syllables which is the same number of syllables that are in the word “seventeen”. Thus tenmerlesh timtiskin is base 131072: (123456)0 i.e.123456 x 131072, which is technically a two digit number in base 131072. Presumably no human being is going to find a base this big to be practical, but it shows that the system can handle the biggest bases with ease.

Tenmerlesh timtiskin plus one = tenmerlesh timtiskin one or tenmerlesh timtiskin si.