How do you pronounce numbers written in different bases? [closed]

Question

The decimal (base 10) number "2" can also be represented as the binary (base 2) number "10".

Let's use binary "10" (equivalent to decimal "2") as an example. I could see a few different ways to go here. Assume that the base doesn't need to be specified, and is understood from the context of the conversation (e.g. two programmers talking about memory addresses would understand that they were using hexadecimal).

It could be "ten", since that is what it looks like. One might even argue that ten, as a concept, refers to a one followed by a zero irrespective of the radix. In other words, ten means "a quantity exactly equal to the base it's represented in".

On the other hand, you could argue that "ten" refers specifically to the quantity; in other words, "1010" in binary, "10" in decimal, and "12" in octal would all be pronounced "ten," and "10" in binary should be pronounced "two".

So how would you pronounce the following numbers?

"10" binary ("2" decimal)

"10" octal ("8" decimal)

"10" hexadecimal ("16" decimal)

"1F" hexadecimal ("31" decimal)

Answer 1

I pronounce your examples "ten", "ten", "ten", and "one ef". I count in hexadecimal, "One, two, three, four, five, six, seven, eight, nine, ay, bee, see, dee, ee, ef, ten, eleven, twelve, ..., one-ee, one-ef, twenty, twenty-one, ..." etc.

I've heard some people make the argument that, as a "number" is a concept that is independent of the numerals and radix used to represent it, that therefore we should read binary 10 as "two", octal 12 as "ten", etc, because that is the concept that these strings of digits represent. I was on another forum once where several people were quite adamant about this, and insisted that anyone who read octal 10 as "ten" was demonstrating profound mathematical ignorance, corrupting the youth, and so forth. I disagree with that idea on two grounds: one philosophical, one practical.

On the philosophical, who says that "thirteen" means "this many: X X X X X X X X X X X X X" and not "the string of digits consisting of a one followed by a three"? There are many possible representations of "this many fingers", including decimal 13, octal 15, Roman numerals XIII, Hebrew symbols yod-gimel, etc etc. Who says that the only correct way to read all these representations is by the word "thirteen"? Are French people "wrong" because they read it as "treize" rather than as "thirteen"? If it's linguistic chauvinism to say that the French are wrong to use French words rather than English words, perhaps it is "radix chauvinism" to say that names derived from the decimal number system are "right" and names derived from any other number system are "wrong". Need I point out that "thirteen" is obviously derived from a string of digits, "1" and "3". To look at (octal) "15" and read it "thirteen" is clearly imposing a decimal-based name on an octal representation.

On more practical terms, trying to read numbers in other bases using names derived from their decimal equivalents quickly becomes wildly impractical. If you insist that octal 10 be read "eight", then presumably we keep counting 11=nine, 12=ten, 13=eleven, 14=twelve, ... 20=sixteen, 21=seventeen, ... 100=sixty-four, ... etc. Imagine trying to read off a series of octal numbers to another person for him to copy. Would you really look at octal 34702 and read it "fourteen thousand seven hundred eighty-six", and then expect the other person to hear this and type in "34702"? Such a process would be very difficult and error-prone. It makes a lot more sense to read it "three four seven zero two" or "thirty-four thousand seven hundred two".

Once you grant that when numbers exceed two or three digits it is most natural and practical to read them using the digits given and not trying to use the same words you would use for "this many" in decimal, it follows that for consistency we should always do this. If I read octal 12 as "ten" but octal 1000 as "one thousand", then we would have to define some cut-off point where we transition from "decimal names" to "octal names". As such a cut-off point would be arbitrary, it would likely be confusing. Better to just consistantly use the natural octal reading.

Answer 2

By convention:

• "one-zero binary" (people rarely say "base 2" in my experience)
• "octal one-zero" or "one-zero octal"
• "hex one-zero"
• "hex one-eff"

If you say "hex ten" to a developer, they will mentally translate it to "hex one-zero" anyway, so you're better off saying "hex one-zero" in the first place.

In general, developers tend to

• pronounce every digit in bases other than decimal
• pronounce groups of four in binary when unambiguous (e.g. "1011" is said "ten-eleven", but "1000" is pronounced "one-zero-zero-zero")

That being said, 0xdeadbeef is always pronounced "dead beef." But then, you've entered the realm of hexspeak.

Answer 3

Pronouncing the hexadecimal letters A through F

The default pronunciation for the letters are simply their English names, "ay, bee, see, dee, ee, eff."

When reading off a hex MAC address, I have both used and heard the NATO phonetic alphabet used for the letters A through F. Since both the speaker and the listener know that a hex string is coming, they will pronounce

1A-48-0F-CF-3B-24

as "One alpha, four eight, zero foxtrot, charlie foxtrot, three bravo, two four."

With hexadecimal numbers, I have also heard a somewhat simplified phonetic alphabet.

A=Abel
B=Baker (or Boy)
C=Charlie
D=Dog
E=Easy
F=Fox

So the above string would be pronounced "One abel, four eight, zero fox, charlie fox, three baker, two four."

Last week, I provisioned a modem and had to pronounce the MAC address of said device. I'm a AE speaker and the hearer was Filipino. I pronounced the address using the simplified phonetic alphabet, and he confirmed the address using the NATO phonetic alphabet.

There are other spelling alphabets used around the world but the NATO phonetic alphabet is the most common.

Pronouncing individual digits versus pronouncing as if they comprised a decimal number

When I have taught computer science classes, I would always pronounce a binary number like 1010 as "one oh one oh, base 2." To pronounce it as if it were a decimal number, "one thousand and ten" seemed to invite confusion. Thus, I would never pronounce 10₂ as "ten, base two" or "ten, binary."

This professor, who teaches cryptography and algorithms, uses a similar convention.

If decimal, just say the number (with the word "decimal" if we're mixing contexts)

If any other base, read the digits and say the name of the base

So I might say, "therefore the answer is one-zero-one binary, or 5 decimal."

I would never call 10 hex "ten". Nor would I call 10 binary "two."

Said professor pronounces, "one zero one," while I might shorten and say "one oh one."

(And please click through to the picture of the T-shirt. It's only funny if you misread "one zero base 2" as "ten.")

One HBO Silicon Valley episode

I taught Computer Science back in the day when mighty dinosaurs ruled the earth, punch cards were on the wane, and floppy disks were still floppy.

Acknowledging that language evolves, here is a link to a blog entitled "How to pronounce hexadecimal", dated 2015. The blog author is Bzarg.

It includes a dialogue from an HBO series, Silicon Valley:

Kid: Here it is: Bit… soup. It’s like alphabet soup, BUT… it’s ones and zeros instead of letters.

Bachman: {silence}

Kid: ‘Cause it’s binary? You know, binary’s just ones and zeroes.

Bachman: Yeah, I know what binary is. Jesus Christ, I memorized the hexadecimal times tables when I was fourteen writing machine code. Okay? Ask me what nine times F is. It’s fleventy-five. I don’t need you to tell me what binary is.

Bzarg goes on to propose a pronunciation convention.

For the hex digits in the units place, pronounce the usual 0-9 and "ay, bee, see, dee, ee, eff."

For the hex digits in the sixteen's place, the author proposes these (based on "fleventy"): "atta, bibbity, city, dickety, ebbity, fleventy."

For four hex digits, Bzarg suggests separating each two digits by "bitey."

So the MAC address above might be:

1A-48 = "abteen bitey forty-eight" 0F-CF = "eff bitey city-eff" 3B-24 = "thirty-bee bitey twenty-four"

Despite Bzarg's heroic efforts (and an echo from http://www.xanthir.com/b4ej0), I have not observed this in practice, even once.

How to pronounce your examples

I think your examples depend on whether you are doing math or doing programming.

In math, numbers in other bases are written with the base following as a subscript. (Please see https://math.stackexchange.com/a/638782/5220 for 1100₂ = C₁₆.) Math also routinely talks about logarithms in base 2, base 10, or natural logarithms like this (with the base following the log as a subscript):

log₂ (pronounced "log base 2")

log₁₀ (pronounced "log base 10")

ln (pronounced "log")

So I would pronounce your numbers in a math context as:

• 10₂, pronounced "one zero base two"
• 10₈, pronounced "one zero base eight"
• 10₁₆, pronounced "one zero base sixteen"
• 1F₁₆, pronounced "one eff base sixteen"

In programming, there is a convention that literals in other bases are preceded with a 0b, 0o, or 0x for binary, octal, or hexadecimal, respectively. (See this proposal for Python "Integer Literal Support and Syntax" at https://www.python.org/dev/peps/pep-3127/).

The proposal is that:

octal literals must now be specified with a leading "0o" or "0O" instead of "0";

binary literals are now supported via a leading "0b" or "0B"; and

provision will be made for binary numbers in string formatting.

(Python, C, C++, and Java already precede hexadecimal literals with "0x").

Their motivation was:

The default octal representation of integers is silently confusing to people unfamiliar with C-like languages. It is extremely easy to inadvertently create an integer object with the wrong value, because '013' means 'decimal 11', not 'decimal 13', to the Python language itself, which is not the meaning that most humans would assign to this literal.

(Note the Pythonic displeasure with the C and C++ convention that writes an 13₈ as 013.)

So with this in mind, if you were pronouncing your examples in a programming context, the literal is preceded by a base marker and the pronunciation follows suit:

• 10₂, written 0b10, pronounced "binary one zero" or "binary one oh"
• 10₈, written 0o10 in Python or 010 in C++, pronounced "octal one zero"
• 10₁₆, written 0x10, pronounced "hex one zero"
• 1F₁₆, written 0x1F, pronounced "hex one foxtrot" or "hex one eff" or "hex one fox"

Answer 4

In notations other than decimal, always read out the symbols, which is what they are.

Do not even call the individual elements as digits when the number system is not binary, decimal or octal because in higher notations, alphabets are also used, which will create the illogical (not technically incorrect, maybe) use of digit.

When we read 'one' in say, hex, we are not referring to a value of unity, only the name of the symbol.

Answer 5

I find the simplest to pronounce any numeral in any base using any symbols would be to organize the numerals in bytes of three. For example

123 456 789 abc def in hexadecimal

I'd call this one, two, three - tera; four, five, six - giga; seven, eight, nine - mega; ay, bee, cee - kilo; dee, ee, eff

This method's advantage is that it makes describing a numeral in any base simple and correct:

001 000 000 000 binary is one giga.
020 000 000 decimal is two, zero mega
a00 000 hexadecimal is ay, zero, zero kilo

One should be careful because 123 giga hexadecimal is not one hundred and twenty-three giga, it should be viewed as:

(1*10²+2*10¹+3)*10⁹ all in hexadecimal notation

Another example, 101 giga binary should be viewed as:

(1*10¹⁰+0*10¹+1)*10⁹ all in binary notation

And lastly for something familiar 456 giga decimal should be viewed as:

(4*10²+5*10¹+6)*10⁹ all in decimal notation

This notation coincidence with four hundred fifty-six giga in decimal notation.

The quantity these numerals represents is another story. For example 144 (one, four, four) is not a number; it is a numeral that could represent a number. Where as numbers one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, thirty, forty, etc are all specific numbers representing a specific quantity in english.

In addition one hundred forty-four and a gross (twelve times twelve) both represent the same number. This same number could be represented by fourteen-ten and four or seven-twenty and four. I could go on and on. Of course all this requires some knowledge of multiplication and addition expect for a gross.

One could say that 10 (one, zero) binary is a number and they would be correct in mathematics. However one zero binary is jargon means nothing in Common English. It must be translated to 'two' if you wish to be understood.

Answer 6

The following joke brought me here: "There are 10 kinds of people in the world. Those who understand binary and those who don't." I used to think that it could only be funny when read...making it perfect for sharing on the Internet. This is because I assumed that the pronunciation in the joke would be "two kinds"

But when it occurred to me that if said out loud as "ten kinds", it is even funnier since binary is to rarely thought of as anything other that being read as the digits aloud. That is, one would expect that number if read in binary to be read as "one zero kinds".

Now, I'm convinced that the proper way to read it "ten kinds". Just as if you are trying to say the following: x x x x x x x x x x x x x x x In base 10, that's 15 (pronounced "thirteen".
In base 8, that's 17 (pronounced "fifteen". In binary, that's 1111 (Pronounced "one thousand, one hundred and eleven).

If you were examining 1111 if it were a base ten number you'd have: 1*10^3 + 1*10^2 + 1*10^1 + 1*10^0 which equals 1111 in base ten If you were examining that same number, 1111, in base 2, you'd do the following base ten calculation: 1*2^3 + 1*2^2 + 1*2^1 + 1*2^0 which equals 15 in base ten.

You are still bound to use tradition base ten terminology to describe the base itself. When saying that each column represents a power of ten you wouldn't say a power of "x x x x x x x x x x". But even the word "decimal" case refers to what is traditionally our Base Ten meaning of "ten".

But once the base is established, it seems to me that the number should be read as the digits appear. Thus 10 is read as ten regardless of the base, etc.

Answer 7

See this 3-minute educational video (from the Schoolhouse Rock series) ...

This is base 12, the two extra digits are pronounced "dek" and "el. The numeral for last finger is written "10" and pronounced "do" (dough, doe). It stands for "dozen" I guess.

Answer 8

We use several pronunciation systems in English for base ten, for example when talking about a telephone extension number with a switchboard operator we pronounce 4567 as "four five six seven", but when talking about dollars the same written number is pronounced "four thousand five hundred sixty-seven". If it's a year it's pronounced "forty-five sixty-seven".

We should bear this in mind when discussing how to pronounce binary, octal, and hexadecimal numbers. Because numbers written in those bases are used primarily in a way that is somewhat like a telephone extension number is used, they get pronounced analogously. Thus "1010" of binary would tend to be pronounced "one zero one zero" or "one oh one oh" or "one nought one nought". Binary is rarely used for talking about numbers of dollars or other things, but when it is, it would make sense to pronounce "1010" as neither "one oh one oh", nor "ten", but rather "eight two", meaning "eight and two", or "mi ti", meaning "mi and ti", where mi is my new word for "eight" and "ti" is my new word for two.

We shouldn't be looking for one right way to pronounce numbers of other bases. My answer here is primarily about how to pronounce (and write out in English words) numbers of other bases when they are to be used for referring to numbers of dollars and other things, but I am not implying that anyone should stop using the pronunciation system that suits their purposes. A computer programmer would probably want to continue using the "telephone number" or "code number" type of pronunciation when talking about machine code in binary or hexadecimal. And there's nothing wrong with that. The same coder might, when counting or calculating in other bases, find my system of pronunciation useful.

The original poster asks:

"On the other hand, you could argue that "ten" refers specifically to the quantity; in other words, "1010" in binary, "10" in decimal, and "12" in octal would all be pronounced "ten," and "10" in binary should be pronounced "two"."

I would pronounce binary 1010, "eight two" or "mi ti". I would pronounce decimal 10, "ten". I would pronounce octal 12, "eight two" or "mi ti" (Yes, it's the same as for binary 1010).

The original poster then asks:

"So how would you pronounce the following numbers?" (My answers are inserted in square brackets).

"10" binary ("2" decimal) ['two' or 'ti']

"10" octal ("8" decimal) ['eight' or 'mi']

"10" hexadecimal ("16" decimal) ['sixteen' or 'ri']

"1F" hexadecimal ("31" decimal) ['sixteen fifteen' or 'sixteen eff' or 'ri fifteen' or 'ri eff' (or 'ri tel' but don't worry about this, newbies. It's explained further down.)]

Note that "sixteen fifteen" means "sixteen plus fifteen" just as "one hundred fifteen" means "one hundred plus fifteen". Likewise, 'ri eff' means 'ri plus eff'. My entire system is based on the English pronunciation of base ten, and is, with a few exceptions, closely analogous to it.

A simple example or two of counting (the first twenty numbers) in binary may be helpful at this point.

Using numerals: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100

Using one of the OP's suggested possibilities: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty.

Using the easiest (beginner) system of pronunciation: 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, sixteen, sixteen one, sixteen two, sixteen two one, sixteen four.

Using my slightly more difficult (at first) system of pronunciation: 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.

Note that 'sixteen' is a bit unwieldy, being two syllables and calls to mind base ten where sixteen is conceptualized as ten plus six. Thus 'ri' which means sixteen, being two the power four (the 'r' of 'ri' indicates the '4' of 2^4) is used instead of 'sixteen'.

Using my more difficult pronunciation system (good if you want to get away from base ten as much as reasonably possible): 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. Note that there's no need to rush to this stage. The point of my system of easily pronounced names for the powers of two is to deal with the big powers of two like 2^4 (sixteen) and bigger, but for the sake of completeness, I created a set of rules for deducing the name of any power of two. That's right, my system contains an infinite number of names for for the infinite number of powers of two. The names used two count to twenty in binary are, "si" which means "one", "ti" which means "two", "ni" which means "four", "mi" which means "eight", and "ri" which, as already explained, means "sixteen".

Arithmetic in base two. Two plus two one equals four one. Another: Two plus four one equals four two one.

The same examples with the new names: Counting in base two: si, ti, ti si, ni, ni si, ni ti, ni ti si. Arithmetic in base two: ti plus ti si equals ni si. Another: ti plus ni si equals ni ti si.

The system can be learned in easy stages, and there's no need to master all of it, or even most of it to get some significant benefit. The basics could be learned in a few minutes.

Below is a detailed explanation that goes into much more depth.

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

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 bases 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 have 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 final 'consonant' sound of 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 writing the numbers 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 linguistic 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 out loud using my fingers in binary or another power of two.

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. In short, am counting in modulo ten, on my fingers.

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^-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 taught at school not usually in much depth, 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 equals tenmerlesh timtiskin one or tenmerlesh timtiskin si.