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Every base is base 10

It's a hilarious witty joke that points out how every base is '$10$' in its base. Like,

\begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align}

My question is if whoever invented the decimal system had chosen $9$ numbers or $11$, or whatever, would this still be applicable? I am confused - Is $10$ a special number which we had chosen several centuries ago or am I missing a point?

Christian Chapman
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Shubham
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    I guess there's a connection between the number of fingers in our hands and the chosen base - I speculate our fingers were also the first portable counting device. This question reminds me also from [this book](http://www.amazon.com/Number-Language-Science-Tobias-Dantzig/dp/0452288118). – Red Banana Jul 05 '12 at 05:45
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    Other number systems have been used in different civilisations. The most famous examples are the Babylonians with base 60, Mayans and Aztecs with base 20. – roninpro Jul 05 '12 at 05:49
  • @roninpro: Even if you have 60 symbols in base 60. 60 == 10 (base 60), provided the use similar symbols and conventions of positional umber system. – Shubham Jul 05 '12 at 05:52
  • Also read [this](http://en.wikipedia.org/wiki/Decimal) – Red Banana Jul 05 '12 at 05:52
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    Oh, he's asking about this phenomena: (3 base 3 = 10), (4 base 4 = 10), (5 base 5 = 10)... – Red Banana Jul 05 '12 at 05:54
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    Under normal conventions, no matter what base you are using you would say "I use base 10" if talking in the same base. You just would say out loud "one zero", and it would be very different from "ten". – Alex Becker Jul 05 '12 at 05:55
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    [Quite apropos...](http://math.stackexchange.com/questions/8734) – J. M. ain't a mathematician Jul 05 '12 at 05:57
  • @GustavoBandeira: Exactly and as glujac explains if we had been using base 16 or 20 we still be asking the same question. – Shubham Jul 05 '12 at 06:01
  • @J.M. He's asking another question. – Red Banana Jul 05 '12 at 06:13
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    @Gustavo: and that's why I said "apropos". (Also, that's why it's a comment, not an answer.) – J. M. ain't a mathematician Jul 05 '12 at 06:16
  • @J.M. Yup. Sorry, I've speculated on words meaning, but I was wrong. – Red Banana Jul 05 '12 at 06:19
  • @roninpro The most interesting one is the civilization that used base 9 - they counting with the gaps between their fingers, rather than their fingers. – Ragib Zaman Jul 05 '12 at 06:29
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    @Ragib: There are only 8 gaps between fingers, four on each hand... – BlueRaja - Danny Pflughoeft Jul 05 '12 at 06:36
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    @BlueRaja-DannyPflughoeft Indeed, "The Yuki people had a quaternary (4-based) counting system, based on counting the spaces between the fingers, rather than the fingers themselves". I will go and revise how to count now. – Ragib Zaman Jul 05 '12 at 06:45
  • And a base-9 system is so much more elegant. Oh well. Say, why assume that whoever invented the decimal system was a guy? – Mr Lister Jul 05 '12 at 14:51
  • @MrLister I rather like base-12. Count finger-joints, using the thumb to point. This way of finger-counting, though people using it still usually thing in base-10, is common on the Indian Subcontinent. Use both hands, and you can have base-24. – Darael Jul 05 '12 at 16:37
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    There are 10 types of people in the world - those who understand binary jokes and those who don't. – JohnL Jul 05 '12 at 16:42
  • It is a lateral question for linguistics: for what languages the word "ten" means also "full" or "completed thing"? For such languages, indeed this word could work in the joke, as 10 means just that, ONE complection of the base number. – arivero Jul 05 '12 at 17:24
  • @JohnL: But the latter one is more applicable. :P – Shubham Jul 05 '12 at 18:00
  • the interesting point is that whatever the base you're using is, you can replace the character and invent something even better. Of course those wont be available in utf-8 but still... For those who want to create a new... Myst (the game) – Nicolas Manzini Jul 05 '12 at 20:40
  • This may be related... http://math.stackexchange.com/q/65760/16332 – AlbeyAmakiir Jul 05 '12 at 22:49
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    No, 3 is a magic number. – andrerpena Jul 06 '12 at 17:39
  • Given that you didn't understand the joke (wow, that sounds mean, but I don't mean it in a negative way), how did you find it to be a "hilarious witty joke"? – Nikolaj-K May 14 '13 at 20:10
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    @JohnL: There are 10 types of people, those 2 mentioned by you & the 10th type, who did not expect a base 3 joke coming... :P – anishsane May 06 '14 at 09:08
  • the only magic numbers are 0 and 1. all others are derived from these. – Chapuller Jul 31 '15 at 06:48
  • FYI You denote base x of k as $k_x$ –  Nov 11 '15 at 05:03
  • It's how digits work: `0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z + /` is the digit sequence (Base64 is the largest I know). $x_x$ is always equal to $10_x$ because $(x-1)_x$ is the largest digit supported on a base. Then, the tens digit is used. In the sequence, `0` is the first digit, and `1` is the second, so $08_{9}$ is the same as $10_8$ (Base 9 chosen to support $08$). It goes like this for base 3: `0` `0` --- `0` `1` --- `0` `2` --- `1` `0` --- etc. It's just that $x-1$ is the maximum digit in a base $x$. – EKons Jun 08 '16 at 17:28
  • @anishsane there are 2 types of people in the world: 1) Those that can extrapolate from incomplete information. – MercyBeaucou Mar 17 '17 at 16:28
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    There are A type of people in the world; One who knows "single digit representation of a 64 radix numbering system", and nine who don't have any idea what this joke is about. – j0h Mar 23 '17 at 22:34

10 Answers10

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Short answer: your confusion about whether ten is special may come from reading aloud "Every base is base 10" as "Every base is base ten" — this is wrong; not every base is base ten, only base ten is base ten. It is a joke that works better in writing. If you want to read it aloud, you should read it as "Every base is base one-zero".

You must distinguish between numbers and representations. A pile of rocks has some number of rocks; this number does not depend on what base you use. A representation is a string of symbols, like "10", and depends on the base. There are "four" rocks in the cartoon, whatever the base may be. (Well, the word "four" may vary with language, but the number is the same.) But the representation of this number "four" may be "4" or "10" or "11" or "100" depending on what base is used.

The number "ten" — the number of dots in ".........." — is not mathematically special. In different bases it has different representations: in base ten it is "10", in base six it is "14", etc.

The representation "10" (one-zero) is special: whatever your base is, this representation denotes that number. For base $b$, the representation "10" means $1\times b + 0 = b$.

When we consider the base ten that we normally use, then "ten" is by definition the base for this particular representation, so it is in that sense "special" for this representation. But this is only an artefact of the base ten representation. If we were using the base six representation, then the representation "10" would correspond to the number six, so six would be special in that sense, for that representation.

ShreevatsaR
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    4 could also be represented as: 1111 (base == 1). – colemik Jul 05 '12 at 08:32
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    @trismarck: Yes I thought of that, so I decided not to make the claim that the list of possible representations was exhaustive. :-) "Base 1" (unary) is also weird, as it is not a positional number system: whereas for other bases $b$ there are $b$ symbols usually denoted $0$ to $b-1$, for unary it is unnatural to use just the digit $0$ (writing $4$ as $0000$ is just weird). A "base 1" representation is not like representation in bases of larger integers. There are also other representations of four of course, such as "IV". – ShreevatsaR Jul 05 '12 at 10:06
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    Isn't the Roman numerical system some sort of a weighted numerical system? - weighted == positional but the base is not constant for every position? ;) (I'm clearly going to the wrong side of this). Additionally, to denote 4, the author could also use a negative base system (or even something more complicated than that). – colemik Jul 05 '12 at 10:43
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    Nice, except that the reason the joke works at all is because the alien doesn't know what "four" is; he doesn't have the concept of a single digit representing that number. So, the number we call "four", he calls "ten" (or his language's equivalent). It would be the equivalent of us meeting an alien race that counted in base-100. Their representation of one hundred would be "10" and they would likely have a simple word like "ten" to describe it. We, on the other hand, would have no concept of whatever words they used between "nine" and "ten" to name the other 90 numbers with a single digit. – KeithS Jul 05 '12 at 14:33
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    @KeithS: Part of what you say is right, but not sure why you say that the alien doesn't know what "four" is: do you say we don't know what "ten" is? There is nothing about the word "ten" that indicates that it occupies two digits in our usual notation. And we do have simple words like "dozen", "score", "hundred", "thousand" and so on, for numbers that are more than one digit long. (In fact many of the world's languages' words for large numbers are older than place-value notation or even writing, which is another illustration that numbers exist independent of their decimal representations.) – ShreevatsaR Jul 05 '12 at 15:53
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    When I said the alien does not know what "four" is, I mean he does not have a word, or even a concept, for whole numbers between 3 and 10 as written in what we would call base 4. He knows what "ten" is, but it's different than what we call "ten"; we call his "ten" "four", and he'd call our "ten" "twenty-two". Again, I refer you to the reverse situation; if we encountered a race that counted in what we'd call base 100, their "ten" is our "one hundred", and they would have a name for the quantity we call "ten" that we would not understand. – KeithS Jul 05 '12 at 16:18
  • All of this is prefaced on the posit that language develops to name numbers based on the counting system. Our names for numbers are based on groups of ten which is the base of our counting system; if our system were base-16, we'd have names for what we now call "ten", "eleven", "twelve", "thirteen", "fourteen", "fifteen" and "sixteen" that are no longer based on base-10; they'd be like "five", "six" and "seven" as we know them, and we'd use them in compound and multi-word number names. As it is we don't have these words and would be confused if someone who did told us we used base "neno". – KeithS Jul 05 '12 at 16:48
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    This is a good answer but I'm surprised that it doesn't mention hands and fingers. Isn't that the real reason that base 10 is commonly used for everyday stuff? – regularmike Jul 05 '12 at 20:16
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    [Pulling a Batman again](http://math.stackexchange.com/a/54568/5363), eh? :) – t.b. Jul 05 '12 at 22:29
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    @regularmike: Yes, our use of base 10 certainly is related to our having 10 fingers, but this is well-known and not the real question of the OP (as I interpreted it) — it is already implicitly assumed as shared background by the comic; the human has ten fingers and uses base ten; the alien has four fingers and uses base four. – ShreevatsaR Jul 06 '12 at 03:42
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    The alien would still need to know what 'one' and 'zero' is, there's no reason to assume he would use the same words or symbols for these. He's probably using base ☺┼ – MatsT Jul 06 '12 at 09:51
  • @ShreevatsaR: you are always allowed to use 1 and 0 for any base. So in unary, the number 101 is 2 (base ten). There are strange bases that you can represent that way, like 0.5 (which looks a lot like binary, but you move the point one digit left and then flip). – dhasenan Jul 06 '12 at 13:55
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    @dhasenan: See http://en.wikipedia.org/wiki/Unary_numeral_system -- "base 1" usually means unary, a system of representing 'n' with 'n' copies of a symbol. It's different from $b \neq 1$, and in particular different from $b = 0.5$ or $b = \phi$, etc. (See http://en.wikipedia.org/wiki/Category:Non-standard_positional_numeral_systems for other interesting representations.) – ShreevatsaR Jul 07 '12 at 06:01
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    @ShreevatsaR yup, I read that too fast. Thanks. – regularmike Jul 10 '12 at 19:22
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    I'm writing it as "base ten" from now on. I'll never write it as "base 10" again. [This changes everything](http://troll.me/images/conspiracy-keanu/this-changes-everything.jpg). – rurouniwallace Aug 06 '13 at 18:07
  • @colemik: Base 1 would not include a digit `1` just as Base 10 does not include a digit `10`. – Jo So Aug 16 '14 at 00:10
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    @ShreevatsaR: Actually `0000` is equal to `0` (not 4 or whatsoever) in any base and there is no meaningful Base 1. Our usual number systems are the polynomial rings in infinitely many variables over groups. Base 1 would be the polynomial ring over the trivial group {0}, and this ring has only a single member (000000000...) – Jo So Aug 16 '14 at 00:23
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    @JoSo: You may notice that I always used "base 1" in quotes — of course it's not one of the general class of "base n" representations, but there does exist such a thing as the unary numeral system: http://en.wikipedia.org/wiki/Unary_numeral_system. I feel I'm repeating the comment I made to user dhasanen above. In the unary numeral system, you can use any symbol you like (either 0 or 1 or X or whatever), and correspondingly 0000 or 1111 or XXXX would represent the number 4. – ShreevatsaR Aug 16 '14 at 16:51
  • Just to be sure - and as a response to @KeithS - isn't the joke just that the human says "Ten" while the alien actually says "OneZero" ? - only the reader confuses them. – BmyGuest Apr 24 '17 at 13:38
  • @KeithS Fun fact: we can pronounce every digit in base 256 with a single syllable. Just encode 16 consonants, 4 vowels, and 4 more consonants. – Will Bickford Jan 08 '18 at 07:43
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The magic of the number 10 comes from the fact that "1" is the multiplicative unit and "0" is the additive unit. The first two-digit-number in positional notation is always 10 and also always denotes the number of digits.

Dirk
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    simply put, 10 is special because it's the lowest two-digit integer in any base. – Lie Ryan Jul 06 '12 at 10:14
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    Feel like, the real question should be, "Is two a magical number?" as in "the first *two*-digit-number.. :) – N Unnikrishnan Aug 10 '15 at 11:22
  • More than "two" digit, 10 is the smallest, non-one-digit number. So nothing very special about two in this case I reckon. :) – Harsh Oct 26 '21 at 23:09
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Yes, ten ( ..... ..... ) is a special number. Not magical but special because it is a very convenient base for species that have ten fingers.

Arguably we can use hands and fingers to encode 1024 numbers using the binary system, but that would be less robust across reading directions and some configurations/gestures are physiologically hard to do.

Heiko Haller
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    Or offensive in the local culture. But even excluding both middle and ring fingers, you still have six left which can comfortably count to 64. – Potatoswatter Jul 05 '12 at 09:52
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    Whyever would you exclude something that only has a cultural offensiveness because we never used it for math position notation. That's just silly. Also, in some cultures, it's the index finger that means what in predominant Western is the middle finger. Aside from that, I came here to comment that if we use knuckle placement we can actually get that up to 2048 or more positions on just two hands ;-) – jcolebrand Jul 05 '12 at 16:58
  • Some civilizations used (and still use) base-eight (octal) because they used the spaces between fingers for counting. Other cultures used base 60 (actually, we still use it for time and geometry) by counting up to 12 on one hand (thumb pointing at a knuckle) and keep track of "iterations" on the other hand (up to 5 dozen, or 60). – David Murdoch Jul 06 '12 at 14:10
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I do not accept your concept of "1-0" as being a number.

The 1-0 you are using is a notation used on different numbers. So, as special the number 10decimal is, the notation 1-0 is not a special number.

To me, it is a special notation.

1-0 is the notation for the number 10decimal.
1-0 is the notation for the number 2binary
1-0 is the notation for the number 8octal
1-0 is the notation for the number 12radix12
1-0 is the notation for the number 13radix13
1-0 is the notation for the number 14radix14
1-0 is the notation for the number 15radix15
1-0 is the notation for the number 16hexadec

So, calling number 10dec a special number because the notation 1-0 is special would be akin to expressing the correlation

cows eat corn. cows are stupid.
Mary eats corn. And therefore, Mary is stupid.

However, you could say that the notation 1-0 denotes a number that is special within each radix. That is saying that every number is a special number in the set of all radix systems.

  • There are innumerable radix systems.
  • There are innumerable numbers.
  • A radix system is denoted by radix(n)
  • where n is a special number within the set of numbers in radix(n) because it is denoted by the notation 1-0radix(n)
  • Therefore, every number is a special number within the radix denoted by that number.
  • So is the notation 1-0-0 special, as is the notation 1-0-0-.......-0

The notation 1-9 is also a special notation, for all radix systems greater than radix(8), because it signifies the special occasion when the number mutates from 1-8 to 1-9 or from 1-A to 1-9

In fact, every notation member of the sets of all possible notations is special, by the virtue that that notation signifies a transition from a lesser value to a greater value, vice versa.

The notation A is also special notation, for all radix systems greater than radix(9). Because it signifies the transition from a numeral digit procession to an alphabetic procession.

Therefore, the number 10dec is indeed a special number not by the virtue of the notation 1-0, but by the virtue of the notation A. Because for all radix systems greater than radix(10), the value 10dec is always denoted by the special notation A. Where A is special because it is a consequence of the end of numeric digit procession into an alphabetic one.

That is like every parent in the world saying "My kid is special".

Blessed Geek
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    I like your answer. Most layman questions about numbers are about confusing number from notation or set of cyphers (so the usual phone numbers, or VAT numbers, which are not numbers at all). In this case, the guy rightly points out that the sequence of symbols for unit and zero is a clever trick. Al-Kwaritz was the inventor, as fair as I know (never read his book). – arivero Jul 05 '12 at 17:29
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    "In this case, the guy rightly points out" - Isn't frustrating that they never presumed a girl rather than a guy came out with this answer? – Blessed Geek May 11 '16 at 15:32
  • I call for the excuse of "non-native English". For instance I have problems to know if "lads" refers to men or women. – arivero May 11 '16 at 20:44
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Your comic is not talking about the number ten, it's talking about the string "10" (read that as "one-zero," not "ten"). "10" ("one-zero") only represents the integer ten in base-ten. In other bases, "10" represents a different number.

In base-nine, the string "10" would represent the integer nine (ten would be "11").

Similarly, in base-eleven, "10" would represent eleven (ten would be represented by a new symbol, traditionally "A").

The point of the comic is the fact that the string "10" in base-n always represents n. There's nothing deeper to it than that.

BlueRaja - Danny Pflughoeft
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One point you may be missing (I did initially) is that the little guy has only two fingers on each hand. Also, he miraculously speeks English, and knows how to distinguish 4 from 10, even though he doesn't know what 4 is.

Marc van Leeuwen
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The fact that humans have 10 fingers in their hands gives to the number 10 special status. Historically are used bases 20 if we count fingers of our hands and feet. Base 60 we use because the number 60 has many divisors. If we suppose that in planet Mars lives intelligent creatures with two ,,hands,, in each hand with 3 ,,fingers,, then their ,,magical,, number probably will be the number 6.

Adi Dani
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Yes, it would still be 10. The base number is always denoted by 10. If you had 11 numbers you would require eleven symbols. Since we already have 10 symbols for the first 10 numbers (0,1,..,9) you would only need one to symbolize the one we call ten. For example, in base 16, the letters A,B,C,D,E,F are used to denote 10, 11, 12, 13, 14 and 15 respectively. So:

10 = A (base 16)

11 = B (base 16)

and so on. You should check : http://en.wikipedia.org/wiki/Radix

glujac
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10 is not magic (see the other answers for the reason), but 1 and 0 are magic (or at least special) : for any number n, we have

  • 0×n=0, since 0 is the neutral element of addition, and therefore the absorbing element of multiplication
  • 1×n=n, since 1 is the neutral element of multiplication.

Therefore, 10 in basis b is always 1×b+0×1=b. Less surprisingly zero and one are always written 0 and 1, no matter the basis, and 100 always is b².

Frédéric Grosshans
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  • By the way this property is not limited to integers, but is also true for any [semiring](http://en.wikipedia.org/wiki/Semiring) where a radix notation makes sense. – Frédéric Grosshans Jul 05 '12 at 16:09
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I've always assumed it was the number of fingers on the human hand that originated the decimal system. I sometimes make people feel better about their age by saying something like, "Hey, if humans has 6 fingers on each hand you'd still be in your thirties."

Aaron Anodide
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