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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,

2 = 10 (base 2)
8 = 10 (base 8)

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 just missing some point?

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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 5 '12 at 5:49
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 5 '12 at 5:55
Quite apropos... –  J. M. Jul 5 '12 at 5:57
@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 5 '12 at 6:45
There are 10 types of people in the world - those who understand binary jokes and those who don't. –  JohnL Jul 5 '12 at 16:42

10 Answers 10

up vote 187 down vote accepted

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.

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4 could also be represented as: 1111 (base == 1). –  colemik Jul 5 '12 at 8:32
@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 5 '12 at 10:06
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 5 '12 at 14:33
@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 5 '12 at 15:53
Pulling a Batman again, eh? :) –  t.b. Jul 5 '12 at 22:29

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.

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+1 for multiplicative/additive unit. I've understood the concept of differently based number systems, but that helps me understand it a little more deeply. –  Phil Jul 5 '12 at 14:04
simply put, 10 is special because it's the lowest two-digit integer in any base. –  Lie Ryan Jul 6 '12 at 10:14

Yes, ten ( ..... ..... ) is a special number. Not magical but special because it 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.

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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 5 '12 at 9:52
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 5 '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 6 '12 at 14:10

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".

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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 5 '12 at 17:29

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.

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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.

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You believe in talking dogs (Dilbert)? :) –  Shubham Jul 5 '12 at 8:59

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.

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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

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So '10' is a magical number but our 10 is different from alien 10. May be the alien is also amused by the fact? –  Shubham Jul 5 '12 at 5:58
Shouldn't you question the alien about his feelings? –  Vladimir Putin Jul 5 '12 at 6:01

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.

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By the way this property is not limited to integers, but is also true for any semiring where a radix notation makes sense. –  Frédéric Grosshans Jul 5 '12 at 16:09

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."

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