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.
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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 $1times b + 0 = b$.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. $begingroup$ @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).
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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'. $endgroup$–Jul 5 '12 at 10:06.
$begingroup$ 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.
$endgroup$–Jul 5 '12 at 14:33. $begingroup$ @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?
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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.) $endgroup$–Jul 5 '12 at 15:53.
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 2binary1-0 is the notation for the number 8octal1-0 is the notation for the number 12radix121-0 is the notation for the number 13radix131-0 is the notation for the number 14radix141-0 is the notation for the number 15radix151-0 is the notation for the number 16hexadecSo, 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. 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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