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My friends and I had a debate:

"Does the word 'ten' have a base?"

My Argument:

  • 'ten' is only 10 in base 10
  • so if i have 10 objects, counting in base 10, when I get to the end of the list, I will have 10 objects
  • in binary I would have 1010 objects, read 'one zero one zero' and not 'one thousand and ten', i.e. 'one thousand and ten' is only 'one thousand and ten' in decimal.
  • in hex I would have A objects, not 'ten'.
  • the word 'ten' only means 10 in decimal and is not a baseless number/word.
  • The only reason 'ten', X (roman numerals), A, 1010 is 10, is because we're translating those numbers to the decimal value 10 -> ten.

His Argument:

  • The word 'ten' is the name of the value 10 in decimal or 1010 in binary or A in hex, and it does not have a base associated with it.
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2 Answers 2

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Your friend is right. "Ten" is indeed a baseless number, which exists whether you write it down in digits or not.

It's worth mentioning, however, that the names we give to numbers are based on the significance of this specific one. I guess the reason for that is the number of fingers each one of us has on both their hands.

Having said that, still, your friend is right.

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I'm not sure I understand the question "does the word 'ten' have a base?" but what your friend says is correct. You can think of 'ten' mapping to $||||||||||$, and then that maps to various representations like $10$, $1010$, and $\text{A}$ depending on the base. And if the base is known the mapping can be reversed from the representation to the word.

If it bothers you to say something like "base 10" because "10" is already in "base 10" by convention, and this is pathologically recursive, then you can consider using "base ten" or "base $||||||||||$" instead.

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