10 vs Ten

What does “ten” mean?

Here are some dictionary definitions:

  1. The number 10. (MacMillan)
  2. The cardinal number equal to 9 + 1. (American Heritage)
  3. Equivalent to the product of five and two; one more than nine; 10. (Oxford)

Superficially, these seem like comparably valid definitions: Ten is the number that comes after nine, that is, 10.

This is a place where natural language conflicts with mathematical rigor. The mathematical value of 10 is not controversial: It is \(b^1\), where \(b\) is the numeric base in which the number is written. Since the default base of mathematics is decimal, 10 generally represents this many objects: oooooooooo.

However, there are other numeric ways to represent that number of objects. Going by integer bases less than decimal, for instance, we have 10102, 2013, 224, 205, 146, 137, 128, and 119. All of these are mathematically valid representations for that number of objects.

In base 6, for instance, 14 represents the number of objects defined by American Heritage as “ten”. Meanwhile, 106 represents this number of objects: oooooo. So MacMillan’s definition and American Heritage’s clash: Is “ten” the English equivalent of 10, regardless of base, or does “ten” represent a number of objects?

Let’s look at this another way. Say there is a culture that, for whatever reason, counts in base 4. Here are their first twenty numbers:

Ugal, dungal, trifo, mandi, mandugal, mandungal, mandrifo, dungmandi, dungmandugal, dungmandungal, dungmandrifo, trifmandi, trifmandugal, trifmandungal, trifmandrifo, yurdal, yurdugal, yurdungal, yurdrifo, yurdamandi.

They may well write these, using their own numerals, as 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110.

When we say “there are ten things there”, we want to be as unambiguous as possible. When dealing with this culture, I don’t think we’d want to leave it open to interpretation that “ten” could refer to oooo or to oooooooooo, depending on the cultural and mathematical savvy of the speaker. Because we happen to come from a language where decimal is the prevailing base, “ten” in English should, by default at least, correspond to “dungmandungal” (that is, 22).

Furthermore, there are some words in our language and related ones that reflect a different base system. In English, we have several special words for two (such as “pair” and “couple”) and twelve (“dozen”, and “gross” for 122); French has remnants of a base twenty system.

Unfortunately, this means that we don’t have ready words for number places in bases other than decimal. Properly speaking, if “ten” means 10 in decimal but 12 in octal, for instance, then 10 in octal is “eight”, while 100 is “sixty four” and 1000 is “five hundred and twelve” (or “five hundred twelve”, but therein lies another rant). There’s no pattern.

We could, of course, make terms up, but given the rarity with which people outside of computer programming and mathematics, and even indeed within those fields, speak out loud about non-decimal numbers, I’m not sure if there’s a strong value in doing so.

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