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I've noticed three different approaches in general use:

  1. Numbers are abstract and numerals are used to express numbers following a particular set of rules. The number of days in two weeks is a number. 14, E, and XIV are numerals showing various ways of rendering fourteen. In this approach, numeral may also have some overlap with the individual characters that make up numerals.

  2. Numerals are nothing more than the individual characters employed to express numbers. Numbers, can be either numbers in the abstract sense, or renderings of those numbers using numerals. In this approach, 14, E, and XIV are numbers no matter what. E could be a numeral, depending on context, but 14 and XIV definitely are not numerals.

  3. The abstract notion of a number is definitely called a number. The characters used to express numbers are definitely numerals. Renderings of numbers using numerals are interchangeably called numerals and numbers.

In formal mathematics, how are the terms number and numeral understood?

I am writing some technical documentation and I want it to be correct. My personal understanding is #2, but I'm open to correction.

trw
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    I don't ever use the word numeral in mathematics. If I did, however, it would mean the symbol used to represent a number, which is understood to be either taken as some primitive (undefined) abstract notion or a well-defined abstract notion, depending on what I'm doing. But, again, I don't actually use the word numeral, I just conflate the abstract idea with its symbol and call them both number. – got it--thanks Oct 18 '14 at 20:33
  • I agree with @gotit--thanks. I do mathematics for a living and I have never had a reason to use the word 'numeral'. I would call the symbolic string (like "153") representing an abstract number simply 'number'. If I need to refer to single characters in the string, I would call them 'characters' or 'digits'. – Joonas Ilmavirta Oct 18 '14 at 20:42
  • Your points are well taken. Thank you! If you call the single characters characters, what term would you use to distinguish them from @ and ü? To me, digit has a specific meaning aside from number and numeral. 14 has two digits and two numerals, XIV has no digits and three numerals. – trw Oct 18 '14 at 20:49

1 Answers1

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Math Is Fun offers a nice distinction between digits, numerals, and numbers, with an analogy to the distinction between letters, words, and the things words represent. By that characterization, "14" and "XIV" are numerals. But as the commenters have pointed out, few mathematicians ever feel much need to use the term "numeral."

Barry Cipra
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