How do numbers differ from numerals?

Question

Tags numbers and numerals appear to cover exactly the same ground: names of numbers, which Wikipedia describes in "Numeral (linguistics)". Both are distinct from grammatical-number, which is about inflection of nouns, verbs, and adjectives for a generally small set of numbers. Is there a difference between numbers and numerals, or would it be a good idea to make them synonyms?

Answer 1

• "Numbers" are ideas

• "Numerals" are the written forms of those ideas, and

• "Digits" are the individual characters used to display those written forms.

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According to the Math Is Fun page, "Numbers, Numerals, and Digits", the distinctions are as follows:

• Number – "a count or measurement, that is really an idea in our minds."

  

• Numeral – "a symbol or name that stands for a number."

• Digit – "A digit is a single symbol used to make numerals."

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^∗ All images taken from article cited.

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EDIT 1

Per @curiousdannii's request, here are some more sources:

1. Oxford Dictionaries Online (ODO)

   • Number:

     An arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations

   • Numeral:

     A figure, symbol, or group of figures or symbols denoting a number.

     A word expressing a number.

   • Digit:

     Any of the numerals from 0 to 9, especially when forming part of a number.

2. Merriam-Webster Online (M-W Online)

   • Number:

     a word or symbol (such as “five” or “16”) that represents a specific amount or quantity

     a number or a set of numbers and other symbols that is used to identify a person or thing

   • Numeral:

     a conventional symbol that represents a number

   • Digit:

     a written symbol for any of the numbers 0 to 9

3. Oxford English Dictionary (OED)

   • Number:

     An abstract entity representing a quantity, used to express how many things are being referred to, or how much there is of some thing or property; an arithmetical value corresponding to a particular quantity of something. Also: an analogous entity or value used in mathematical operations without reference to actual things.

   • Numeral:

     A word denoting or expressing a number.

     A figure or character (or a group of these) denoting a number. Also used reductively of a person.

   • Digit:

     A whole number less than ten; any of the nine or (including zero) ten Arabic numerals representing these, a series of which is used to represent other numbers in decimal notation. Cf. article n. 9.

4. Encyclopædia Britannica (EB)

   • Number:

     any of the positive or negative integers, or any of the set of all real or complex numbers ...

   • Numeral:

     any of various sets of symbols ... to represent numbers

   • Digit:

     ... numbers from the group 0, 1, ..., 𝑏−1 [where 𝑏 is the base.]

5. Clark, John R.. “Number, Numeral, and Operation”. The Arithmetic Teacher 7.5 (1960): 222–230. Web... [Paper]^(1,2)

   • Number:

     Number is an idea, an abstraction.

   • Numeral:

     A numeral is a symbol which, by agreement, represents a number.

   • Digit:

     The ten number symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 ...

6. Vogler. "Defining Numerals, Numbers, and Digits", Ask Dr. Math, The Math Forum, Drexel University²

   • Number:

     ... number is much more general, and refers to a concept rather than a way of writing. Integers are numbers, but so are complex numbers, as well as 𝑒 and 2π and √7. And ⁴⁵߼₄ is a number, and it is the same number no matter what numerals you might use to describe that number (such as 11.25 [in decimal] or XLV/IV [in Roman numerals] or 101101/100 in binary).

   • Numeral:

     Numeral is a way of representing a number.

   • Digit:

     Digit generally refers to a part of a numeral, but I don't think that it has to be contextualized by place value, and it certainly doesn't have to be in base 10. For example, I've heard of questions like "What is the ones digit of this number when written in base 64?" In that case, the "digit" can be any number from 0 to 63. On the other hand, when solving certain types of problems, I might say that the variable 𝑎 is a "digit," meaning an integer from 0 to 9 (when the base 10 is understood), and in this case it doesn't have a place value.

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Footnotes

1. Account registration required to view full article. (Registration is free.)

2. As the caveat on the 1^st page says,

   This paper deals with the writer's notion of the nature of arithmetic,ߞwith the concepts and principles which determine the meaning and rationale of arithmetic.

As such, this should not be treated as purely objective fact. However, not only are the given definitions supported by the other sources I've provided, but the author includes the rationale behind their defining the terms that way. Take it as you will.

Answer 2

In my mind, numerals are only numbers as written in digits (0, 1, 2 etc), whereas numbers would refer to spoken numbers or numbers written in words (one etc). Having distinct tags seems okay to me. (Note I haven't checked if they are used in this way.)