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Numbers and Numerals


Date: 11/20/98 at 11:14:53
From: Will Crall
Subject: Numbers and numerals

Dear Dr. Math,

What is the difference between numbers and numerals?

Will (6th Grade Student)


Date: 11/20/98 at 16:55:35
From: Doctor Peterson
Subject: Re: Numbers and numerals

Hi, Will. 

The difference is sort of like the difference between a person and his 
name. You are a person, and there is just one of you. But different 
people probably call you by different names. Your name isn't you, but 
it represents you.  

In the same way, a number is a thing that we talk about in math, such 
as "three," which is hard to define exactly. I might say it's the 
abstract property or "threeness" that is shared by any set of three 
things. A numeral, on the other hand, is any name or symbol for that 
number, such as "3" or "III" or "11 (binary)" or whatever animal face 
is used to represent a three in Mayan carvings. You can't see a three, 
but you can see things that can be described by the number 3, and you 
can see the numeral "3" that is used to represent it.

We have to be careful sometimes not to confuse the numeral with the 
number. For example, a numeral "12" may have two digits, but the number 
12 could be represented by numerals with other numbers of digits, or by 
numerals where the concept of digit is meaningless. We often get 
questions like "are there any numbers after a trillion?" when people 
really mean, "are there any names for numbers bigger than a trillion?"  
Similarly, the fact that no one can ever write out the numeral for the 
number we call "pi" completely doesn't mean the number doesn't exist!

On the other hand, we do often use the word "number" to mean "numeral" 
when it isn't important to make that distinction. I don't get upset if 
someone says he's writing a number, when it's really a numeral he 
wrote.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   


Date: 01/29/2002 at 04:48:01
From: Am
Subject: Numerals and numbers

What is the exact difference between numerals and numbers?


Date: 01/29/2002 at 09:07:49
From: Doctor Peterson
Subject: Re: Numerals and numbers

Hi! I did a little extra research to check up on my understanding 
here, and was a little surprised!

A number is an abstract concept; a numeral is a way to express a 
number, usually in writing. For example, the number 5 can be thought 
of as the concept of "fiveness" which all sets of five objects have in 
common; it can be expressed using numerals such as 5, V, |||||, five, 
101 (base 2), and so on.

Since dictionary writers are often able to express meaning better 
than I can, I looked up "number" and "numeral" at m-w.com (Merriam-
Webster), and found

    number: 1 a (1) : a sum of units : TOTAL (2) : COMPLEMENT 1b
    (3) : an indefinite usually large total <a number of members
    were absent> <the number of elderly is rising> (4) plural :
    a numerous group : MANY (5) : a numerical preponderance
    b (1) : the characteristic of an individual by which it is
    treated as a unit or of a collection by which it is treated
    in terms of units (2) : an ascertainable total <bugs beyond
    number>
    c (1) : a unit belonging to an abstract mathematical system
    and subject to specified laws of succession, addition, and
    multiplication; especially : NATURAL NUMBER (2) : an element
    (as Y') of any of many mathematical systems obtained by
    extension of or analogy with the natural number system
    ...
    4 a : a word, symbol, letter, or combination of symbols
    representing a number b : a numeral or combination of
    numerals or other symbols used to identify or designate
    <dialed the wrong number> 

    numeral: a conventional symbol that represents a number

I was not surprised at how hard it is to define number, or by the fact 
that one definition tells us that "number" is often used to mean 
"numeral"; I was surprised that "numeral" is defined so briefly and 
without examples to clarify, for example, whether a numeral is a 
single digit ("1") or the whole "number" ("123"). In my understanding, 
it is the latter. Here are American Heritage's definitions (from 
www.bartleby.com):

    number: 1. Mathematics a. A member of the set of positive
    integers; one of a series of symbols of unique meaning in a
    fixed order that can be derived by counting. b. A member of
    any of the further sets of mathematical objects, such as
    negative integers and real numbers. 2. numbers Arithmetic.
    3a. A symbol or word used to represent a number. b. A numeral
    or a series of numerals used for reference or identification:
    his telephone number; the apartment number. 

    numeral: 1. A symbol or mark used to represent a number. 

I like this definition of number better; but "numeral" still gets an 
inadequate definition. My old paper copy is a little better:

    numeral: 1. A symbol, such as a letter, figure, or word used
    alone or in a group to denote a number. See Arabic numeral,
    Roman numeral.

    Arabic numerals. The numerical symbols 1, 2, 3, 4, 5, 6, 7, 8,
    9, and 0.

    Roman numeral. One of the letters employed in the ancient
    Roman system of numeration...

So they are clear that a numeral is a single symbol (or word, such as 
"one"), not the whole "number." But I think it is common in teaching 
to use the word "numeral" to refer to any representation of a number 
(as I suggested above) rather than just to the individual symbols of 
which it is made, particularly since the symbols can be combined in 
different ways (as in base 10 and base 2, which use some of the same 
set of symbols). In the cuneiform system, all numbers are made up of 
"wedges" | and > ; are those the numerals, or combinations like 
>|| that were made in specific patterns? I think the "single symbol" 
definition is hard to apply in general, and for our purposes is not 
sufficient. I don't always trust lexicographers to catch the nuances 
of mathematical usages, so you can take the dictionary references with 
a grain of salt.

One more dictionary reference partially justifies my answer; here's 
what the Academic Press Dictionary of Science and Technology (listed 
in our FAQ) says for "numeral":

    a written representation of a fixed numerical quantity;
    in particular, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

So the term is used _most specifically_ of digits, but _more 
generally_ of the whole representation of any number. I like that 
view.

Here are two previous answers from our archives, which are clearer 
than what I've said, being untainted by dictionary nonsense:

   Numbers and Numerals
   http://mathforum.org/dr.math/problems/moore9.6.97.html   

   The History of Numbers and Numerals
   http://mathforum.org/dr.math/problems/jacobs12.8.98.html   

We all agree that a numeral is a representation of a number.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

    
Associated Topics:
Elementary Measurement
Elementary Number Sense/About Numbers

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