Defining Numerals, Numbers, and Digits

Date: 05/20/2004 at 12:04:18
From: Tim
Subject: Is it a number, numeral, or digit

What is the meaning of each of the terms "numeral", "number", 
and "digit"?  How are they different?

I have always thought that:

1. A "numeral" is one of ten Hindu-Arabic symbols used to denote a 
quantity in the context of our base-ten system; a graphemic 
representation of the abstract quantities 0,1,2,3,4,5,6,7,8,9 (e.g. 
the numeral 8 is composed of two circles, one above the other, 
slightly squashed together to touch by almost one diameter).

2. A "number" is a physical quantity of something that can be a 
spoken or written word or a symbol such as pi, root 2, or a numeral 
if it has a referent attached to it making it a "magnitude" (e.g 
there are 26 letters in the alphabet).

3. A "digit" is a numeral when contextualized by place value in the 
base ten system (e.g. 462 has three digits).

I am close to having sleepless nights about this so your help would 
be much appreciated.  Thanks, Tim  

Date: 05/20/2004 at 14:29:20
From: Doctor Vogler
Subject: Re: Is it a number, numeral, or digit

Hi Tim,

While this may seem like a math question, in truth it is not; it is an
English question, because you are not asking for any math, but only
definitions of words.  English is not as absolute a study as math is;
different people will give you different ideas as to what is correct.
 Case in point:  If you don't know the gender of a person, do you say
that something that belongs to this person is "his" or "hers" or
"theirs" or "his or hers" or what?  Different people will tell you
different things, even people who study these topics professionally.

You ask for the definitions of these three words.  I would recommend a
dictionary, but remember that even dictionaries have differences 
between them.  As for my own personal opinion, I think that any
differences between those three words is not important.  But if you 
have to make a distinction, then this is what I would advise:

Numeral is a way of representing a number.  It certainly should not be
limited to the Hindu-Arabic symbols, since this would make the phrase
"Roman numerals" meaningless.  You are referring to a distinction
between whether the "numerals" are the individual symbols that make up
a number (e.g. 0 through 9) or the collection of symbols that comprise
the whole number (such as 321.45).  I think either definition is fine,
for different situations, so I wouldn't make a fuss about it.

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.

Finally, 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 2*pi and sqrt(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 or XLV/IV or 101101/100 in binary).

But that's just my opinion, and other speakers of English might have
other ideas.

If you have any questions about this, please write back, and I will
try to offer further suggestions.

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