Numbers: Cardinal, Ordinal, Nominal?

Date: 10/25/1999 at 07:55:36
From: Mr. Tomlinson
Subject: Math

My math class is learning about cardinal, ordinal, and nominal 
numbers. The definition given in the book tells us that cardinal 
numbers tell how many (12 shirts per box), ordinal numbers tell 
position or order (1st place, 5th in line) and nominal numbers name 
things (number on a jersey, a telephone number). 

So far so good? That's what we thought until we were asked to 
determine which group of numbers "time" would fit into. We've had a 
lively discussion here at the International School in Brussels and are 
hoping that you can help clarify our dilemma.

Respectfully,
Mr. Tomlinson

Date: 10/25/1999 at 08:13:48
From: Doctor Jerry
Subject: Re: Math

Hello Mr. Tomlinson,

Well, time would be like numbers used in measuring length, area, 
volume, mass, or other physical or geometric quantities.

     16 seconds, 12.5 centimeters, 1344.5 cubic meters.

I'm a mathematician but I've never heard of nominal numbers nor felt 
the need of a name. Cardinals or ordinals are familiar and useful.

What would pi be? It's just a real number, no other designation needed 
(except for some purposes one wants to know that pi is not an 
"algebraic number" but is a "transcendental" number).

I think about the set R of real numbers as a given object.  We can use 
them with attached units to measure something or we can associate them 
with points on a line or we can form them into pairs to model the 
Euclidean plane or into triples to model Euclidean three space or ...

Sorry that I wasn't able to give a crisp, direct answer.

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

Date: 10/25/1999 at 09:03:02
From: Mr. Tomlinson
Subject: Re: Math

Thank you for your quick reply to our last question (what kind of 
number is "time" - cardinal, ordinal or nominal) but of course, my 
class had additional thoughts on the matter.

Yes we agree that time (30 seconds, 2 minutes, etc.) could be 
considered a cardinal number but what do you do with the "12th hour" 
or the "18th century?" Wouldn't this type of time be considered an 
ordinal number? And if they are ordinal numbers, then what about 
showing up for a meeting at "12:30" p.m.?

Respectfully,
Mr. Tomlinson

P.S. The term "nominal" numbers is from the Harcourt Brace Math text 
for 5th grade. Are there nominal numbers or not? If there are, and 
they do in fact name things, then aren't you naming something  (an 
hour, for example) when you say "12 o'clock"?

Date: 10/25/1999 at 16:26:11
From: Doctor Peterson
Subject: Re: Math

Hi, Mr. Tomlinson.

Like Dr. Jerry, I've never heard of "nominal numbers"; and for that 
matter, I haven't heard "cardinal" and "ordinal" numbers, in the 
elementary sense, used in higher math. In my mind they're really more 
a matter of English than of math - the words are worth knowing, to 
describe how we use numbers in our language, but we don't really do 
anything mathematical with ordinal numbers (in this sense). You may be 
interested in reading about these two terms in the more specialized 
sense:

   http://mathworld.wolfram.com/CardinalNumber.html   

I suspect that some text writer fairly recently felt a need to respond 
to questions like yours from students, wondering whether, say, a phone 
number or a uniform number is cardinal or ordinal, and for that reason 
made up a new category, "nominal," where the number is purely 
arbitrary and has no implications of number or sequence. That makes 
some sense, though I'm not sure it really contributes anything to our 
understanding of numbers. The fact is that "cardinal" and "ordinal" 
aren't meant to cover every possible use of a number in the first 
place, so there's no real need to worry about it.

Since time does involve sequence - you can compare or subtract two 
times, which is meaningless with telephone numbers - I would have to 
say, with Dr. Jerry, that time is not a mere "nominal" number, but 
fits whatever category you use for other measurements such as height. 
You're counting hours (or feet), so it fits the meaning of "cardinal" 
(except that "cardinal" usually only applies to whole numbers, which 
can be counted discretely, rather than to real numbers and continuous 
measurements of time or distance).

In addition, there is a difference between talking about an elapsed 
time of "1:25" (one hour and twenty-five minutes) and an actual time 
like "1:25" (twenty-five minutes after one o'clock). This is similar 
to the difference between a distance or interval (5 miles) and a 
location or coordinate (milepost 5, or the 50-yard line). The latter 
are used as names of a place; but rather than calling them "nominal" 
for that reason, I might call them a variety of ordinal, since they 
mean the same thing as "5th mile." I have to admit I can see elements 
of all three categories in a form such as "mile 5" or "5 o'clock"; and 
I'm reluctant to force it into one of two or three categories when 
none of them really fits. I think it's really a waste of time to try 
to classify every application of numbers this way.

On the other hand, "12th hour" and "18th century" are clearly 
ordinals; there you are very explicitly counting a position in a 
sequence. There's nothing wrong with the fact that we can use both 
cardinal and ordinal numbers in talking about time, any more than it's 
wrong to talk of both "5 students" and "the fifth student."

Because I'm curious about this terminology, I searched the web and ran 
across a couple of references to it. It turns out that the terms 
"ordinal" and "nominal" are used in statistics, an area in which I 
have little experience; here's a site that explains that usage:

Statistical Support - University of Newcastle upon Tyne, U.K.
http://www.ncl.ac.uk/ucs/statistics/common/documentation.html   
   Follow the link to "Some Common Statistical Terms."

The meaning of these categories is a little different from what you 
are discussing; it claims only to categorize statistical variables, 
not all uses of numbers, and "nominal" variables don't even have to be 
numbers. There are four categories, "nominal," "ordinal," "interval," 
and "ratio," with increasing mathematical content in terms of the 
operations that can be applied (=, >, -, /), and roughly corresponding 
to "set," "ordered set," "group," and "field."

I also found the following lesson plan on the subject at your grade 
level, which likewise has four categories, the last two being called 
"natural" and "cardinal," which correspond quite closely to the 
statistical categories. He calls street addresses "ordinal," but 
unfortunately never mentions time. (I think he would call 1 o'clock a 
"natural" number, along with Celsius temperature.)

edu-orchard.net - Intermediate (4-6) Math Lesson Plans
What Are Numbers? (5 or 6), Fred Jacquot
http://www.edu-orchard.net/PROFESS/LESSON/MATH/MATH46/ma46fjbl.html   

In this presentation the categories make some sense, though I've never 
heard their names used in quite this way; but I think the point of it 
is not to introduce important terms that the students will ever see 
again, but to get them thinking about how numbers are used. If that's 
the purpose of your text's discussion too, maybe you can get them 
thinking even more by trying to decide together whether they have been 
given too few categories to choose from, and letting them come up with 
their own category for coordinates (times and mileposts) if they think 
it's needed. After all, math is not always a matter of following known 
rules; sometimes we have to think for ourselves and invent new 
categories or concepts by looking for patterns or parallels in need 
of a name. A discussion like this can give them a more realistic 
picture of what mathematicians (or, in this case, perhaps linguists or 
philosophers) do.

I'd be interested to hear how your text defines the terms. If they are 
anything like those in the last reference, I would call both "30 
minutes" and "1:25" cardinal numbers (or Jacquot's "natural"); I'd 
still call "20th century" ordinal, though a case could be made against 
it under these rules.

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