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### Numbers: Cardinal, Ordinal, Nominal?

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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
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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/
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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

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"?
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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
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).

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."

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

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/
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Associated Topics:
Elementary Calendars/Dates/Time
Elementary Definitions
Middle School Calendars/Dates/Time
Middle School Definitions