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/
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