## Whole numbers

Sue Van Hattum wrote in response to a recent post:

I’d like to know what you think of my ‘abuse of terminology’. I teach at a community college, and I sometimes use incorrect terms (and tell the students I’m doing so), because they feel more aligned with common sense.

To me, and to most students, the phrase “whole numbers” sounds like it refers to anything that doesn’t need fractions to represent it, and should include negative numbers. (It then, of course, would mean the same thing that the word integers does.) So I try to avoid the phrase, mostly. But I sometimes say we’ll use it with the common sense meaning, not the official math meaning.

Her comments brought up a couple of things I want to blather about.

### Official meaning

There is no such thing as an "official math meaning".  Mathematical notation has no governing authority and research mathematicians are too ornery to go along with one anyway.  There is a good reason for that attitude:  Mathematical research constantly causes us to rethink the relationship among different mathematical ideas, which can make us want to use names that show our new view of the ideas.  An excellent example of that is the evolution of the concept of "function" over the past 150 years, traced in the Wikipedia article.

What some "authorities" say about "whole number":

• MathWorld  says that "whole number" is used to mean any of these:  Any positive integer, any nonnegative integer or any integer.
• Wikipedia also allows all three meanings.
• Webster's New World dictionary (of which I have been a consultant, but they didn't ask me about whole numbers!) gives "any integer" as a second meaning.
• American Heritage Dictionary give "any integer" as the only meaning.
• Someone stole my copy of Merriam Webster.

### Common Sense Meaning

Mathematicians think about and talk any particular kind of math object using images and metaphors.  Sometimes (not very often) the name they give to a math object embodies a metaphor.  Examples:

• A complex number is usually notated using two real parameters, so it looks more complicated than a real number.
• "Rings" were originally called that because the first examples were integers (mod n) for some positive integer, and you can think of them as going around a clock showing n hours.

Unfortunately, much of the time the name of a kind of object contains a suggestive metaphor that is bad,  meaning that it suggests an erroneous picture or idea of what the object is like.

• A "group" ought to be a bunch of things.  In other words, the word ought to mean "set".
• The word "line" suggests that it ought to be a row of points.  That suggests that each point on a line ought to have one next to it.  But that's not true on the "real line"!

Sue's idea that the "common sense" meaning of "whole number" is "integer" refers, I think, to the built-in metaphor of the phrase "whole number" (unbroken number).

I urge math teachers to do these things:

• Explain to your students that the same math word or phrase can mean different things in different books.
• Convince your  students to avoid being fooled by the common-sense (metaphorical meaning) of a mathematical phrase.

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Peter says:

I don’t know if there’s any historical connection, but in German, the *only* word for integer is “ganze Zahl”, i.e., literally “whole/complete/full/intact number”.

Also, wikipedia’s talk page for integers is a fun read — among other things discussing the French terminology.

Finally, a small comment. There is such a thing as official math notation (and meaning!), just like there is such a thing as orthography.

But just like orthography is only legally binding for a very small part of writing (e.g., school exams and legal documents), so is standardized mathematical notation.

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Sue VanHattum says:

(I kept waiting for a reply where I made the comment. Since I hadn’t subscribed to your blog yet, I only saw this when I was trying to find a way to close the tab I had open and subscribed to comments.)

It’s ironic that I never looked up the phrase ‘whole numbers’. All of the lower level textbooks that I’ve used define whole numbers as starting at either 0 or 1 (I can never remember). Good to know many place one would look for a definition agree with me.

I think it’s important to start from our intuition, so new mathematical ideas have a ‘hook to hang on’. But I do also attempt to “convince my students to avoid being fooled by the common-sense (metaphorical meaning) of mathematical phrases”.

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SixWingedSeraph says:

Sorry, I intended to put a reply to your message saying I had made a post about it, but I forgot to. I have now done that in case anyone else sees it.

One of the annoying things about Word Press is that if you open Gyre&Gimble, the most recent post does not show comments. You have to click on the title before they appear.

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some guy on the street says:

With our host’s indulgence, playing with etymologies:

As it happens, “integer” is also from the latin for “whole” or “unbroken” (even “untouched”), which does also suggest not fractured; that is, not a fraction — so is the math noun “integral”, though in the rest of English it’s used as an adjective. It also amuses me that a frequent use of integrated circuits is for numerical approximations of integrals using “integer” arrithmetic, (which is really arrithmetic modulo some large power of 2).