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Just recently, I attempted to answer a question involving "whole numbers", but discovered that my long-held assumption (that they're the same as the integers), is not universal.

[In fact, it seems I owe a retraction for whenever I've snickered as a result of people claiming there are no "whole number" solutions to $a^n+b^n=c^n$ when $n>2$.]

Question: When it comes to the "whole numbers", who uses which definition?

I'm thinking geographically. E.g., since I've always equated whole numbers and integers, perhaps it's an Australian thing (or perhaps it's a "me" thing).

Question: Are there any rules-of-thumb as to who uses what definition?

This matters, because when I'm teaching, I sometimes say "whole numbers" while meaning "integers", and expect others to arrive at the same conclusions as me.

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In my middle school and high school, the whole numbers were defined as the positive integers. –  Alex Becker Apr 29 '12 at 23:11
In Argentina we call whole numbers (numeros enteros) to $n \in \Bbb N \wedge n>0$. Also, why the downvote folks? –  Pedro Tamaroff Apr 29 '12 at 23:13
I think that Alex Becker’s experience is very common in the U.S. I may have run into a few people who were taught in grade school to include $0$, but that seems to be less common. I would NEVER use the term in a post-secondary setting. –  Brian M. Scott Apr 29 '12 at 23:23
In India, where I did my schooling, natural numbers meant $\{1,2,3,\ldots\}$ whereas whole numbers meant $\{0,1,2,3,\ldots\}$. –  user17762 Apr 29 '12 at 23:28

3 Answers 3

up vote 5 down vote accepted

The Wikipedia page claims that "whole numbers" can refer to the integers, or the nonnegative integers, or the strictly positive integers. A hidden comment gives a few examples of each usage, which I reproduce below:

Whole number as nonnegative integer:

Whole number as positive integer:

  • The Math Forum, in explaining perfect numbers, describes whole number as "an integer greater than zero".
  • Eric W. Weisstein. "Whole Number." From MathWorld—A Wolfram Web Resource. (Weisstein's primary definition is as positive integer. However, he acknowledges other definitions of whole number, and is the source of the reference to Bourbaki and Halmos above.)

Whole number as integer:

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From Wikipedia, you have that

Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.

Whole numbers may refer to:

  1. Natural numbers in sense (1, 2, 3, ...) — the positive integers
  2. Natural numbers in sense (0, 1, 2, 3, ...) — the non-negative integers

So it seems it is rather not good to use such terminology and rather stick to the more descriptive positive/negative/non-negative integers, naturals, etc.

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Almost nobody uses the terminology "whole numbers." Usually, people refer to {0, 1, 2, ...} as the natural numbers (and sometimes they don't include 0). In general, whatever text you're using will provide a definition and that will be consistent within the text but not necessarily outside it.

This is the case with many mathematical definition--internal consistency within texts, but no guarantee of universal adoption. Generally, however, the definitions of a given concept is similar enough across texts that the same methods of proof apply regardless.

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The first sentence isn’t really true: the term is quite common in grade school. –  Brian M. Scott Apr 29 '12 at 23:18
@BrianM.Scott Indeed. Even more to contrast them with fractional numbers, when integer can't really convey this. –  Pedro Tamaroff Apr 29 '12 at 23:19
@PeterTamaroff Why not just say positive integers? –  Pacerier May 11 '12 at 16:19
@Pacerier Those are the natural numbers. I don't follow. –  Pedro Tamaroff May 11 '12 at 21:15
@Pacerier Because for a kid "whole" makes more sense, it conveys the idea that we can think of $4$ or $1$ as a whole entity, such as 4 cakes, 1 cake, while we use fractions to convey the partitioned whole. –  Pedro Tamaroff May 12 '12 at 13:57

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