Never thought about it before, but it's mildly amusing.

http://www.etymonline.com/index.php?term=integral

Integral comes from the Latin meaning "forming a whole."

Integer is from the Latin "integer" which means "whole," BUT the "whole numbers" are not the same as the Integers. Rather, they are a subset of the Integers.

N = Set of Natural numbers = {1, 2, 3, ... }
W = Whole numbers = Natural #s plus 0 = {0, 1, 2, 3, ... }
I = Integers = { ..., -3, -2, -1, 0, 1, 2, 3, ... }

Some books use "J" instead of "I" to represent the Integers. Usually, books use a typeface called "blackboard bold" (just looked it up) with letters that represent sets. I note without evidence or explanation that the meanings of the terms "whole number" and "natural number" may have changed in the last 40 or 50 years.

According to Wikipedia, the natural numbers are either the positive integers or the non-negative integers; the integers are the natural numbers including zero and the negatives of the non-zero; and the whole numbers are defined variously as the non-negative integers, the positive integers, or all integers.

That's interesting. I was aware of at least one older text that swapped the definitions of whole and natural numbers. Never seen any other use of the term Integer other than the one I gave.

But my authoritative math source agrees with wiki.
http://mathworld.wolfram.com/WholeNumber.html

I don't think there is an accepted standard.

From what I gather:
'Integers' is an all encompassing term.

negative = ...,-3,-2,-1
positive = 1,2,3,....
non negative = 0,1,2,3,...
non positive = ...,-3,-2,-1,0

Whole numbers can be either 'positive' and 'non negative'.
But by this reckoning 'non positive' is equivalent to 'negative'.
very convoluted.

Not quite. Non-positive includes zero, negative does not.

All but one of the books I've ever read use the definitions I gave; however, I vaguely recall the kids I tutor using a book that used different definitions. Seems like it was only this year or last year. I'm tempted to think that either the textbook author just didn't bother checking up before writing the book or the author is older and using a definition he learned back in ancient times.

Either way, I prefer "positive", or "non-negative", etc. Less ambiguous. And I know now when I read the terms, I need to find where the author defines them or where he has used them in a way that the meaning is clear.

not to muddy the waters, or anything (heh), but:

"Signed zero is zero with an associated sign. In ordinary arithmetic, −0 = +0 = 0. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero)."
-wiki

so, you may say that 0 is neither positive or negative (in your set theory discussions); or you may say zero is *both positive and negative (in your floating point discussions).

also, consider this statement: zero is an even integer, although it is neither positive nor negative.

As a side note, it can be shown that the rational numbers, i.e., the numbers that can be represented as the ratio of two integers, have a one-to-one correspondence to the integers, or for that matter, to the natural numbers.

Yes. If you're not using two's complement arithmetic, you end up with the bizarre situation that there are two different representation for zero. OTOH, two's complement takes care of that. I think most systems nowadays use two's complement, so this is seldom an issue. Even the IEEE floating point uses two's complement for both the mantissa and exponent. I suspect, but am not certain, that this is a big issue with legacy systems and no issue at all with current systems so long as they follow the standard.

I was thinking about discussing rational numbers next, because that issue actually came up in a discussion a few months back - and it also helps to explain the term to my tutees (or whatever you might call the students I tutor).

Yes, they are called rational numbers because they can be expressed as the ratio (division) of two integers, not because they are more sensible or useful than irrational numbers.

Typically, authors use the symbol Q (in blackboard bold) or just a bold R to represent the rational numbers. According to wiki, the Q stands for "quotient" which makes sense, though I never
considered it before. The real numbers are typically represented by a cursive R.

You're right, of course, that the cardinality (size) of the Rationals is the same as the Integers. Also interesting is the fact that although there are an infinite number of integers (and rationals); the set of irrationals is infinitely larger.

Irrelevant note. I think my kids' (tutor subjects again) books these days refer to the Integers by the symbol Z. I don't remember learning it that way and I don't understand the reason for it. I just this second wiki'd and the Z comes from the German word "Zahlen" which means "numbers."

Not quite. Non-positive includes zero, negative does not.

True that!

What I should've said was

But by this reckoning 'non positive' should be equivalent to 'negative'.