Clearly, whole numbers specify how many elements there are in a collection while decimal numbers specify how much of a substance there is in a lump---but only after a unit of that substance has been chosen. But then whether we use a whole number or a decimal number depends on the chosen unit:

0.004 Kilometers is a whole number of meters, namely 4.

0.00004 Kilometers is a whole number of centimeters namely 4,

But it gets worse: While

0.00004 KiloDollars is a whole number of centiDollars, namely 4?

can we really say that

0.004 KiloPeople is a whole number of Peoples, namely 4,

But then what about

0.00004 KiloPeople is a whole number of centiPeople, namely 4?

Where do you draw the line between whole and decimal and how do you explain it to very raw beginning students who want to understand? (Saying that, here, 4 is really the decimal number 4.0 does not really help.)

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    $\begingroup$ I don't agree with your first sentence's use of the word "clearly." I generally use whole numbers to refer to the counting numbers (including zero) and decimal to indicate a number that is written in its base 10 representation, often with a decimal point. So, e.g., if I wanted to refer to a number like 0.5, 0.333..., 0.12345..., but not e.g. 1, 2, 3, ..., then I might refer to it as a non-integer decimal, or say a decimal that is not a whole number. $\endgroup$ May 27 '17 at 18:49
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    $\begingroup$ But, doesn't saying $4$ is the decimal number $4.0$ help? Every counting number is a decimal, but, not every decimal is a counting number. Perhaps the error is in thinking that only counting numbers can be used to count. As your examples point out, it depends what we're counting. $\endgroup$ May 27 '17 at 20:49
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    $\begingroup$ -1. Question is based on faulty assumptions. OP said in May 27 comment was willing to take out faulty assumption but has not done so. Numerous questions included; specific question is unclear. $\endgroup$ Jun 3 '17 at 5:35
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    $\begingroup$ Aside (?): decimals are not numbers, they are numeral -- a system of notation. 4 is, for example, a way to write the whole number four in decimal notation. $\endgroup$
    – user797
    Jun 3 '17 at 21:51
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    $\begingroup$ The number 4 is a real real number. Really. It is also used to count 4 things. I fail to understand this discussion. Sorry I missed your comment a week back schremmer. $\endgroup$ Jun 4 '17 at 4:56

"Counting" (leading to counting numbers) is a special case (with ambiguities) of "measuring", where the role of "the measure/unit" is more visible. Obviously (I think) the natural unit implied in "counting" situations is some relevant-atomic unit (such as "whole, operational person", rather than not-so-functional smaller-part of a person).

That is, counting implicitly measures with unit the smallest/atomic feasible/operational measure (often so universally implicit that it is beyond discussion).

A fancier analogue arises when more advanced undergrads are first exposed to the idea that infinite sums (a.k.a. "series") fall under the umbrella of "integrals", but with "counting measure"... and that discrete sets have at least one natural regular, positive Borel measure, namely, the counting measure.

  • $\begingroup$ 1. That is what I was alluding to in my opening sentence and so, of course, I agree and I like the particular tinge you are giving to it. 2. But how do you respond to the very raw beginning students who ask "Why can't we say 0.04 DekaPeople since we can say 0.04 KiloPeople? Somehow, that 0.04 DekaPeople = 0.4 People and 0.04 KiloPeople = 40 People does not help: their view is that once we operate in the decimal-metric system, there should be no recourse to extraneous considerations and things should not depend on whether the "denominator" is People or liters of milk. $\endgroup$
    – schremmer
    Jun 3 '17 at 4:57
  • $\begingroup$ @schremmer, I'd argue that without "recourse to extraneous considerations" the arithmetic still makes sense, sure, but relevance/applicability can sometimes suffer. Context matters. $\endgroup$ Jun 3 '17 at 13:37
  • $\begingroup$ Of course, context is essential as happens most of the time. These, though, are so-called developmental students and are very hard to get to take logic into consideration. But then, once they start, naturally, they get hung-up on things like that. I try to tell them that they will always be able to tell from the "denominator", to which they agree, but they still insist that "there ought to be a rule" independent of whether we are talking People of liters of milk. That is what I don't know how to answer. $\endgroup$
    – schremmer
    Jun 4 '17 at 3:17
  • $\begingroup$ @schremmer, you might tell them that not everything (even in math) can be reduced to a list of unambiguous rules. I realize there are various developmental situations, but, still, I try to assure students at all levels that they should not suspend their own critical judgement... but/and that they have a responsibility for using it, rather than just using magical thinking or invoking inexplicable "rules". $\endgroup$ Jun 4 '17 at 19:03
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    $\begingroup$ My response to a question like "Why can't we say 0.04 DekaPeople [0.4 people]" is that we certainly can say something like that. E.g. Question: What is the population density in the Falkland Islands per square kilometer? Answer: 0.26 people. link $\endgroup$ Jun 8 '17 at 2:05

Why can't we say "0.04 People" since we can say "0.04 KiloPeople"?

Some quantities (e.g., people) are discrete quantities and some (e.g., meters, dollars) are continuous quantities.

The following discussion is from here. (I've emphasized the words "natural number" and "decimal.")

Classification of quantities

A quantity is either discrete or continuous. A discrete quantity is the magnitude of a countable set (one whose elements are “mutually separated and individually distinct”). Its numeral value is a natural number (“division into a quantity less than a unit cannot be considered”) and its unit is clear at the start. An example of a discrete quantity is “three boys.”

A continuous quantity is the magnitude of a “continuum” (“a continuous entity which can be divided into any number of smaller parts” such that “any two such entities can be combined into a larger one”). Its numeral value (a decimal or a fraction) and its unit “have not been determined a priori.” An example of a continuous quantity is “three dollars.”

A continuous quantity is either extensive or intensive. The former expresses breadth or magnitude (such as area or weight); the latter expresses quality or intensity (such as density or speed). An extensive quantity has additivity: the attribute of the union of two bodies is equal to the sum of the attributes of the two bodies. An intensive quantity does not have additivity. For example, the weight of two bodies is necessarily the sum of their weights, but the speed of two bodies is not necessarily the sum of their speeds.

The text is written for mathematics educators, but it can be reworded to be more easily understood by beginners.)

My original answer (included here for context) which the OP pointed out did not address the intended question:

Some quantities, such as, say, $1/3$ liter, have decimal representations ($0.\overline{3}$ liters) but no whole number representations.

  • $\begingroup$ What does this have to do with my question? $\endgroup$
    – schremmer
    May 29 '17 at 5:03
  • $\begingroup$ Your question was "Where do you draw the line between whole and decimal and how do you explain it to very raw beginning students who want to understand?" I am proposing that you draw the line when the decimal representation does not terminate and that this example should be clear to "very raw beginning" students. $\endgroup$ May 29 '17 at 5:08
  • $\begingroup$ @The very raw beginners I am dealing with have no idea of what a decimal may represent, let alone of a decimal representation that does not terminate. Besides, 1/3 liter of milk is 1, which is a whole number that numerates the things _ denominated_ by of which it takes 3 to make a liter of milk so here is your whole number representation. In any case, that has little to do with the original question. $\endgroup$
    – schremmer
    May 29 '17 at 16:11
  • $\begingroup$ So how about $\sqrt{2}$ meters, the length of the hypotenuse of an isosceles right triangle with each leg of length $1$ meter? Would you agree that it has a decimal representation but not a whole number representation? $\endgroup$ May 30 '17 at 0:22
  • $\begingroup$ Of course but what does it have to do with the original question? You are still answering a question I never asked. The question I asked turns around: Why can't we say "0.04 People" since we can say "0.04 KiloPeople"? $\endgroup$
    – schremmer
    May 30 '17 at 6:33

I think the confusion is largely a consequence of the fact that many people find the prefixes of the metric system (kilo-, centi-, etc.) unfamiliar, and find decimals (even terminating ones) less intuitive than the "vulgar fractions" they represent.

If somebody asked me "How can 0.004 Kilometers, a decimal number, be the same as 4 meters, a whole number"? (as the OP mentions in the comments below his question), I would respond with something like this:

Are you also bothered by the fact that $1/2$ a dozen eggs, a fraction, is the same as 6 eggs, a whole number?

What would come next depends on the questioner's response. But let's assume they respond with something like: "Okay, I guess I get that. But why can I say '0.04 kilopeople' but I can't say '0.04 people'?" In that case, I would respond with:

Are you also bothered by the fact that you can boil half a dozen eggs, but you can't boil half an egg?

The point of these responses, to be clear, is not to shut down the conversation with a zinger, but rather to bring to the surface what the underlying issues are: "1 kilopeople" means the same as "1000 people", and you can have half of a thousand people in just the same way that you can have half of a dozen eggs. On the other hand you can't have $1/7$ of a thousand people, in exactly the same way that you can't have $1/7$ of a dozen eggs.

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    $\begingroup$ My problem with a question like "Why can't we say '0.04 People'", is that it seems to me like we certainly can say that. For example, it could be the population density per square kilometer in a certain region. In fact: 0.04 people actually is exactly the population density (per km^2) in the Svalbard and Jan Mayen islands of Norway. link. $\endgroup$ Jun 8 '17 at 1:53
  • $\begingroup$ @mweiss Developmental students who begin to ask questions do not like being answered with a question. They would dismiss your "Are you also bothered ..." as a "teacher trick". Later, in the discussion, of course, they would have no objection to your line of reasoning and would in fact go along with it. However, what I think their question is really about, as I commented to Paul Garrett, is: "once we operate in the decimal-metric system, there should be no recourse to extraneous considerations and things should not depend on whether the "denominator" is People or liters of milk." $\endgroup$
    – schremmer
    Jun 9 '17 at 2:39

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