Difference between whole numbers and decimal numbers

Question

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.)

Answer 1

"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.

Answer 2

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.)

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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.

Answer 3

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.