How you you class all positive, non-zero numbers? (real, natural, whole, rational, etc)?

You pretty much already have your answer; you would call them positive real numbers. It's absolutely standard terminology. One writes R⁺ to indicate the set of positive real numbers.

Positive implies non-zero. Zero is not positive anyway. There is no reason to say "positive" and then add "non-zero." If, for some reason, your students are confused, you should not add redundant terminology, you should teach them that zero is not positive.

Whole, rational, etc. are not helpful. The area of a circle is frequently not rational or whole or anything of the sort.

So really, it's just "positive real numbers."

Please note, however, that for objects that are not geometrically "easy" it is not so simple as "multiply two nonzero numbers to get the area." Perhaps for your purposes it is, but you know, in general it isn't.

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Response to additional information:

First, please review the definitions of "positive "and "even":

x is positive means x>0. So 0 is not positive. No number is greater than itself. Obviously.

x is even if x/2 is an integer. You seem to think that because 0 is even, it is positive. This is not true (you are a math teacher??).

You also don't seem to grasp that, first of all, an object CAN have zero area (a line has no area). You should probably tell your students that areas are non-negative (positive or zero) as opposed just positive.

But area is a measure of a set (a shape is a set of points contained in some specific place), and a measure is defined to be non-negative. There is no such thing as negative area.

And none of this has anything to do with "leaving the Euclidean universe." These things are true in general, whether or not we restrict ourselves to the real plane.

Related to area, but not the same, is the notion of a "signed measure" which is essentially a way to measure area and determine whether it is positive or negative. Integrals are more or less a rudimentary example of a signed measure. But areas do not have this property and are never negative.

Perhaps your math book assumes you remember this from your last course. Did you check the glossary in the back of the book? Did you look up or google these words? Oh well ..... Natural or counting numbers ate the ones that we count with (1,2,3,4, etc) Whoe numbers are the natural numbers AND 0 (0,1,2,3, etc) Integers are (. . . , -3,-2,-1,0,1,2,3, . . ) Rational numbers are all the numbers that can be written as the quotient of two integers OR as either a repeating or terminating decimal. That would include all those listed above as well as all the fractions like 3/4, 5/7, etc Irrational numbers are those that cannot be written that way. Sqrt3, for instance Real numbers include all those listed so far Imaginary numbers are those that are square roots of negative numbers. Like 3i, which is the square root of -9.

There is no simple term. The only set that has positive numbers only is the natural numbers, and they are whole numbers.

The only way I can think of to say it is "all real numbers bigger than 0".

All positive numbers are real numbers. If you use the term positive, then non-zero is somewhat redundant.

Natural does not include the fractions and decimals, so you just have to say all positive, non-zero numbers.

natural numbers a.k.a. counting numbers

natural