I'm sure there are more elegant approaches, but I would start like this:
A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations.
It is not well defined and should not be used as basis for other definitions.
"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.
Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.
The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.
The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.
This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.
I understand very well that the peano axioms for example are logical and not circular.
My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).
So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.
Comments
I'm sure there are more elegant approaches, but I would start like this:
A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations. It is not well defined and should not be used as basis for other definitions.
"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.
Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.
The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.
The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.
This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.
TBH this is the worst explanation of anything I have ever seen.
and now the time has come for Intro to Complex Analysis, good luck not leaving everyone completely dumbfounded.
Also, a natural number is not “imaginary” in any sense. (E.g. you do not imagine the number of fingers on your hand - you discover it, by counting.)
Personally I find these definitions a bit circular.
Equations like "a + x = b", "ax = b" and "sin(x) = 0" have no meaning unless you have an understanding of what the terms mean, or could mean.
You can easily define + on natural numbers and * on integer numbers without running into problems! Thus there is no cirularity here.
I understand very well that the peano axioms for example are logical and not circular.
My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).
So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.