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.