>Only for rational numbers. Doesn't work for real and complex numbers.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.