I am afraid I did not make myself clear. I don't mean that multiplying by irrational numbers is repeated addition. What I mean is that multiplication by irrational (or even non-integers) was defined to generalize the earlier notion of multiplying by positive integers.
Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").
That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.
Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").
That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.