rationalize the complex number multiplication rule For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of  $(a,b)$ and $(c,d)$, why do we define a product of $(a,b) * (c,d) = (ac-bd, ad+bc)$?
I wanted to start here and introduce the $i$ and polar expression of complex number as a consequence.
You could explain it in terms of rotation.

Suppose you have a point $(a,b)$ in the plane. If you multiply it by a real number $s$ then you scale the length but preserve the direction, $(sa,sb)$. What if we want to multiply two vectors together in a way that multiplies their lengths together and adds their polar angles?

This would allow us to rotate points just by multiplying them... etc





Personally I feel that introducing $i$ and building from there is a more natural approach though.
The complex number (a, b) should actually be a + bi; where $i^2 = -1$.

Therefore, $(a,b) * (c,d) = … = (ac-bd) + (ad+bc)i =  (ac-bd, ad+bc)$.