Definition of multiplication intuition (Set theory)

Question

We define how to multiply two integers using set theory as follows [(a,b)][(c,d)] = [(ac + bd,ad + bc)] what is the intuition behind defining multiplication like this?

Answer 1

The pair $(a, b)$ (where $a$ and $b$ are both positive integers is supposed to represent the integer $a - b$. (The pair $(a+1, b+1)$ is an equally good representative, as are many other pairs). We know, from our experience with numbers in everyday life, that $$ (a-c)(b-d) = ab + cd -ad - bc = (ab+cd) - (ad + bc) $$ This tells us that $(ab+cd, ad+bc)$ would be a good representation for the product...provided that both entries of the ordered pair are positive integers. Fortunately, they are in fact both positive, so we're good.

(Your book may specify "nonnegative" rather than "positive" ... both can be made to work).

You might want to ask yourself "What's wrong with choosing $(ab-ad, cd - cb)$ as the representation?" as a way to verify that you understand what's going on here.