Good page. Helped me a lot with the geometric representation.
Those 3 basic forms
are also the basis for mental arithmetic. Lets take your example:
If you wanted to multiply 38 * 42 you could use that form with y=10.
Answer 16*100 - 4 = 1596 like magic.
For 63 * 57
63*57 = 36 * 100 -9 = 3591
With a little practice and algebra, you can do lots of arithmetic in your head.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.