What is difference between the "usual multiplication" and multiplication? I have been reading books in the algebra, and I noticed that some books use terms "usual multiplication" and "usual addition".  Do they carry different meaning that multiplication and addition?  If "usual" means something, is there such things as "usual division and subtraction"?
 A: "Usual multiplication" in this sense refers to the elementary definition of multiplication which is when a number $a$ is added to itself multiple times (we'll call the frequency $b$), which is thought as repeated addition: $a \times b = \underbrace{a + \cdots + a}_{\text{$b$ times}}$. However, just "multiplication" in this sense refers to the combining of matrices, vectors or any other sort of quantities under certain rules (probably used most in abstract algebra) to get a certain product. Basically, "usual" refers to elementary. I'm sure "usual subtraction" and "usual division" do exist like "usual addition" and "usual multiplication" because they all refer to the elementary use of these operations. Do note that in abstract algebra, we only use the addition and multiplication operations because subtraction and division can be rewritten in addition and multiplication respectively.
So, the bottom line is..."usual multiplication" is elementary multiplication, which is NOT the same thing as mutliplication in abstract algebra. Same goes for addition.
