What are the properties of multiplications with their meanings?

Answer 1

The properties of multiplication need to be considered in the context of the set over which this operation is defined.

For most number systems, multiplications is

Commutative: x*y = y*x for all x and y

Associative: (x*y)*z = x*(y*z) so that , without ambiguity the expression can be written as x*y*z for all x, y and z

Distributive property over addition or subtraction:

x*(y+z) = x*y + x*z for all x, y and z

Identity Element: There exists a unique element, denoted by 1, such that

1*x = x = x*1 for all x

Zero element: there is an element 0, such that x*0 = 0 for all x.

In some sets, an element x also has a multiplicative inverse, denoted by x-1 such that x*x-1 = x-1*x = 1 (the identity).