Simon X. answered • 03/18/20
Higher Level Math Tutor
Tricky question, Blue.
I'm going to assume we're working in a commutative ring with unity, where we define 1 as the multiplicative identity. We define two as the identity added to itself, and we want to know what two multiplied by two is, of course, working in our ring with arbitrary binary operations plus and times. So since the ring is commutative, we can simply apply the distributive law in any order to obtain
(1 + 1)*(1 + 1)
=(1)*(1+1) + (1)*(1+1) by the distributive law
=((1)*(1) + (1)*(1)) + ((1)*(1) + (1)*(1)) by two more applications of the distributive law
=(1+1)+(1+1) since 1 is the multiplicative identity.
=(2)+(1+1) by definition
=(2+1)+1 since addition is associative
=(3)+1 defining 3 as 2+1
=4 (defining 4 as 3+1).
So the answer is 4. (Don't take this seriously please).
Hope this helped,
Simon
