Question

Let A = {a + √5b: a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A.

Answer 1

Let A = a + √5b and B = c + √5d, where a, b, c, d ∈ M.

Now A * B ={a + √5b)(c + √5d) 

= ac + √5ad + √5bc + √5b √5d 

= (ac + 5bd) + √5(ad+ bc) ∈ A 

Where a, b, c, d ∈ Z 

So * is a binary operation.