Question

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

Answer 1

Let A = `"a" + sqrt(5) "b"` and B = `"C" + sqrt(5)"d"`

Where a, b, c, d ∈ M.

Now A * B = `("a" + sqrt(5)"b")("c" + sqrt(5)"b")`

= `"ac" + sqrt(5)"ad" + sqrt(5)"bc" + sqrt(5)"b" sqrt(5)"d"`

= (ac + 5bd) + `sqrt(5)`(ad+ bc) ∈ A

Where a, b, c, d ∈ Z

So * is a binary operation.