Is the product of two complex numbers unique?

Question

I mean when you multiply two given complex numbers, their answer is only one. Unique in a sense that the answer can only be one and no other answer is possible.

Answer 1

Depends on what you mean by unique. If I take 2 x 3, that always equals 6. But so does 1 x 6, so it's not a unique result for all numbers.

Similarly complex numbers always multiply to a specific number, but other pairs of complex numbers can also end up with that result.

Answer 2

Yes.

(a + bi)(c + di) = (ac - bd) + (ad + bc)i. This gives an exact method for computing the product of two complex numbers. No other complex number is equal in value to this product.

Answer 3

Yes

let them be a+bi and c+di

product = (a+bi)(c+di)
= ac + adi + bci - db
= (ac-bd) + i(ad+bc)

which is unique for unique pair