Question

1. Perform the indicated operations, simplify, and write the results in the form x + iy:
(i) (3 3 4i) + ((5 + 7i)

Answer 1

Complex numbers

Definition. A complex number z is defined to be an ordered pair of real numbers (a, b) that satisfies the following condition (i) and the following laws of operations (ii) and (iii)

(i) (a, b) = (c, d) iff a = c, b = d

(ii) (a, b) + (c, d) = (a + c, b + d)

(iii) (a, b) . (c, d) = (ac - bd, ad + bc) .

Solution.

We have to add the complex numbers (3 + 3i) and (5 + 7i).

Then, (3 + 3i) + (5 + 7i)

= 3 + 3i + 5 + 7i

= 3 + 5 + 3i + 7i

= 8 + 10i

This is in the form (x + iy).

Another approach will give the same result.

(3 + 3i) = (3, 3) & (5 + 7i) = (5, 7)

Thus, (3 + 3i) + (5 + 7i)

= (3, 3) + (5, 7)

= (3 + 5, 3 + 7)

= (8, 10)

= 8 + 10i