Complex Number Operations
Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative_ECE301Fall2008mboutin, commutative_ECE301Fall2008mboutin and distributive_ECE301Fall2008mboutin laws of algebra, together with the equation i 2 = −1:
- Addition: $ \,(a + bi) + (c + di) = (a + c) + (b + d)i $
- Subtraction: $ \,(a + bi) - (c + di) = (a - c) + (b - d)i $
- Multiplication: $ \,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i $
- Division: $ \,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,, $
where c and d are not both zero.
Examples
- Addition: $ \,(5 + 7i) + (6+4i) = (5 + 6) + (7 + 4)i = 11 + 11i $
- Subtraction: $ \,(2 + 3i) - (4 + i) = (2 - 4) + (3 - 1)i = -2 + 2i $
- Multiplication: $ \,(2 + 3i)(1 + 7i) = 2*1 + 1*3i + 2*7i + 3i*7i = -19 + 17i $
- Division: $ \,\frac{(1 + 2i)}{(2+4i)} = \frac{(1*2 + 2 * 4)}{(2^2 + 4^2)} + \frac{(2 * 2 - 1 * 4)}{(2^2 + 4^2)} = \frac{1}{2} $
