The form a + bi
A complex number is written z = a + bi, where a and b are ordinary real numbers. We call a the real part and b the imaginary part (note: the imaginary part is the real number b, not bi). When b = 0 you have a plain real number; when a = 0 you have a pure imaginary number. So the reals sit inside the complex numbers as a special case.
Adding and subtracting are easy: just combine real parts with real parts and imaginary parts with imaginary parts, exactly like collecting like terms in algebra.
(3 + 5i) + (4 - 2i) = (3 + 4) + (5 - 2)i = 7 + 3i (3 + 5i) - (4 - 2i) = (3 - 4) + (5 + 2)i = -1 + 7i
Multiplying with FOIL
To multiply two complex numbers, treat them like two binomials and use FOIL. The only new step: wherever i^2 appears, replace it with -1 and tidy up.
(3 + 2i)(4 - 5i) F: 3·4 = 12 O: 3·(-5i) = -15i I: 2i·4 = 8i L: 2i·(-5i) = -10i^2 = -10(-1) = +10 = 12 + 10 + (-15i + 8i) = 22 - 7i
The conjugate and division
The complex conjugate of z = a + bi is z-bar = a - bi: same real part, flipped sign on the imaginary part. To divide, multiply the top and bottom by the conjugate of the denominator. This is exactly rationalizing the denominator — it clears i from the bottom because (a + bi)(a - bi) is real.
(4 + i) / (2 - 3i)
multiply top & bottom by conjugate 2 + 3i:
numerator: (4 + i)(2 + 3i) = 8 + 12i + 2i + 3i^2
= 8 + 14i - 3 = 5 + 14i
denominator: (2 - 3i)(2 + 3i) = 2^2 + 3^2 = 4 + 9 = 13
= (5 + 14i)/13 = 5/13 + (14/13)i