Adding (3+2i) and (1-4i)

Complex Analysis · Axiom Academy

EXAMPLE Adding (3 + 2i) and (1 + 4i) Add two complex numbers by combining their real parts and their imaginary parts separately. Add the complex numbers (3 + 2i) and (1 + 4i) . Combine like terms — real with real, imaginary with imaginary — to write the sum in the form a + bi . Each complex number is an arrow in the plane: real part → horizontal, imaginary part → vertical. Lay the second arrow tip-to-tail on the first and the combined arrow lands on the sum, 4 + 6i. Nice work — you added two complex numbers by treating their real and imaginary parts as separate, like terms. Add componentwise: (a + bi) + (c + di) = (a + c) + (b + d)i — reals with reals, imaginaries with imaginaries. Here: the real parts give 3 + 1 = 4 and the imaginary parts give 2 + 4 = 6 . Result: (3 + 2i) + (1 + 4i) = 4 + 6i . Geometrically this is just tip-to-tail vector addition — i acts as a separate direction from the real axis, so the two components never mix.

This is the written version of the interactive lesson above. See the full Complex Analysis course.