Graphically Add and Subtract Complex Numbers

This video introduces the idea of adding and subtracting complex numbers using their coordinates on the complex plane.

Each complex number has a real part and an imaginary part and occupies a unique point on the complex plane. The numbers can be added or subtracted using two different methods. The first method is to write the numbers out into their respective real and imaginary parts and then use complex arithmetic to find the sum or difference. Namely, add or subtract the real parts and perform the same operation on their imaginary parts.

The second method is to add or subtract the complex numbers by representing them as vectors. Their vectors point from the origin to the coordinate point of the complex number and have both a magnitude (length) and a direction (angle from the positive real axis). If adding, place the beginning of the second vector on the end or tip of the first vector. After combining, draw a new vector from the origin to the end of the second vector. The new vector points to the sum of the two complex numbers. For subtraction, find the opposite of the second vector by looking at the opposite of its real and imaginary parts. Then add that opposite vector to the first vector.

Both methods will arrive at the same answer and can be used to check your work.

Khan Academy exercise to practice: