Multiplication on the Complex Plane

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we review two ways to locate a point on a plane. Then, using complex numbers, one of these two ways is used to multiply numbers on the complex plane.

Multiplication on the Complex Plane

Think about the days before we had Smartphones and GPS. If you had to describe where you were to a friend, you might have made reference to an intersection. Usually, the intersection is the crossing of two streets. So you might have said, ''I am at the crossing of Main and Elm.''

In this lesson we review this idea of the crossing of two lines to locate a point on the plane. Another approach uses a radius and an angle. Then, we naturally extend these ideas to the complex plane and show how to multiply two complex numbers.

Locating a Point on the Plane

A plane has two dimensions. To locate a point on the plane, we need two numbers. We can use the Cartesian form of (x, y). The horizontal axis is usually labeled the x-axis while the vertical axis, is usually labeled the y-axis. For example:


A point located at (-4, 3)
A_point_located_at_(-4, 3)


In this Cartesian form, the point is at the intersection of the line x = -4 and the line y = 3.

We can also locate this same point using the polar form of (r, θ). The radius, r, is a line from the origin, (0, 0), to the point while the angle, θ, is a rotation measured counter-clockwise with respect to the positive x-axis. In polar form:


The same point at (5, 143.13 degrees)
The_same_point_at_(5, 143.13_degrees)


In this polar form, r = 5 and θ = 143.13o.

To convert from one form to the other we use:


r=sqrt(x^2+y^2),theta=atan(y/x),x=rcos_theta,y=rsin_theta


For example, for r = 5 and θ = 143.13o, to find x:


x=rcos_theta=-4


Complex Numbers and the Complex Plane

Complex numbers also have two parts called real and imaginary. The complex plane is analogous to the x-y plane. The horizontal axis is labeled ''Real'' while the vertical axis is usually labeled ''Imag'' (short for imaginary).

The Cartesian form of a complex number is a + bi where a is the real part and b is the imaginary part. The polar form of a complex number is reiθ. The conversion formulas to go from one form to the other:


r=sqrt(a^2+b^2),theta=atan(b/a),a=rcos_theta,b=rsin_theta


This is all very similar to the usual two-dimensional plane, but what does this have to do with multiplying complex numbers? Well, a lot really. When adding or subtracting complex numbers, the Cartesian form is ideal. For multiplication or division, the polar form is preferable.

Let's multiply two complex numbers which are already in polar form: 2ei30o and 3ei45o.

The product is 6ei75o. All we do is multiply the radii, 2 times 3, and add the angles, 30 + 45:


(2e^i30)(3e^i45)=2(3)e^i(30+45)=6e^i75


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