Multiplication of Complex Numbers

Distributing Products Across Real and Imaginary Terms

Multiplying two complex numbers is similar to multiplying two binomial algebraic expressions. We can use the distributive property of multiplication over addition.

Start with $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$, then distribute each term of the first factor across the second.

Because $i^2 = -1$, replace the quadratic imaginary term:

Next, group the real terms and the imaginary terms:

So, the general formula for complex number multiplication is:

Two Ways to Calculate the Same Product

Let $z_1 = 2+i$ and $z_2 = 1-2i$. Find $z_1 \times z_2$.

Solution:

Using the distributive property:

Use $i^2=-1$:

Or using the general formula with $x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2$:

Both methods give the same result, $4-3i$.

Exercise

This problem multiplies two complex numbers, one written in rectangular form and one with a fractional real part. Expand the product with the distributive law, replace $i^2$ with $-1$, and collect the real and imaginary parts.

Let $z_1 = 1+i$ and $z_2 = \frac{1}{2} - 2i$. Find $z_1 \times z_2$.

Worked Solution

Using the distributive property:

Using the general formula with $x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2$: