Multiplication of Complex Numbers

Multiplying Two Complex Numbers

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

Let's see how to multiply $z_1 = x_1 + iy_1$ by $z_2 = x_2 + iy_2$.

Remember that $i^2 = -1$, so we can substitute:

Now, let's group the real and imaginary parts:

So, the general formula for complex number multiplication is:

Calculation Example

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

Solution:

Using the distributive property:

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

The result is the same!

Exercise

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

Answer Key

Using the distributive property:

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