How to Multiply Complex Numbers

Here you will learn how to multiply complex numbers both in Cartesian form and in polar form.

Cartesian Form

In Cartesian form you multiply complex numbers together, term by term. This is done in the same manner as for multiplication of real algebraic expressions with parentheses. When you multiply complex numbers, you need to remember that the imaginary unit i has the property i2 = 1.

Formula

Multiplication in Cartesian Form

Let z1 = a + bi and z2 = c + di be complex numbers, then:

z1 z2 = (a + bi) (c + di) = ac + adi + bci + bdi2 = (ac bd) + (ad + bc)i.

The product of two complex numbers is a new complex number. Mathematically, this is formulated as the set of complex numbers being closed under multiplication.

Example 1

Find z1 z2 and z2 z3 for the complex numbers z1 = 3i, z2 = 5 + 2i and z3 = 1 i

You find z1 z2 by multiplying z1 with both terms in z2:

z1 z2 = 3i (5 + 2i) = 3i 5 + 3i 2i = 15i + 6i2 = 6 + 15i.

In order to find z2 z3, you multiply both terms in z2 with both terms in z3:

= z2 z3 = (5 + 2i) (1 i) = 5 1 + 5 (i) + 2i 1 + 2i (i) = 5 5i + 2i 2i2 = (5 + 2) + (5 + 2) i = 7 3i.

z2 z3 = (5 + 2i) (1 i) = 5 1 + 5 (i) + 2i 1 + 2i (i) = 5 5i + 2i 2i2 = (5 + 2) + (5 + 2) i = 7 3i.

Polar Form

If you write complex numbers using the complex exponential function, you can multiply complex numbers by using normal power rules.

Formula

Multiplication in Polar Form

Let z1 = r1ei𝜃1 and z2 = r2ei𝜃2 be complex numbers, then

z1 z2 = r1ei𝜃1 r 2ei𝜃2 = (r1 r2) ei𝜃1+i𝜃2 = r1r2ei(𝜃1+𝜃2).

When you multiply complex numbers in polar form, you multiply the norms and add the arguments. This can be visualized in the complex plane:

Geometric visualization of multiplication of complex numbers.

Multiplication of complex numbers can be thought of as a rotation and a scaling in the complex plane. For instance, multiplication with the imaginary unit i corresponds to a rotation of 90° or π 2 radians because i has norm 1 and argument π 2.

Example 2

Find z3 = r3ei𝜃3 = z1 z2 when z1 = 2eiπ 3 and z2 = 4eiπ 6

The norm of z3 is found by multiplying the norms of z1 and z2:

r3 = 2 4 = 8.

The argument of z3 is found by adding the arguments of z1 and z2:

𝜃3 = π 3 + π 6 = π 2.

The product z3 is then:

z3 = 8eiπ2 = 8i.

Properties of Multiplication

As with the real numbers, the commutative, associative and distributive properties for multiplication are true for complex numbers.

Rule

Commutative, Associative and Distributive Properties

For all complex numbers z1, z2 and z3, the commutative property

z1 z2 = z2 z1,

the associative property

(z1 z2) z3 = z1 (z2 z3) ,

and the distributive property

z1 (z2 + z3) = z1 z2 + z1 z3

all hold.

The commutative and the associative properties state that you can freely change the order of numbers and parentheses as long as the calculation consists only of multiplication. The distributive property states that you can multiply complex numbers into parentheses and factorize complex numbers from parentheses.

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