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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 $i^{2} = - 1$ .

Formula

Multiplication in Cartesian Form

Let

z_{1} = a + b i

and

z_{2} = c + d i

be complex numbers, then:

\begin{array}{l} z_{1} \cdot z_{2} & = (a + b i) (c + d i) \\ = a c + a d i + b c i + b d i^{2} \\ = (a c - b d) + (a d + b c) i . \end{array}

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

Example 1

Find $z_{1} \cdot z_{2}$ and $z_{2} \cdot z_{3}$ for the complex numbers $z_{1} = 3 i$ , $z_{2} = 5 + 2 i$ and $z_{3} = 1 - i$

You find $z_{1} \cdot z_{2}$ by multiplying $z_{1}$ with both terms in $z_{2}$ : $\begin{array}{l} z_{1} \cdot z_{2} & = 3 i \cdot (5 + 2 i) \\ = 3 i \cdot 5 + 3 i \cdot 2 i \\ = 15 i + 6 i^{2} \\ = - 6 + 15 i . \end{array}$

In order to find $z_{2} \cdot z_{3}$ , you multiply both terms in $z_{2}$ with both terms in $z_{3}$ :

\begin{array}{l} z_{2} \cdot z_{3} \\ = (5 + 2 i) \cdot (1 - i) \\ = 5 \cdot 1 + 5 \cdot (- i) \\ + 2 i \cdot 1 + 2 i \cdot (- i) \\ = 5 - 5 i + 2 i - 2 i^{2} \\ = (5 + 2) + (- 5 + 2) i \\ = 7 - 3 i . \end{array}

\begin{array}{l} z_{2} \cdot z_{3} & = (5 + 2 i) \cdot (1 - i) \\ = 5 \cdot 1 + 5 \cdot (- i) + 2 i \cdot 1 + 2 i \cdot (- i) \\ = 5 - 5 i + 2 i - 2 i^{2} \\ = (5 + 2) + (- 5 + 2) i \\ = 7 - 3 i . \end{array}

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 $z_{1} = r_{1} e^{i 𝜃_{1}}$ and $z_{2} = r_{2} e^{i 𝜃_{2}}$ be complex numbers, then

\begin{array}{l} z_{1} \cdot z_{2} & = r_{1} e^{i 𝜃_{1}} \cdot r_{2} e^{i 𝜃_{2}} \\ = (r_{1} \cdot r_{2}) e^{i 𝜃_{1} + i 𝜃_{2}} \\ = r_{1} r_{2} e^{i (𝜃_{1} + 𝜃_{2})} . \end{array}

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 $90^{\circ}$ or $\frac{π}{2}$ radians because $i$ has norm $1$ and argument $\frac{π}{2}$ .

Example 2

Find $z_{3} = r_{3} e^{i 𝜃_{3}} = z_{1} \cdot z_{2}$ when $z_{1} = 2 e^{i \frac{π}{3}}$ and $z_{2} = 4 e^{i \frac{π}{6}}$

The norm of $z_{3}$ is found by multiplying the norms of $z_{1}$ and $z_{2}$ :

r_{3} = 2 \cdot 4 = 8 .

The argument of $z_{3}$ is found by adding the arguments of $z_{1}$ and $z_{2}$ :

𝜃_{3} = \frac{π}{3} + \frac{π}{6} = \frac{π}{2} .

The product $z_{3}$ is then:

z_{3} = 8 e^{i \frac{π}{2}} = 8 i .

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 $z_{1}$ , $z_{2}$ and $z_{3}$ , the commutative property

z_{1} \cdot z_{2} = z_{2} \cdot z_{1},

the associative property

(z_{1} \cdot z_{2}) \cdot z_{3} = z_{1} \cdot (z_{2} \cdot z_{3}),

and the distributive property

z_{1} \cdot (z_{2} + z_{3}) = z_{1} \cdot z_{2} + z_{1} \cdot z_{3}

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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