Multiplying Complex Numbers
2026-02-28 01:12 Diff
https://brightchamps.com/en-us/math/numbers/multiplying-complex-numbers

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Last updated on December 10, 2025

A complex number is written in the form z = a + ib, where i is the imaginary unit (i2 = -1) and a and b are real numbers. It involves applying the distributive property to compute the product of two complex numbers. In this article, we will learn about the multiplication of complex numbers.

What is the Multiplication of Complex Numbers?

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What are the Properties of the Multiplication of Complex Numbers?

The multiplication of complex numbers follows similar properties to the real numbers. The properties are 

  • Commutative property
     
  • Associative property
     
  • Distributive property

Commutative property: The order of multiplication does not affect the result. For example, \(z_1 \times z_2 = z_2 \times z_1 \) 

Associative property: The associative property of complex numbers states that the order of grouping the complex numbers doesn't change the result. That is, \(z_1 \times (z_2 \times z_3) = (z_1 \times z_2) \times z_3 \) 

Distributive property: The distributive property states that z1 \(z_1 \times (z_2 + z_3) = z_1 z_2 + z_1 z_3 \)

How to Multiply Complex Numbers in Cartesian Form

Now, let’s learn how to multiply complex numbers in Cartesian form. In this form, we multiply the complex number term by term. 
 

That is, if\(z_1 = a + ib \) and \(z_2 = c + id \)

Then \(z_1 \times z_2 = (a + ib)(c + id) \)

= \(ac + a (id) + ib (c) + i^2 bd \)

= \((ac - bd) + i(ad + bc) \)

So, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)

For example, find the product of \(z_1 = 3 + 2i \) and \(z_2 = 1 + 4i \)
z1z2 = (3 + 2i) (1 + 4i)

Using the formula for multiplying complex numbers, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)

\((3 + 2i)(1 + 4i) = ((3 \times 1) - (2 \times 4)) + i((3 \times 4) + (2 \times 1)) \)

=\((3 - 8) + i (12 + 2) \)

= \(-5 + 14i\)

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How to Multiply Complex Numbers in Polar Form

When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments.

That is, when multiplying \(z_1 = r_1 e^{i\Theta_1} \) and \(z_2 \)= \(r_2 e^{i \Theta_2} \)

\(z_1 \times z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)

For example, multiplying \(z_1 = 2 e^{i \pi/6} \) and \(z_2 = 3 e^{i \pi/4} \)

\(z_1 z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)

Here, r1 = 2

r2 = 3

Θ1 = \(\frac{\pi}{6} \)

Θ2 = \(\frac{\pi}{4} \)

Multiplying moduli \((r_1 r_2) = 2 \times 3 = 6 \)

Adding the arguments = \(\frac{\pi}{6} + \frac{\pi}{4} \)

= \(\frac{2\pi}{12} + \frac{3\pi}{12} \)

= \(\frac{5\pi}{12} \)

So, \(2 e^{i \pi/6} \times 3 e^{i \pi/4} = 6 e^{i 5\pi/12} .\)

Multiplication Of Complex Numbers with Real and Imaginary Numbers

The multiplication formula for complex numbers is  \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When multiplying the complex number by a real number, let's consider b as 0. That is \(a\) \((c + id) = ac + i a d \). 

For example: \(-5i (2 + 3i) \)
 

Using the formula, \(a (c + id) = ac + i a d \)

Here, a = -5i

c = 2

d = 3i

Substituting the value, \(-5i (2 + 3i) = (-5i \times 2) + (-5i \times 3i) = -10i + (-15 i^2) \)

Substituting \(i^2 = -1 \)

=\(-10i + 15 = 15 - 10i \). 

Squaring Complex Numbers

The formula for multiplying complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When squaring a complex number, consider a = c and b = d, and then the formula becomes;

\((a + ib)^2 = (a \times a - b \times b) + i(ab + ba) \)

\((a^2 - b^2) + i (2ab) \)

For example, squaring \(4 + 5i \)

\((a + ib)^2 = (a^2 - b^2) + i(2ab) \)

So, \((4 + 5i)^2 = (4^2 - (5i)^2) + i(2 \times 4 \times 5) \)

= \((16 - 25 \times i^2) + 40i \)

= \(16 - 25 + 40i \)

= \(-9 + 40i \)

Multiplicative Inverse Of Complex Numbers

The multiplicative inverse of a complex number \(z = a + ib\) is another complex number \(z^{-1} \), that is, \(z \times z^{-1} = 1 \). For a complex number \(z = a + ib \), the multiple inverse \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)

Here, \(z = a - ib \), so, \(|z| = \sqrt{a^2 + b^2} \)

The formula for the multiplicative inverse: \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)

Where, conjugate:\(z = a - ib \), and the modulus of the complex number is \(|z| = \sqrt{a^2 + b^2}.\)

Tips and Trips to Master Multiplying Complex Numbers

Learn effective strategies to multiply complex numbers accurately, both in standard and polar forms, and understand their geometric interpretations.

  • Always use the formula \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \).
     
  • Separate real and imaginary parts before multiplying.
     
  • Practice squaring complex numbers to see patterns.
     
  • Use the Argand plane to visualize multiplication geometrically.
     
  • Convert to polar form to simplify multiplication: \(r_1 e^{i\Theta_1} \cdot r_2 e^{i\Theta_2} = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \).
     
  • Teachers should help students build a foundation before diving deep into multiplying complex numbers. Making sure that they are fluent with the distributive property and binomial multiplication (FOIL), since we apply the same ideas while multiplying complex numbers.
     
  • Parents should review the key rule \(i^2 = -1\) and have them practice rewriting expressions such as \(5i^2 = -5\) to avoid later sign mistakes.
     
  • Parents can help young learners by encouraging them to write intermediate steps instead of doing everything mentally. It is easier for them to track the source of each term, and they can find errors more easily.
     
  • Teachers can note some of the structures students may often see and highlight them to learners, so they can easily understand and practice them until they feel almost automatic in calculating.

Common Mistakes and How to Avoid Them in Multiplying Complex Numbers

The students usually make mistakes when multiplying complex numbers, and mostly they repeat the same mistakes. In this section, we will discuss some common mistakes and ways to avoid them in multiplying complex numbers. 

Real-World Applications of Multiplying Complex Numbers

Complex number multiplication is used in various fields, like engineering, quantum mechanics, computer graphics, etc. In this section, we will discuss them in detail. 

  • Electrical engineering - Multiplying complex numbers is used to calculate impedance, voltage, and current in alternating current (AC) circuits, where signals have both magnitude and phase.
     
  • Signal Processing - Complex number multiplication helps in analyzing and manipulating signals in communication systems, such as modulation and filtering.
     
  • Control Systems - Engineers use complex numbers to represent system dynamics; multiplying them helps in understanding system stability and response.
     
  • Quantum mechanics - Complex numbers represent quantum states, and their multiplication is used in computing probability amplitudes and transformations.
     
  • Rotations in 2D space - Multiplying complex numbers is used to rotate vectors or points in a plane, which is useful in computer graphics, robotics, and navigation.

Problem 1

Find the product of (2 + 3i) × (4 + 5i)

Okay, lets begin

The product of\( (2 + 3i) × (4 + 5i) is -7 + 22i.\)

Explanation

To multiply the complex number, we use the formula;

\((a + ib) (c + id) = (ac - bd) +  i(ad + bc)\). 

Here, a = 2, b = 3, c = 4, and d = 5

\((ac - bd) = 2 × 4 - 3 × 5 = 8 - 15 = -7\)

\((ad + bc) = 2 × 5 + 3 × 4 = 10 + 12 = 22\)

So, the product is \(-7 + 22i\)

Well explained 👍

Problem 2

Find the product of (1 + i) and (1-i) using the polar form

Okay, lets begin

The product is 2.

Explanation

To find the product in polar form, we first convert each complex number to polar form. 

For 1 + i, 

\(r = \sqrt{a^2 + b^2} \)

= \(\sqrt{1^2 + 1^2} = \sqrt{2} \)

\(\Theta = \tan^{-1}\left(\frac{b}{a}\right) \)

= \(\tan^{-1}\left(\frac{1}{1}\right) = 45^\circ \)

Therefore, \(1 + i = \sqrt{2} \big(\cos 45^\circ + i \sin 45^\circ\big) \)

For 1 - i, 

\(r = \sqrt{1^2 + (-1)^2} \)

=√2

\(\Theta = \tan^{-1}\left(\frac{-1}{1}\right) = -45^\circ \)

Therefore,\(1 - i = \sqrt{2} \big(\cos(-45^\circ) + i \sin(-45^\circ)\big) \)

Here, \(r_1 r_2 = \sqrt{2} \times \sqrt{2} = 2 \)

\(\Theta_1 + \Theta_2 = 45^\circ + (-45^\circ) = 0^\circ \)

\((1 + i)(1 - i) = 2 \big(\cos 0^\circ + i \sin 0^\circ\big) = 2 \).

Well explained 👍

Problem 3

Find the product of √5 (cos15° + i sin15°) and √5 (cos 30° + i sin30°)

Okay, lets begin

The product of \(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \) and \(\sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) \) = \(5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)

Explanation

When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments. 

Here, \(r_1 = \sqrt{5}, \quad r_2 = \sqrt{5} \)

\(\Theta_1 = 15^\circ, \quad \Theta_2 = 30^\circ \)

\(r_1 r_2 = \sqrt{5} \times \sqrt{5} = 5 \)

\(\Theta_1 + \Theta_2 = 15^\circ + 30^\circ = 45^\circ \)

Therefore,\(\sqrt{5} \big(\cos 15^\circ + i \sin 15^\circ\big) \text{ and } \sqrt{5} \big(\cos 30^\circ + i \sin 30^\circ\big) = 5 \big(\cos 45^\circ + i \sin 45^\circ\big).\)

Well explained 👍

Problem 4

Find the product of (3 -4i) and (1 +2i)

Okay, lets begin

The product of (3 -4i) and (1 +2i) is \(11 + 2i.\)

Explanation

To multiply the complex number, we use the formula;

\((a + ib) (c + id) = (ac - bd) +  i(ad + bc)\). 

Here, a = 3, b = -4, c =1, d = 2

\((ac - bd) = ((3 × 1) - (-4 × 2)) = 3 + 8 = 11\)

\((ad + bc) = ((3 × 2) + (-4 × 1)) = 6 - 4 = 2\)

Therefore, the product of (3 -4i) and (1 +2i) is \(11 + 2i.\)

Well explained 👍

Problem 5

Find the square of the complex number z = 3 + 4i

Okay, lets begin

\((3 + 4i)^2 = -7 + 24i .\)

Explanation

\((a + ib)^2 = (a^2 - b^2) + i (2ab) \)

= \((3^2 - (4i)^2) + i (2 \times 3 \times 4) \)

= \((9 - 16) + i (24) \)

= \(-7 + 24i.\)

Well explained 👍

FAQs on Multiplying Complex Numbers

1.What is a complex number?

The complex number is a number that has a real and an imaginary part. It can be written in the form a + ib.

2.What is the multiplication of complex numbers formula?

The formula to multiply complex numbers, a + ib and c +  id is (a + ib)(c + id) = (ac -bd) + i(ad + bc)
 

3.What is the formula to multiply complex numbers in polar form?

The formula to multiply the complex numbers in polar form is r1r2(cos(Θ1 + Θ2) + sin(Θ1 + Θ2))

4.What is an argument in polar form?

The argument is the angle between the positive real axis and the line connecting the origin. The argument can be calculated using the formula, Θ = tan-1(b/a)

5.What is the value of i²?

6.What real-life applications can I explain to my child?

Applications include calculating AC circuit currents, analyzing signals in communication systems, representing rotations in computer graphics, and understanding transformations in quantum mechanics.

7.How can I help my child practice multiplying complex numbers at home?

Encourage them to solve step-by-step using the formula \((a+ib)(c+id)=(ac−bd)+i(ad+bc)\), start with simple numbers, then move to larger numbers or polar forms.

8.At what stage should my child learn to multiply complex numbers?

Typically, students learn about complex numbers in higher secondary (grades 11-12) or during early college courses in mathematics, physics, or engineering.

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.