Complex Numbers
2026-02-28 09:07 Diff
https://brightchamps.com/en-us/math/numbers/complex-numbers

1398 Learners

Last updated on December 2, 2025

A complex number is a number that has two parts: a real part (like regular numbers) and an imaginary part, which includes i, where i² = –1. It is written as a + bi, where 'a' represents the real part and 'b' represents the imaginary part. Examples include 3 + 2i and – 5 – i. Complex numbers are useful in math, physics, and engineering to solve problems that real numbers alone cannot.

What are Complex Numbers?

What Are Numbers? 🔢 | Fun Explanation with 🎯 Real-Life Examples for Kids | ✨BrightCHAMPS Math

▶

The idea of complex numbers started long ago with the Greek mathematician Hero of Alexandria in the 1st century. He tried to take the square root of complex number and negative number, but since people at that time didn’t understand negative roots, he changed the negative number into a positive one before solving it. Many years later, in the 16th century, the Italian mathematician Gerolamo Cardano formally introduced complex numbers while working on equations involving negative roots.
 

A complex number is written in the form a + bi, where a is the real part and b is the imaginary part. Complex numbers help us solve problems that include the square root of a negative number—something regular real numbers can’t do. Today, tools like a complex number calculator make it much easier for students and learners to work with these numbers, check their answers, and understand the concept more clearly.
 

What is i?

The i in the complex number is called ‘iota’. It is used to represent the imaginary part of the complex number. Thereby, it helps to find the square root of negative numbers as the value of i2 = –1. 

Example:

√(–9) = 3i

√(–16) = 4i

We use i to turn negative square roots into meaningful expressions in complex numbers.

Properties of a Complex Number

In order to understand more about complex numbers, first let’s learn about its properties.

  • If x, y are real numbers and x + iy = 0 then x = 0, y = 0. 

    That is, \(  x + iy = 0 = 0 + i0   \)

    According to the definition of equality for complex numbers, the real parts and imaginary parts must be equal. Therefore, we conclude that:

    x = 0, y = 0

  • If \(x + iy = u + iv,\) then x = u, and y = v

    Given that x, y, u and v are real numbers, we know:

    \( x + iy = u + iv\)

    By definition of complex number equality, their real and imaginary components must be equal, leading to: 

    x = u, y = v

  • For any three complex numbers u, v, and z, the following fundamental laws hold:
     
  1.  Commutative Law: \( u + v = v + u\)
    \( u . v = v. u\)
     
  2. Associative Law:\( (u + v) + z = u + (v + z)\)
    \( (u . v) . z = u . (v . z)\)
     
  3. Distributive Law: \(x . (v + z) = x . v + x . z\)
  • The product of two conjugate complex numbers is a real number. Let z = x + iy, where x and y are real. The conjugate is:

    z = x – iy

    Multiplying z by its conjugate:

    \(z . z = (x + iy) (x – iy)\)

    Expanding using the difference of squares: 

    x² – i²y²

    Since i² = – 1, this simplifies to: 

    x² + y²

    Which is a real number.

Explore Our Programs

Classification of Complex Numbers

The standard form of a complex number is given by z = a + ib, where a, b ∈ ℝ and i (iota) represents the imaginary unit. Based on the values of a (the real part) and b (the imaginary part), complex numbers can be categorized in

  • Zero Complex Numbers: If a = 0 and b = 0, the complex number is called a zero complex number. The only example of this is 0. 
     
  • Purely Real Numbers: If a ≠ 0 and b = 0, the number is considered a purely real number, meaning it has no imaginary component. Examples include 2, 3, 5, 7, and all the other real numbers. 
     
  • Purely Imaginary Numbers: If a = 0 and b ≠ 0, the number is considered a purely imaginary number, as it has no real part. Examples include \(– 7i, – 5i, – i, 5i, 7i,\) and other numbers with only an imaginary component. 
     
  • General Complex Numbers: If a ≠ 0 and b ≠ 0, the number is classified as a complex number that includes both real and imaginary parts. Examples include \((–1 – i), (1 + i), (1 – i), (2 + 3i),\) and similar expressions. 

Different Forms of Complex Numbers

Complex numbers can be represented in different forms, each with its own advantages depending on the application. The three most common representations are Rectangular (standard) Form, Polar Form, and Exponential Form. These forms allow for easier calculations in various fields.

  • Rectangular Form: Also known as the Standard Form, this representation of a complex number is written as a + ib, where a and b are real numbers.

    Examples:\( (5 + 5i), – 7i, (– 3 – 4i),\) etc.

  • Polar Form of complex number: In this representation, a complex number is expressed in terms of its magnitude (r) and angle () relative to the positive x-axis. It is written as: 

    r [cos θ + i sin θ]

    Here, r represents the distance from the origin, and is the angle between the radius vector and the positive x-axis. 

    Examples: \([cos 2 + i sin 2], 5 [cos 6 + i sin 6]\), etc.

  • Exponential Form: This form uses Euler’s Formula to express a complex number as: 

    \(re^{iθ} \space \text{or}\space r·e^{iθ}\)

    Where r is the magnitude of complex number and θ is the angle. 

    Examples: \(e^{i0}, \quad e^{i\frac{\pi}{2}}, \quad e^{i\frac{5\pi}{6}} \) etc.

Geometrical Representation of Complex Numbers

A complex number z = a + ib is uniquely represented by the point P (a, b) on the complex plane. Similarly, every point on the complex plane corresponds to a unique complex number.

Complex Plane: The plane where complex numbers are uniquely represented is known as the Complex Plane, also called the Argand Plane or Gaussian Plane. 

The Complex Plane has two axes, they are:

  • X-axis or Real Axis
     
  • Y-axis or Imaginary Axis

X-Axis (Real Axis)
 

  • Every purely real complex number is represented by a unique point on this axis.
     
  • The real part, Re(z), of any complex number is plotted along this axis.
     
  • For this reason, the axis is known as the Real Axis.

Y-Axis (Imaginary Axis)
 

  • Every purely imaginary complex number is represented by a unique point on this axis. 
     
  • The imaginary part, Img(z), of any complex number is plotted along this axis.
     
  • Therefore, the Y-axis is referred to as the imaginary Axis. 

Steps to Represent Complex Numbers on a Complex Plane

To plot a complex number z = a + ib on the complex plane, follow these rules:

  • The real part of z, denoted as Re(z) = a, is assigned as the X-coordinate of the point P. 
     
  • The imaginary part of z, denoted as Im(z) = b, is assigned as the Y-coordinate of the point P.
     
  • Thus, the complex number z = a + ib corresponds to the point P (a, b) on the complex plane. 

Modulus and Argument of Complex Number

The modulus of a complex number represents its absolute value of a complex number and is defined as the distance from the origin to the corresponding point on the complex plane. It is also known as the magnitude of the complex number. 


For a complex number z = a + ib, the modulus is calculated as:


z = √(a² + b²)


The argument of a complex number is the angle θ between its vector and the positive real (x) axis. 


Mathematically, for z = a + ib, it is given by:

 
  \(\theta = \arctan\left(\frac{b}{a}\right) \)

Operations on Complex Numbers

Complex numbers follow specific mathematical operations similar to real numbers, but with additional rules for handling the imaginary unit i. These operations include addition, subtraction, multiplication, and division, are carried out using fundamental algebraic properties. Mastering these operations is crucial for handling complex numbers in both mathematical and real-world applications. 

            
Example: (3 + 2i) + (1 + 4i) = 4 + 6i.

  • Subtraction: Subtract the real parts and the imaginary parts separately.


Example: (3 + 2i) – (1 + 4i) = 2 – 2i


Example: (3 + 2i) (1 + 4i) = 3 + 12i + 2i + 8i2 = 3 + 14i – 8 = – 5 + 14i.

  • Division: To divide a complex number by another, multiply both numerator and denominator by the conjugate of the denominator, then simplify.
     

Example: (3 + 2i) / (1 + 4i) = (3 + 2i) (1 - 4i) / (1 + 4i)(1 – 4i) = (11 - 10i)/17.

Trips and Tricks to Master Complex Numbers

Solving complex numbers can be difficult, but with the right approach, it can be easy and simple. Here are some tips and tricks to master the complex numbers:

  1. Memorize the powers of i:
    \(i = √–1\)
    \(i2 = –1\)
    \(i3 = –i \)
    \(i4 = 1\)
     
  2. When adding or subtracting two complex numbers, add or subtract the real parts and the imaginary parts separately and then combine both.
     
  3. When solving for large power of i, simplify the power in factor of 4, for easy calculation.
     
  4. Remember the values of some common expressions:
    \((1 + i)2 = 2i\)
    \((1 – i)2 = –2i\)
    \(w3 = 1\)
    \(1 + w + w2 = 0\)
     
  5. Always rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Common Mistakes of Complex Numbers and How to Avoid Them

Complex numbers can be a bit tricky to learn, and making mistakes while learning about them is quite normal for students. Here are the top five mistakes of complex numbers and how to avoid them.

Real Life Applications of Complex Numbers

Complex numbers are more than just numbers or theoretical math, they are essential in technology, science, and engineering. Here are some of the real applications of complex numbers. 

  • Electrical Engineering: Complex numbers are used in AC circuit analysis, where voltage and current are represented as complex numbers to simplify calculations involving phase differences.
     
  • Signal processing: They help in processing audio, radio, and TV signals by representing waves mathematically. They are also used in fluid dynamics.
     
  • Computer graphics and animation: They are used to performing transformations, rotations, and scaling of objects in 2D and 3D graphics.
     
  • Physics: Complex numbers are used in quantum mechanics to describe wave functions, in electromagnetism to solve field equations, and in fluid dynamics.
     
  • Control systems: Engineers use complex numbers to evaluate stability and performance in systems such as airplane autopilot and robotic motion control. 

Problem 1

What is (3 + 2i) + (4 – 5i)?

Okay, lets begin

 7 – 3i
 

Explanation

Add the real parts: 

\(3 + 4 = 7\)

Then add the imaginary parts: 

\(2i + (– 5i) = – 3i\)

Well explained 👍

Problem 2

What is (2 + 3i) x (1 – i)?

Okay, lets begin

5 + i 
 

Explanation

Use the distributive property: 

 
\((2 + 3i)(1 – i) = 2 – 2i + 3i – 3i² = 2 + i + 3 = 5 + i\).

Well explained 👍

Problem 3

What is the conjugate of 4 – 7i?

Okay, lets begin

 4 + 7i

Explanation

The conjugate of a complex number a + bi is found by changing the sign of the imaginary part, so 4 – 7i becomes 4 + 7i.
 

Well explained 👍

Problem 4

Find the magnitude (modulus) of – 3 + 4i.

Okay, lets begin

5

Explanation

The modulus of a + bi is given by 


a2+ b2

Substituting a = – 3 and b = 4, we get 


\((-3)2 + (42)  = 9 + 16 = 25 = 5\)

Well explained 👍

Problem 5

Solve 3 + 2i / 1 - i.

Okay, lets begin

(1 + 5i) / 2
 

Explanation

Multiply by the conjugate of the denominator

\((3 + 2i) (1 + i) / (1 - i) (1 + i)\)

The denominator simplifies to 12 – (–1) = 2. Expanding the numerator:

\(3 + 3i + 2i + 2i²\)

= \(3 + 5i – 2\)

=\( 1 + 5i.\)

Dividing by 2 gives (1 + 5i) / 2.
 

Well explained 👍

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.