Complex Numbers

Introduction

We are familiar with real numbers in mathematics those that can be plotted on the number line. So, what does a number get factored and end up in when it contains a square root of a negative number? That’s why we need complex numbers.

Complex numbers expand the number world and allow solutions that would otherwise be impossible with real numbers. From solving quadratic equations to learning about electricity and waves in science, you rely on complex numbers.

In this blog, we are going to discuss what complex numbers are, why do we need to use them and read some complex numbers examples that help simplify the understanding.

 

Table of Contents

• What Are Complex Numbers?
• Complex Numbers: Meaning in Simple Terms
• Parts of a Complex Number
• Complex Numbers Examples
• Representation of Complex Numbers on the Argand Plane
• Operations with Complex Numbers
• Modulus and Conjugate of Complex Numbers
• Applications of Complex Numbers in Real Life
• Common Mistakes with Complex Numbers
• Conclusion
• Frequently Asked Questions on Complex Numbers

 

What Are Complex Numbers?

A complex number is a number made up of two parts:

• A real part

• An imaginary part

It is written in the form:

z = a + bi

Where:

• a is the real part

• b is the imaginary part

• i is the imaginary unit, defined as i² = -1

So, complex numbers include numbers like 3 + 4i, 5 – 2i, or even 0 + 6i.

 

Complex Numbers: Meaning in Simple Terms

Let’s break it down even more simply.

Imagine you are trying to find the square root of -4. In real numbers, that’s impossible. But with complex numbers, we say:

√-4 = √4 × √-1 = 2i

Here, i is the imaginary unit. So, a complex number lets you combine these imaginary values with real ones, giving us a new kind of number system.

This expanded system helps in solving equations that have no real solution.

 

Parts of a Complex Number

In the number z = a + bi, we have:

• a → Real part

• b → Imaginary part

• bi → Imaginary component

• z → Complex number

Example:

For z = 7 - 3i,

• Real part = 7

• Imaginary part = -3

 

Complex Numbers Examples

Let’s look at some complex numbers examples to understand this better:

• 5 + 2i → Real part = 5, Imaginary part = 2

• -3 - 7i → Real part = -3, Imaginary part = -7

• 0 + 4i → Real part = 0, Imaginary part = 4

• 2 – 0i → A real number, still a complex number!

Even real numbers like 3 or -5 are technically complex numbers with the imaginary part 0.

 

Representation of Complex Numbers on the Argand Plane

Complex numbers can be visualized like points or vectors on a plane called the Argand Plane.

Here’s how:

• The x-axis represents the real part

• The y-axis represents the imaginary part

So the number 3 + 4i is plotted at (3, 4).

This is helpful in visualizing operations like addition and multiplication of complex numbers.

 

Operations with Complex Numbers

Addition

Add real parts and imaginary parts separately.
(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i

Subtraction

Subtract real and imaginary parts.
(5 + 6i) - (3 + 2i) = (5 - 3) + (6 - 2)i = 2 + 4i

Multiplication

Use algebraic multiplication and i² = -1.
(2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i² = 8 + 22i - 15 = -7 + 22i

Division

Multiply numerator and denominator by the conjugate of the denominator.
We’ll cover conjugates next.

 

Modulus and Conjugate of Complex Numbers

Modulus (or absolute value)

The modulus of z = a + bi is:
|z| = √(a² + b²)

It shows the distance from the origin to the point on the Argand Plane.

Example:
|3 + 4i| = √(9 + 16) = √25 = 5

Conjugate

The conjugate of z = a + bi is a - bi.

Used to simplify division and other operations.

Example:
Conjugate of 5 + 2i is 5 - 2i

 

Applications of Complex Numbers in Real Life

You might wonder where these numbers are actually used. Here are some real-life applications of complex numbers:

Electrical Engineering

Complex numbers help in calculating voltage, current, and impedance in AC circuits.

Quantum Physics

They describe wave functions, making sense of complex physical phenomena.

Control Systems

Used in designing stable systems in aerospace, robotics, and automation.

Signal Processing

Complex numbers allow for precise control over audio, video, and radar signals.

Computer Graphics

They help in creating fractals, rendering, and transformations.

Despite the name, complex numbers solve real-world problems!

 

Common Mistakes with Complex Numbers

• Forgetting i² = -1
  It’s crucial to simplify correctly.

• Mixing real and imaginary parts
  Treat them like separate values when adding or subtracting.

• Not applying the conjugate when needed
  Especially important in division.

• Assuming complex numbers aren’t real math
  They’re just as real, just part of a broader system!

 

Conclusion

Complex numbers are a beautiful and powerful idea in maths. They are beyond ordinary real numbers and they can help us solve problems that were previously unsolvable. All the way from simple algebra to the art of applying complex numbers in the "white heat" of a crisis in science and engineering, the realm of complex numbers turns the caller on to a new dimension of problem-solving and understanding analysis

So now you’ve known the complex numbers meaning, have seen some complex numbers examples and know how to operate with them. Whether you are a student or an inquiring general reader, complex numbers are a stage on the way up to the math Olympiad.

 

Related Topics 

Large Number - Explore how to read, write, and understand large numbers with ease, perfect for mastering place value and big calculations!

Roman Numbers - Decode the ancient Roman number system and learn how to convert between Roman and modern numerals in minutes!

Type of Numbers - Understand the different types of numbers natural, whole, integers, rational, and more made simple and fun for learners!

 

Frequently Asked Questions on Complex Numbers

1. What is the standard form of a complex number?

A) a × b
B) a + bi
C) a – b
D) a/b

 

2. Is the number 3i a complex number?

A) No, it’s only imaginary
B) Yes, with real part 3
C) Yes, with real part 0
D) No, it’s irrational

 

3. Can the number 4 be considered a complex number?

A) No, it’s real only
B) Yes, because it has an imaginary part
C) Yes, as 4 + 0i
D) No, complex numbers must have i

 

4. Which of the following is not a complex number?

A) 2
B) 0 + 5i
C) 3 + 4i
D) None of the above

 

5. Who is considered the father of complex numbers?

A) Isaac Newton
B) Carl Gauss
C) Gerolamo Cardano
D) Albert Einstein

Explore more math concepts and examples now to strengthen your foundation in algebra and complex numbers!