Complex Numbers Lesson: Definition, Types, Formula, and Examples

Lesson Overview

• What Is a Complex Number?
• The Structure of a Complex Number
• Types of Complex Numbers
• What Is the Conjugate of a Complex Number?
• Complex Number Formulas
• Calculating Complex Numbers
• Complex Numbers Examples

Complex numbers combine a real and an imaginary part, providing a comprehensive way to perform operations like addition, subtraction, multiplication, and division, even for equations with no real solutions.

What Is a Complex Number?

A complex number is a type of number that extends the concept of real numbers by introducing an imaginary component. It combines both a real part and an imaginary part, which allows it to represent quantities that cannot be expressed with real numbers alone.

A complex number is written in the form x + iy, where:

• x and y are real numbers (like 2, -5, 1/3, or π).

• i is the imaginary unit, defined as the square root of -1 (i² = -1).

The Structure of a Complex Number

Fig: Structure of a complex number

A complex number is a number that combines a real part and an imaginary part. It's expressed in the form:

x + iy

where:

• x represents the real part. This is an "ordinary" number you're familiar with, like 2, -5, 1/3, or π (pi).

• y represents the imaginary part. This is also a real number, but it's multiplied by the imaginary unit i.

• i is the imaginary unit, which is defined as the square root of -1. This means i² = -1.

Here are a few examples:

Types of Complex Numbers

Take This Quiz:

12 - MATHS- UNIT 3. COMPLEX NUMBERS

What Is the Conjugate of a Complex Number?

The conjugate of a complex number is another complex number that is closely related to it. It is formed by changing the sign of the imaginary part of the original complex number, while the real part remains unchanged.

How to find the conjugate:

To find the conjugate, you simply change the sign of the imaginary part. If the complex number is a + bi, its conjugate is a - bi.

Examples:

• Simple Complex Numbers:
  • The conjugate of 3 + 2i is 3 - 2i.
  • The conjugate of -5 - 4i is -5 + 4i.

• Complex Numbers with Fractions or Decimals:
  • The conjugate of 1/2 + (3/4)i is 1/2 - (3/4)i.
  • The conjugate of -2.5 + 1.7i is -2.5 - 1.7i.

• Purely Real or Imaginary Numbers:
  • The conjugate of 7 (which is 7 + 0i) is simply 7 (or 7 - 0i).
  • The conjugate of -3i (which is 0 - 3i) is 3i (or 0 + 3i).

• Complex Numbers with Radicals:
  • The conjugate of √2 + √3 i is √2 - √3 i.

Complex Number Formulas

Just like real numbers, we can add, subtract, multiply, and divide complex numbers. Here's how:

Take This Quiz:

Complex And Imaginary Numbers Quiz

Calculating Complex Numbers

Let's see how to do calculations with complex numbers.

Addition

Add real and imaginary parts separately. Think of it like combining like terms.

• Why? Real and imaginary parts are different. They can't be combined directly.

• How?
  1. Identify the real and imaginary parts.
  2. Add the real parts.
  3. Add the imaginary parts.
  4. Write the answer as a + bi..
     

• Examples:
• (2 + 3i) + (1 - i) = (2 + 1) + (3 - 1)i = 3 + 2i
• (5 - 4i) + (-2 + 2i) = (5 - 2) + (-4 + 2)i = 3 - 2i
  

Subtraction

Subtract real and imaginary parts separately.

• Why? Treat the real and imaginary parts as different types.

• How?
  1. Identify the real and imaginary parts.
  2. Subtract the real parts.
  3. Subtract the imaginary parts.
  4. Write the answer as a + bi.
     

• Examples:
• (4 + 3i) - (1 + i) = (4 - 1) + (3 - 1)i = 3 + 2i
• (6 - 2i) - (3 - 5i) = (6 - 3) + (-2 + 5)i = 3 + 3i
  

Multiplication

Use the distributive property (FOIL). Remember that i² = -1.

• Why FOIL? Treat complex numbers like binomials.
  
• Why i² = -1? This simplifies the answer.
  
• How?

1. Multiply using FOIL.
2. Combine terms.
3. Replace i² with -1.
4. Write the answer as a + bi.

• Examples:
• (1 + 2i)(2 - i) = 2 - i + 4i - 2i² = 2 + 3i + 2 = 4 + 3i
• (3 - i)(1 + 2i) = 3 + 6i - i - 2i² = 3 + 5i + 2 = 5 + 5i
  

Division

Multiply the top and bottom by the conjugate of the denominator.

• Why the conjugate? This removes i from the denominator.

• How?
• Find the conjugate of the denominator.
• Multiply the top and bottom by the conjugate.
• Expand and simplify.
• Write the answer as a + bi.
  

• Examples:

1. (2 + i) / (1 - i) 

=(2 + i)(1 + i) / (1 - i)(1 + i) 

= (2 + 3i + i²) / (1 - i²)

= (1 + 3i) / 2 = 1/2 + 3/2 i

2. (3 - 2i) / (2 + i) 

= (3 - 2i)(2 - i) / (2 + i)(2 - i) 

= (6 - 7i + 2i²) / (4 - i²) 

= (4 - 7i) / 5 

= 4/5 - 7/5 i

Complex Numbers Examples

Here are some examples of complex numbers, showcasing different combinations of real and imaginary parts:

Simple Complex Numbers

• 2 + 3i
• -5 - i
• 4 + i
• -1 - 7i

Complex Numbers with Fractions or Decimals

• 1/2 + (3/4)i
• -2.5 + 1.7i
• 0.8 - 0.3i

Purely Real or Imaginary Numbers

• 7 (This is also a complex number, with an imaginary part of 0: 7 + 0i)
• -3i
• √2i
• -πi (where π is pi, approximately 3.14159)

Complex Numbers with Radicals

• √2 + √3 i
• -√5 - 2√2 i