Introduction to Complex Numbers
Understanding the fundamentals of complex numbers and their applications
๐ Introduction
Complex numbers are important to grasp as they extend our understanding of numbers beyond the real number line. They are used in various fields including engineering, physics, and applied mathematics to solve problems that cannot be addressed using just real numbers.
Further Mathematics students must be confident with Complex Numbers as they are a fundamental topic in the syllabus.
๐ฏ Learning Objectives
By the end of this lesson, you will be able to:
- Know what a complex number is
- Understand the concept of imaginary numbers
- Perform basic operations with complex numbers
- Notice the geometric representation of complex numbers and other terms.
๐ Key Concepts
What are Complex Numbers?
Complex numbers are numbers that have both real parts and imaginary parts. They're given as $a + bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary number.
The real part is just any known number, like 1, -423, 234.42, $ \pi $, etc.
An 'imaginary part' is the part that has the imaginary number.
An imaginary number is denoted (or written as) '$i$' and it equals $ \sqrt{-1} $.
This means that $ i^2 = -1 $.
๐ก Tip - Good to know
Complex numbers are usually given as $z = a + bi$.
$Re(z)$ means 'Real parts of ($z$)'
$Im(z)$ means 'Imaginary parts of ($z$)'
Since $z = a + bi$, then: $$Re(z) = a \quad \text{and} \quad Im(z) = bi$$
$i^2 = -1$ because: $$i = \sqrt{-1}$$
$$i^2 = (\sqrt{-1})^2$$
The power of 2 cancels and gets rid of the square root, leaving us with $-1$.
$z \in \mathbb{C}$ simply means that '$z$' is a complex number.
$\mathbb{C}$ is the set of all complex numbers, and $\in$ simply means 'a member of', so $z \in \mathbb{C}$ means '$z$ is a member of complex numbers'.
๐ Example 1:
Question:
A complex number $z = 3 + 2i$ is given. What are the real and imaginary parts?
Answer:
The real part $Re(z) = 3$
The imaginary part $Im(z) = 2$.
Addition and Subtraction of Complex Numbers
To add or subtract complex numbers, you work on the real and imaginary parts separately. Here's an example with addition:
๐ Example 2:
Question:
Complex numbers $z = 4 + 5i$ and $w = 2 + i$ are given. Solve $z + w$ and $z - w$.
Answer:
For addition:
$z + w$ means: $(4 + 5i) + (2 + i)$
So, lets get the real and imaginary parts of both, and add them.
For $z$:
Real part of $z$ is: $Re(z) = 4$
Imaginary part of $z$ is: $Im(z) = 5$
For $w$:
Real part of $w$ is: $Re(w) = 2$
Imaginary part of $w$ is: $Im(w) = 1$
So, we add the real parts and the imaginary parts separately:
โ
Final Answer: $z + w = (4 + 2) + (5 + 1)i = 6 + 6i$
For subtraction:
$Re(z) - Re(w) = 4 - 2 = 2$
$Im(z) - Im(w) = 5 - 1 = 4$
โ
Final Answer: $z - w = (4 - 2) + (5 - 1)i = 2 + 4i$
๐ Worked Example:
๐ Example 1:
Question: $z = 3 - i, w = 5 + i$. Solve $z + w$
Solution:
Combine the real and imaginary parts:
Which is equivalent to:
$$\text{because } Re(z) = 3, Re(w) = 5, Im(z) = -1, Im(w) = 1$$
0 multiplied by anything is zero, so:
๐ช Practice Problems
Try these problems to test yourself. I suggest you check your answer immediately after completing the question.
๐ฏ Practice Questions A
(A) (2 + 3$i$) + (4 + 5$i$) =
(B) (1 + 7$i$) + (3 + 2$i$) =
(C) (5 + 2$i$) + (โ1 + 6$i$) =
(D) (0 + 4$i$) + (2 + 0$i$) + (6 - 4$i$) =
๐ Show Solutions
(A) 6 + 8$i$
(B) 4 + 9$i$
(C) 4 + 8$i$
(D) 8 + 0$i$ = 8
๐ฏ Practice Question 2
(A) (6 + 8$i$) โ (2 + 3$i$)=
(B) (10 + 5$i$) โ (3 + 2$i$)=
(C) (7 + 6$i$) โ (2 + 4$i$)=
(D) (9 + 3$i$) โ (3 + 0$i$) โ (2 โ 1$i$) =
๐ Show Solutions
(A) 4 + 5$i$
(B) 7 + 3$i$
(C) 5 + 2$i$
(D) 4 + 2$i$
๐ Summary
๐ฏ Key Takeaways
- Equate imaginary and real parts separately.
- Writing out real and imaginary parts separately can help avoid mistakes.
- Practice with different forms of complex numbers.
๐ What's Next?
Next, we will explore multiplication and division of complex Numbers.