Introduction to Complex Numbers
Discover the imaginary unit i, represent complex numbers in a + bi form, and visualise them on the Argand diagram.
The Imaginary Unit i
In the real number system, there is no solution to the equation x2 = −1. To solve such equations, mathematicians introduced the imaginary unit:
i = √(−1) so i2 = −1
Powers of i follow a cyclic pattern:
i1 = i
i2 = −1
i3 = −i
i4 = 1
Cartesian Form: a + bi
A complex number z is written in the form:
z = a + bi
where a is the real part (Re(z)) and b is the imaginary part (Im(z)).
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. For example, if 2 + 3i = a + bi, then a = 2 and b = 3.
Every real number is a complex number with b = 0, and every purely imaginary number has a = 0.
The Argand Diagram
Complex numbers can be represented as points on a plane called the Argand diagram (or complex plane). The horizontal axis represents the real part and the vertical axis represents the imaginary part.
For example, z = 3 + 2i is plotted at the point (3, 2):
The distance from the origin to z is the modulus |z| = √(a2 + b2), and the angle from the positive real axis is the argument arg(z).
Key Vocabulary
Imaginary Unit (i)
Defined as √(−1). Its square is −1.
Complex Number
A number of the form a + bi where a and b are real numbers.
Argand Diagram
A coordinate plane used to represent complex numbers graphically.
Modulus
The distance from the origin to a complex number: |z| = √(a2 + b2).
Worked Examples
Simplify i7.
Step 1: Divide the power by 4: 7 ÷ 4 = 1 remainder 3.
Step 2: So i7 = i3 = −i.
Answer: i7 = −i.
Express √(−9) in terms of i.
Step 1: √(−9) = √(9) × √(−1) = 3 × i.
Answer: √(−9) = 3i.
Find the modulus of z = 3 − 4i and plot z on the Argand diagram.
Step 1: |z| = √(32 + (−4)2) = √(9 + 16) = √25 = 5.
Step 2: On the Argand diagram, plot the point (3, −4): 3 along the real axis and −4 along the imaginary axis (fourth quadrant).
Answer: |z| = 5. The point lies in the fourth quadrant of the complex plane.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is i2 equal to?
Question 2
What is the real part of z = 5 − 3i?
Question 3
Simplify √(−25).
Question 4
What is the modulus of z = 5 + 12i?
Question 5
On an Argand diagram, where is the complex number −2 + 3i plotted?
Key Concepts Summary
- ● The imaginary unit i satisfies i2 = −1, with powers cycling every 4.
- ● A complex number has the form z = a + bi with real part a and imaginary part b.
- ● The Argand diagram plots complex numbers as points (a, b) on a plane.
- ● The modulus |z| = √(a2 + b2) measures the distance from the origin.
- ● Every real number is a complex number with imaginary part equal to zero.