The Complex Plane
The complex plane (Argand diagram) represents complex numbers geometrically, with the real part on the horizontal axis and the imaginary part on the vertical axis.
What You Need to Know
Key Concept Diagram
A complex number z = a + bi is plotted as the point (a, b) on the Argand diagram
The modulus |z| = √(a² + b²) is the distance from the origin to the point
The argument θ is the angle the line from the origin makes with the positive real axis
Complex conjugates z and z* are reflections of each other across the real axis
Key Vocabulary
Argand Diagram
A two-dimensional diagram where complex numbers are plotted as points using real and imaginary axes
Modulus
The distance of a complex number from the origin on the Argand diagram, |z| = √(a²+b²)
Argument
The angle θ in radians or degrees that the complex number makes with the positive real axis
Complex Conjugate
The complex number formed by changing the sign of the imaginary part: conjugate of a+bi is a−bi
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the modulus of the complex number z = 3 + 4i?
Question 2
Where is the complex number −2 + 0i plotted on the Argand diagram?
Question 3
What is the complex conjugate of 5 − 3i?
Key Concepts Summary
- ●A complex number z = a + bi is plotted as the point (a, b) on the Argand diagram
- ●The modulus |z| = √(a² + b²) is the distance from the origin to the point
- ●The argument θ is the angle the line from the origin makes with the positive real axis
- ●Complex conjugates z and z* are reflections of each other across the real axis