The Complex Plane and Modulus
Simple Explanation
A complex number z = a+bi can be plotted as a point (a,b) on the "complex plane" — the real part along a horizontal axis, the imaginary part along a vertical axis. The modulus |z| is the distance from the origin to this point: |z| = √(a²+b²).
Why Do We Need It?
Visualizing complex numbers as points (or arrows) gives geometric meaning to algebra that would otherwise feel abstract, and the modulus is the foundation for the trigonometric form coming up next.
See It
A point at (3,4) on the complex plane, labelled z = 3+4i, with an arrow from the origin showing its modulus of 5, and dashed lines showing the real and imaginary components
Formula
The Modulus of a Complex Number
|z| = √(a² + b²), for z = a + bi
The modulus is the distance from the origin to the point representing z on the complex plane — found using the Pythagorean theorem on the real and imaginary parts.
- a, b
- — the real and imaginary parts of z
- |z|
- — the modulus (magnitude) of z, always a non-negative real number
When to use it: Whenever the size (magnitude) of a complex number is needed, independent of its direction.
Worked Example
Find the modulus of a complex number
Find the modulus of z = 3+4i.
Why Does This Work?
Plotting z=a+bi as the point (a,b), the segment from the origin to that point, together with the horizontal segment of length a and vertical segment of length b, forms a right triangle — the modulus |z| is exactly the hypotenuse of that triangle, so the Pythagorean theorem gives |z|=√(a²+b²) directly.
Real-Life Example
The magnitude of an AC voltage or current
An electrical engineer represents an AC voltage as a complex number, but ultimately needs its actual measurable magnitude.
The modulus of that complex number gives exactly the physically meaningful magnitude of the voltage, independent of its phase.
Practice
Find |z| for z = −6+8i.
MediumCommon mistake
Forgetting that the modulus formula always uses the SQUARES of both parts (which are always non-negative) — |z| is never found by simply adding or subtracting a and b directly.
Quick Review
- z = a+bi plots as the point (a,b) on the complex plane.
- |z| = √(a²+b²) — the distance from the origin, via the Pythagorean theorem.
- The modulus is always a non-negative real number.