Argand Diagram — Definition, Formula & Examples

An Argand diagram is a graph that represents complex numbers as points on a two-dimensional plane, where the horizontal axis shows the real part and the vertical axis shows the imaginary part.

An Argand diagram is a Cartesian coordinate system in which the complex number $z = a + bi$ is mapped to the point $(a, b)$ in $\mathbb{R}^2$ , with the horizontal axis representing $\operatorname{Re}(z)$ and the vertical axis representing $\operatorname{Im}(z)$ .

Key Formula

z = a + bi \quad \longleftrightarrow \quad (a,\, b)

Where:

$a$ = Real part of the complex number (horizontal coordinate)
$b$ = Imaginary part of the complex number (vertical coordinate)
$i$ = The imaginary unit, where $i^2 = -1$

How It Works

To plot a complex number

z = a + bi

, treat

a

as the

x

-coordinate and

b

as the

y

-coordinate. Move

a

units along the real (horizontal) axis and

b

units along the imaginary (vertical) axis. The origin represents

0 + 0i

. The distance from the origin to the plotted point equals the modulus

|z| = \sqrt{a^2 + b^2}

, and the angle measured counterclockwise from the positive real axis is the argument of

z

Worked Example

Problem: Plot the complex number

z = 3 + 4i

on an Argand diagram and find its modulus.

Identify coordinates: The real part is 3 and the imaginary part is 4, so the point is

(3, 4)

a = 3, \quad b = 4

Plot the point: From the origin, move 3 units right along the real axis and 4 units up along the imaginary axis. Mark the point

(3, 4)

Find the modulus: The modulus is the distance from the origin to the point.

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Answer: The point

(3, 4)

represents

z = 3 + 4i

on the Argand diagram, and its modulus is 5.

Visualization

Why It Matters

Argand diagrams turn abstract complex arithmetic into visible geometry. In physics and electrical engineering, plotting impedances and phasors on the complex plane is routine. Mastering this representation in precalculus prepares you for polar form, Euler's formula, and signal analysis.

Common Mistakes

Mistake: Swapping the axes by putting the imaginary part on the horizontal axis and the real part on the vertical axis.

Correction: The real axis is always horizontal and the imaginary axis is always vertical, matching the convention

(a, b)

for

a + bi