Argand Plane

The Argand plane (also called the complex plane) is a two-dimensional coordinate system that gives every complex number a unique geometric home. Just as a real number corresponds to a point on a number line, a complex number z = a + bi corresponds to a point in a plane — with the real part a measured along the horizontal axis and the imaginary part b measured along the vertical axis. This powerful idea transforms algebra into geometry, letting us see complex numbers rather than just manipulate them symbolically. Before the Argand plane was introduced, complex numbers were treated with suspicion — mathematicians called them 'imaginary' precisely because they seemed to have no physical meaning. The geometric representation changed everything. Operations like addition become vector addition, multiplication becomes a rotation and scaling, and taking the modulus becomes finding a distance. Suddenly the 'impossible' square root of −1 had a perfectly concrete location: one unit up the vertical axis. In practice, the Argand plane underpins electrical engineering (impedance in AC circuits), quantum mechanics (wave functions), signal processing, and fractal geometry. Mastering this representation is therefore not just an abstract exercise — it is the gateway to some of the most important mathematics and physics of the modern world.

Key takeaways

Set Up the Axes
Plot a Complex Number
The Modulus — Distance from the Origin
The Argument — Angle from the Positive Real Axis
Polar Form
The Complex Conjugate on the Diagram

Set Up the Axes

Draw two perpendicular axes intersecting at the origin O. Label the horizontal axis the Real axis (Re) and the vertical axis the Imaginary axis (Im). Every complex number z = a + bi will be plotted as the point P(a, b) in this plane, exactly as you would plot the Cartesian point (a, b).

Plot a Complex Number

To plot z = 3 + 4i, move 3 units to the right along the real axis and 4 units upward along the imaginary axis, then mark the point. The number 5 − 2i would be plotted at (5, −2): 5 units right and 2 units down. Pure real numbers like 7 lie on the horizontal axis; pure imaginary numbers like −3i lie on the vertical axis.

The Modulus — Distance from the Origin

The modulus |z| of a complex number z = a + bi is the distance from the origin to the point P(a, b), calculated using the Pythagorean theorem: |z| = √(a² + b²). For z = 3 + 4i, |z| = √(9 + 16) = √25 = 5. The modulus is always a non-negative real number and represents the 'size' or 'magnitude' of the complex number.

The Argument — Angle from the Positive Real Axis

The argument arg(z) is the angle θ (measured anticlockwise) from the positive real axis to the line segment OP. It is found using tan θ = b/a, but care is needed about which quadrant the point lies in. For z = 3 + 4i, θ = arctan(4/3) ≈ 53.1°. For z = −1 + i, a = −1 and b = 1, so the point is in the second quadrant and θ = 180° − 45° = 135°.

Polar Form

Once you know the modulus r = |z| and argument θ = arg(z), any complex number can be written in polar form as z = r(cos θ + i sin θ), often abbreviated z = r cis θ. This connects the Argand diagram directly to trigonometry and leads naturally to Euler's remarkable formula z = re^(iθ). Polar form makes multiplication and division of complex numbers geometrically transparent: multiply the moduli and add the arguments.

The Complex Conjugate on the Diagram

The complex conjugate of z = a + bi is z̄ = a − bi. On the Argand plane, z̄ is the reflection of z in the real axis. This geometric insight immediately explains why z · z̄ = a² + b² = |z|², a purely real, non-negative number — the product of a point and its mirror image always lands on the positive real axis.

The Key Equation

|z| = \sqrt{a^2 + b^2}, \quad \arg(z) = \theta = \arctan\!\left(\frac{b}{a}\right)

z — Complex number (dimensionless): The complex number z = a + bi being represented on the Argand plane.
a — Real part (dimensionless): The horizontal coordinate; Re(z). It is the coefficient of the real component.
b — Imaginary part (dimensionless): The vertical coordinate; Im(z). It is the coefficient of i in z = a + bi.
|z| — Modulus (dimensionless): The distance from the origin O to the point P(a, b) in the Argand plane.
\theta — Argument (radians or degrees): The angle measured anticlockwise from the positive real axis to the line OP.

Worked Example

Problem. Plot the complex numbers z₁ = −2 + 3i and z₂ = 4 − i on the Argand plane. Then find the modulus and argument of z₁, and write z₁ in polar form.

|z_1| = \sqrt{13},\quad \arg(z_1) \approx 123.7^\circ,\quad z_1 = \sqrt{13}\,e^{i\cdot 2.159}

Answer. Modulus = √13 ≈ 3.606; Argument ≈ 123.7° (second quadrant); Polar form z₁ = √13 (cos 123.7° + i sin 123.7°).

Figure

Diagram

Origins · 1 minJean-Robert Argand and how this was discovered

Jean-Robert Argand

1768–1822 · French-Swiss

In 1806, Argand — an amateur mathematician who ran a Parisian bookshop — published a short pamphlet describing how complex numbers could be represented as points in a plane. He introduced the concepts of modulus and what we now call the argument, and even used the diagram to prove the triangle inequality for complex numbers. Although Caspar Wessel had independently published the same idea in 1799 (in Danish, so it went largely unnoticed), and Carl Friedrich Gauss used the idea privately from 1796, Argand's name became attached to the plane through historical accident and the advocacy of later writers. Argand was entirely self-taught in mathematics and had no professional academic position. His 1806 pamphlet was published anonymously, and its authorship was only established years later after a literary detective story involving the mathematician Legendre.

Modulus of a Complex Number — The modulus |z| = √(a² + b²) is the distance from the origin in the Argand plane and is central to interpreting complex numbers geometrically.
Polar Form of Complex Numbers — The Argand plane provides the geometric foundation for writing z = r(cos θ + i sin θ), connecting modulus and argument directly to coordinates.
Complex Conjugate — On the Argand plane the conjugate z̄ = a − bi is the reflection of z in the real axis, giving geometric meaning to an otherwise algebraic operation.
The Unit Circle — The unit circle |z| = 1 in the Argand plane is the set of all complex numbers with modulus 1, linking complex numbers to trigonometry and Euler's formula e^(iθ) = cos θ + i sin θ.