🌀 Complex Numbers
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The Complex Plane

Visualize complex numbers geometrically using the Argand plane.

The complex plane (Argand plane) represents complex numbers geometrically. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number a + bi is plotted at the point (a, b).

∣a+bi∣=a2+b2(modulus = distance from origin)|a + bi| = \sqrt{a^2 + b^2} \quad \text{(modulus = distance from origin)}

The modulus (or absolute value) |z| of a complex number z = a + bi is the distance from the origin to the point (a, b). The argument θ is the angle the line from the origin makes with the positive real axis.

θ=arg⁡(z)=arctan⁡(ba)z=∣z∣(cos⁡θ+isin⁡θ)\theta = \arg(z) = \arctan\left(\frac{b}{a}\right) \qquad z = |z|(\cos\theta + i\sin\theta)

Complex Plane Features

Real axis: horizontal (Re)Imaginary axis: vertical (Im)Point (a, b) represents a + bi|z| = distance from originarg(z) = angle from positive real axis
📍Modulus of 3 + 4i
|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5. This is the distance from the origin to the point (3, 4) in the complex plane.
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Remember This!

Adding complex numbers in the plane follows the same parallelogram rule as adding vectors. Multiplying by i rotates a complex number 90° counterclockwise in the plane.

✏️ Try It!

What is the modulus of the complex number −5 + 12i?

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