Argand Diagrams

This lesson introduces the Argand diagram, which provides a geometric representation of complex numbers. Visualising complex numbers as points (or vectors) in a plane is a powerful tool that connects algebra with geometry and opens the door to understanding modulus, argument, and loci.

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The Argand Diagram

An Argand diagram is a Cartesian plane where:

• The horizontal axis represents the real part (the "Real axis" or Re-axis)
• The vertical axis represents the imaginary part (the "Imaginary axis" or Im-axis)

The complex number $z = a + bi$z=a+bi is represented by the point $(a, b)$(a,b) or equivalently by the position vector from the origin to $(a, b)$(a,b).

Complex number	Point on Argand diagram
$3 + 2i$3+2i	$(3, 2)$(3,2)
$-1 + 4i$−1+4i	$(-1, 4)$(−1,4)
$5$5	$(5, 0)$(5,0)
$-3i$−3i	$(0, -3)$(0,−3)
$0$0	$(0, 0)$(0,0)

Key Point: Real numbers lie on the real axis, purely imaginary numbers lie on the imaginary axis.

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Geometric Interpretation of Operations

Addition