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

The sum $z_1 + z_2$z1​+z2​ corresponds to vector addition — the parallelogram rule. If you draw vectors from the origin to $z_1$z1​ and $z_2$z2​, the sum $z_1 + z_2$z1​+z2​ is the diagonal of the parallelogram formed by these two vectors.

Subtraction

The difference $z_1 - z_2$z1​−z2​ can be thought of as the vector from $z_2$z2​ to $z_1$z1​. Geometrically, $|z_1 - z_2|$∣z1​−z2​∣ is the distance between the points $z_1$z1​ and $z_2$z2​ in the Argand diagram.

Conjugation

The conjugate $\bar{z}$zˉ is the reflection of $z$z in the real axis. If $z$z is above the real axis, $\bar{z}$zˉ is the same distance below.

Negation

$-z$−z is the rotation of $z$z through $180°$180° about the origin (equivalently, the reflection in the origin).

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Worked Example: Plotting Complex Numbers

Plot the following on an Argand diagram: $z_1 = 3 + 2i$z1​=3+2i, $z_2 = -1 + 4i$z2​=−1+4i, $z_1 + z_2$z1​+z2​, $\bar{z}_1$zˉ1​.

• $z_1 = (3, 2)$z1​=(3,2): point 3 right, 2 up
• $z_2 = (-1, 4)$z2​=(−1,4): point 1 left, 4 up
• $z_1 + z_2 = 2 + 6i = (2, 6)$z1​+z2​=2+6i=(2,6): point 2 right, 6 up
• $\bar{z}_1 = 3 - 2i = (3, -2)$zˉ1​=3−2i=(3,−2): reflection of $z_1$z1​ in the real axis

You can verify the parallelogram rule: the vectors to $z_1$z1​ and $z_2$z2​ form two sides of a parallelogram, and $z_1 + z_2$z1​+z2​ is the opposite corner.

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Distance in the Argand Diagram

The distance between two complex numbers $z_1 = a + bi$z1​=a+bi and $z_2 = c + di$z2​=c+di is:

$|z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2}$∣z1​−z2​∣=(a−c)2+(b−d)2​

This is exactly the Euclidean distance formula.

Example: Find the distance between $z_1 = 3 + i$z1​=3+i and $z_2 = -1 + 4i$z2​=−1+4i.

$|z_1 - z_2| = |(3 + i) - (-1 + 4i)| = |4 - 3i| = \sqrt{16 + 9} = 5$∣z1​−z2​∣=∣(3+i)−(−1+4i)∣=∣4−3i∣=16+9​=5

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Multiplication by i

Multiplying a complex number by $i$i rotates it 90° anticlockwise about the origin:

$i \cdot (a + bi) = -b + ai$i⋅(a+bi)=−b+ai