1.6.1 Geometry of Complex Numbers

Geometry of Complex Addition & Subtraction

What does addition look like on an Argand diagram?

In Cartesian form two complex numbers are added by adding the real and imaginary parts
When plotted on an Argand diagram the complex number z₁+ z₂ is the longer diagonal of the parallelogram with vertices at the origin, z₁, z₂ and z₁+ z₂

What does subtraction look like on an Argand diagram?

In Cartesian form the difference of two complex numbers is found by subtracting the real and imaginary parts
When plotted on an Argand diagram the complex number z₁- z₂ is the shorter diagonal of the parallelogram with vertices at the origin, z₁, - z₂ and z₁- z₂

What are the geometrical representations of complex addition and subtraction?

Let w be a given complex number with real part a and imaginary part b
- $w = a + b i$
Let z be any complex number represented on an Argand diagram
Adding w to z results in z being:
- Translated by vector $(a b)$
Subtracting w from z results in z being:
- Translated by vector $(- a - b)$

Worked Example

Consider the complex numbers z₁ = 2 + 3i and z₂ = 3 - 2i.

On an Argand diagram represent the complex numbers z₁, z₂, z₁+ z₂and z₁- z₂.

Geometry of Complex Multiplication & Division

What do multiplication and division look like on an Argand diagram?

The geometrical effect of multiplying a complex number by a real number, a, will be an enlargement of the vector by scale factor a

For positive values of a the direction of the vector will not change but the distance of the point from the origin will increase by scale factor a
For negative values of a the direction of the vector will change and the distance of the point from the origin will increase by scale factor a

The geometrical effect of dividing a complex number by a real number, a, will be an enlargement of the vector by scale factor 1/a

For positive values of a the direction of the vector will not change but the distance of the point from the origin will increase by scale factor 1/a
For negative values of a the direction of the vector will change and the distance of the point from the origin will increase by scale factor 1/a

The geometrical effect of multiplying a complex number by i will be a rotation of the vector 90° counter-clockwise

i(x + yi) = -y + xi

The geometrical effect of multiplying a complex number by an imaginary number, ai, will be a rotation 90° counter-clockwise and an enlargement by scale factor a

ai(x + yi) = -ay + axi

The geometrical effect of multiplying or dividing a complex number by a complex number will be an enlargement and a rotation

The direction of the vector will change

The angle of rotation is the argument

The distance of the point from the origin will change

The scale factor is the modulus

What does complex conjugation look like on an Argand diagram?

The geometrical effect of plotting a complex conjugate on an Argand diagram is a reflection in the real axis

The real part of the complex number will stay the same and the imaginary part will change sign

Worked Example

Consider the complex number z = 2 - i.

On an Argand diagram represent the complex numbers z, 3z, iz, z* and zz*.