1.9.1 Geometry of Complex Numbers

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

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

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 $(- a - b)$

Take extra care when representing a subtraction of a complex number geometrically
- Remember that your answer will be a translation of the shorter diagonal of the parallelogram made up by the two complex numbers

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.

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

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

Make sure you remember the transformations that different operations have on complex numbers, this could help you check your calculations in an exam

Consider the complex number z = 2 - i.

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

DP IB Maths: AA HL

1.9.1 Geometry of Complex Numbers