1.5.3 Introduction to Argand Diagrams

What is the complex plane?

• The complex plane, sometimes also known as the Argand plane, is a two-dimensional plane on which complex numbers can be represented geometrically
• It is similar to a two-dimensional Cartesian coordinate grid
• The x-axis is known as the real axis (Re)
•  The y-axis is known as the imaginary axis (Im)
• The complex plane emphasises the fact that a complex number is two dimensional
• i.e it has two parts, a real and imaginary part
• Whereas a real number only has one dimension represented on a number line (the x-axis only)

What is an Argand diagram?

• An Argand diagram is a geometrical representation of complex numbers on a complex plane
• A complex number can be represented as either a point or a vector
• The complex number x + yi is represented by the point with cartesian coordinate (x, y)
• The real part is represented by the point on the real (x-) axis
• The imaginary part is represented by the point on the imaginary (y-) axis
• Complex numbers are often represented as vectors
• A line segment is drawn from the origin to the cartesian coordinate point
• An arrow is added in the direction away from the origin
• This allows for geometrical representations of complex numbers

What are complex roots?

• A quadratic equation can either have two real roots (zeros), a repeated real root or no real roots
• This depends on the location of the graph of the quadratic with respect to the x-axis
• If a quadratic equation has no real roots we would previously have stated that it has no real solutions
• The quadratic equation will have a negative discriminant
• This means taking the square root of a negative number
• Complex numbers provide solutions for quadratic equations that have no real roots

How do we solve a quadratic equation when it has complex roots?

• If a quadratic equation takes the form ax² + bx + c = 0 it can be solved by either using the quadratic formula or completing the square
• If a quadratic equation takes the form ax² + b = 0 it can be solved by rearranging
• The property i = √-1 is used
  • 
• If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs
  • If z = m + ni (n ≠ 0) is a root of a quadratic with real coefficients then z* = m - ni is also a root
• The real part of the solutions will have the same value as the x coordinate of the turning point on the graph of the quadratic

• When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
  • To solve these you can use the quadratic formula

How do we factorise a quadratic equation if it has complex roots?

• If we are given a quadratic equation in the form az² + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0 we can use its complex roots to write it in factorised form
• Use the quadratic formula to find the two roots, z  = p + qi and z* = p - qi
• This means that z – (p + qi) and z – (p – qi) must both be factors of the quadratic equation
• Therefore we can write az² + bz + c = (z – (p + qi))( z – (p - qi))
• This can be rearranged into the form (z – p – qi)(z – p + qi)