DP IB Maths: AI HL

Revision Notes

1.5.3 Introduction to Argand Diagrams

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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

1-8-1-ib-aa-hl-argand-diagram-diagram-1

Exam Tip

  • When setting up an Argand diagram you do not need to draw a fully scaled axes, you only need the essential information for the points you want to show, this will save a lot of time

Worked example

a)
Plot the complex numbers z= 2 + 2i  and z2 = 3 – 4i as points on an Argand diagram.

1-8-3-ib-hl-aa-argand-diagrams-we-a

b)
Write down the complex numbers represented by the points A and B on the Argand diagram below.

1-8-3-ib-hl-aa-argand-diagrams-we-b

Complex Roots of Quadratics

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
    • square root of negative a end root equals square root of a cross times negative 1 end root equals square root of a cross times square root of negative 1 end root
  • If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs
    • If z = p + qi (q ≠ 0) is a root of a quadratic with real coefficients then z* = p - qi 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 = a(z – (p + qi))( z – (p - qi))
    • This can be rearranged into the form a(z p qi)(z p + qi)

Exam Tip

  • Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation
    • You should get 0 if correct! [Note: 0 is equivalent to 0 plus 0 bold i]

Worked example

Solve the quadratic equation z2 - 2z + 5 = 0 and hence, factorise z2 - 2z + 5.

1-9-3-ib-aa-hl-complex-roots-we-solution-1-a

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Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.