Roots of Complex Numbers (Edexcel A Level Further Maths: Core Pure)

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Roots of Complex Numbers

How do I find the square root of a complex number?

  • The square roots of a complex number will themselves be complex:
    • i.e. if z squared equals a plus b straight i then z equals c plus d straight i
  • We can then square (c plus d straight i) and equate it to the original complex number (a plus b straight i), as they both describe z squared:
    • a plus b straight i equals open parentheses c plus d straight i close parentheses squared
  • Then expand and simplify:
    • a plus b straight i equals c squared plus 2 c d straight i plus d squared straight i squared
    • a plus b straight i equals c squared plus 2 c d straight i minus d squared
  • As both sides are equal we are able to equate real and imaginary parts:
    • Equating the real components: a equals c squared minus d squared  (1)
    • Equating the imaginary components: b equals 2 c d  (2)
  • These equations can then be solved simultaneously to find the real and imaginary components of the square root
    • In general, we can rearrange (2) to make fraction numerator b over denominator 2 d end fraction equals c and then substitute into (1)
    • This will lead to a quartic equation in terms of d; which can be solved by making a substitution to turn it into a quadratic
  • The values of d can then be used to find the corresponding values of c, so we now have both components of both square roots (c plus d straight i)
  • Note that one root will be the negative of the other root
    • g. c plus d straight i and  negative c minus d straight i

How do I use de Moivre’s Theorem to find roots of a complex number?

  • De Moivre’s Theorem states that a complex number in modulus-argument form can be raised to the power of n by
    • Raising the modulus to the power of n and multiplying the argument by n
  • When in modulus-argument (polar) form de Moivre’s Theorem can then be used to find the roots of a complex number by
    • k equals 0 comma blank 1 comma blank 2 comma space horizontal ellipsis space comma space n minus 1
    • Recall that adding 2π to the argument of a complex number does not change the complex number
    • Therefore we must consider how different arguments will give the same result
    • Taking the nth root of the modulus and dividing the argument by n
    • If z blank equals blank r left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis blankthen blank n-th root of z blank equals blank open square brackets r open parentheses cos invisible function application left parenthesis theta plus 2 pi k right parenthesis plus isin invisible function application left parenthesis theta plus 2 pi k right parenthesis close parentheses close square brackets to the power of 1 over n end exponent
    • This can be rewritten as space n-th root of z blank equals blank r to the power of 1 over n end exponent open parentheses cos invisible function application left parenthesis fraction numerator theta blank plus blank 2 pi k over denominator n end fraction right parenthesis plus isin invisible function application left parenthesis fraction numerator theta blank plus blank 2 pi k over denominator n end fraction right parenthesis close parentheses
  • This can be written in exponential (Euler’s) form as 
    • For space z to the power of n equals r straight e to the power of straight i theta end exponent,  z equals blank n-th root of r straight e to the power of fraction numerator theta plus 2 pi k over denominator n end fraction straight i end exponent
  • The nth root of complex number will have n roots with the properties:
    • The five roots of a complex number raised to the power 5 will create a regular pentagon on an Argand diagram
    • The eight roots of a complex number raised to the power 8 will create a regular octagon on an Argand diagram
    • The n roots of a complex number raised to the power n will create a regular n-sided polygon on an Argand diagram
    • The modulus is n-th root of r for all roots
    • There will be n different arguments spaced at equal intervals on a circle centred about the origin
    • This creates some geometrically beautiful results

Examiner Tip

  • de Moivre's theorem makes finding roots of complex numbers very easy, but you must be confident converting from Cartesian form into Polar and Euler's form first
    • You can use your calculator to convert between forms

Worked example

a)
Find the square roots of 5 + 12i, giving your answers in the form a + bi.

pKIYVs8H_al-fm-1-2-4-roots-of-cn-we-solu-1

b)
Solve the equation z cubed equals negative 4 plus 4 square root of 3 straight i giving your answers in the form r (cosθ + isinθ).

LAXY4Oza_al-fm-1-2-4-roots-of-cn-we-solu-2

Roots on an Argand Diagram

What are roots of unity?

  • Roots of unity are solutions to the equation z to the power of n equals 1 where n is a positive integer
  • For the equation z to the power of n equals 1 there are n roots of unity
    •  z equals straight e to the power of fraction numerator 2 straight pi k over denominator n end fraction straight i end exponentwhere k = 0, 1, 2, …, n-1
      • This is given in the formula booklet
  • These can be written 1, ω, ω², …, ωn-1
    • Where omega equals straight e to the power of fraction numerator 2 straight pi over denominator n end fraction straight i end exponent
  • The sum of the roots of unity is zero
    • 1 plus omega plus omega squared plus... plus omega to the power of n minus 1 end exponent equals 0
  • They can be used to find all the roots of the equation z to the power of n equals r straight e to the power of straight i theta end exponent
    • Find one root normally alpha equals n-th root of r straight e to the power of fraction numerator straight i theta over denominator n end fraction end exponent
    • Then the n distinct roots can be found by multiplying α by each root of unity
      • α, αω, αω², …, αωn-1 

What are the geometric properties of roots of complex numbers?

  • The n roots of any non-zero complex number r straight e to the power of straight i theta end exponent lie on a circle on an Argand diagram
    • The centre will be the origin
    • The radius will be n-th root of r
  • The n roots of unity lie on the unit circle centred about the origin
  • Regular polygons can be created by joining consecutive roots of a complex number with straight lines

edexcel-al-fm-cp-1-2-4-unity

How can I use roots of unity to solve geometric problems?

  • Roots of unity can be used to solve problems involving regular polygons centred about the origin
  • Coordinates of vertices (x, y) can be considered as complex numbers x + yi
  • If you know one vertex (x, y) you can find the others by multiplying the complex number representing the given vertex by each root of unity
    • x + yi, (x + yi)ω, (x + yi)ω², …,  (x + yi)ωn-1
    • If you write the vertex using exponential form r straight e to the power of straight i theta end exponent it can make the multiplications easier
    • Then you can just add fraction numerator 2 straight pi over denominator n end fraction to the argument to get the next vertex
  • Write all vertices in Cartesian form to get the coordinates

Examiner Tip

  • You can use your calculator to convert between polar and cartesian forms which may speed up your working
    • Just be aware of questions that may ask you to not use “calculator technology” where you need to show full working (but can still use calculator to check!)

Worked example

An equilateral triangle has its centre at the origin of a Cartesian plane. One of its vertices is at the point left parenthesis 9 comma negative 3 right parenthesis. Find the coordinates of the other two vertices.

al-fm-1-2-4-roots-on-argand-diagram-we-s

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Amber

Author: Amber

Expertise: Maths

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.