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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 1. Number & Algebra / 1.9 Further Complex Numbers / 1.9.4 De Moivre's Theorem


1.9.4 De Moivre's Theorem


De Moivre's Theorem

What is De Moivre’s Theorem?

  • De Moivre’s theorem can be used to find powers of complex numbers
  • It states that forspace z space equals space r space cis space theta,  z to the power of n space equals space left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of n blank equals r to the power of n left parenthesis cos invisible function application n theta plus isin invisible function application n theta right parenthesis blank
    • Where
      • z ≠ 0
      • r is the modulus, |z|, r ∈ ℝ+
      • θ  is the argument, arg z, θ ∈ ℝ
      • n ∈ ℝ
  • In Euler’s form this is simply:
    • open parentheses r straight e to the power of straight i theta end exponent close parentheses to the power of n equals blank r to the power of n straight e to the power of straight i n theta end exponent
  • In words de Moivre’s theorem tells us to raise the modulus by the power of n and multiply the argument by n
  • In the formula booklet de Moivre’s theorem is given in both polar and Euler’s form:
    • left square bracket r left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of n equals r to the power of n left parenthesis cos invisible function application n theta plus isin invisible function application n theta right parenthesis equals r to the power of n straight e to the power of straight i n theta end exponent equals r to the power of n blank cis blank n theta blank

How do I use de Moivre’s Theorem to raise a complex number to a power?

  • If a complex number is in Cartesian form you will need to convert it to either modulus-argument (polar) form or exponential (Euler’s) form first
    • This allows de Moivre’s theorem to be used on the complex number
  • You may need to convert it back to Cartesian form afterwards
  • If a complex number is in the form z equals r open parentheses cos invisible function application open parentheses theta close parentheses minus isin invisible function application open parentheses theta close parentheses close parentheses then you will need to rewrite it as z equals r open parentheses cos invisible function application open parentheses negative theta close parentheses plus isin invisible function application open parentheses negative theta close parentheses close parenthesesbefore applying de Moivre’s theorem
  • A useful case of de Moivre’s theorem allows us to easily find the reciprocal of a complex number:
    •  1 over z equals 1 over r left parenthesis cos invisible function application left parenthesis negative theta right parenthesis plus isin invisible function application left parenthesis negative theta right parenthesis equals 1 over r straight e to the power of negative straight i theta end exponent blank
    • Using the trig identities cos(-θ) = cos(θ) and sin(-θ) = - sin(θ) gives
    • 1 over z equals z to the power of negative 1 end exponent equals r to the power of negative 1 end exponent left square bracket cos invisible function application open parentheses theta close parentheses minus isin invisible function application open parentheses theta close parentheses right square bracket equals blank 1 over r left square bracket cos invisible function application open parentheses theta close parentheses minus isin invisible function application left parenthesis theta right parenthesis right square bracket
  • In general
    • z to the power of negative n end exponent equals r to the power of negative n end exponent left square bracket cos invisible function application open parentheses negative n theta close parentheses plus isin invisible function application open parentheses negative n theta close parentheses right square bracket equals blank r to the power of negative n end exponent left square bracket cos invisible function application open parentheses n theta close parentheses minus isin invisible function application open parentheses n theta close parentheses right square bracket blank blank blank

 

Exam Tip

  • You may be asked to find all the powers of a complex number, this means there will be a repeating pattern
    • This can happen if the modulus of the complex number is 1
    • Keep an eye on your answers and look for the point at which they begin to repeat themselves 

Worked Example

Find the value of begin mathsize 16px style open parentheses fraction numerator square root of 3 over denominator 6 end fraction plus 1 over 6 straight i close parentheses to the power of negative 3 end exponent end style,  giving your answer in the form a + bi.

o~JlLuvG_1-9-3-ib-aa-hl-de-moivres-theorem-we-solution-1

Proof of De Moivre's Theorem

How is de Moivre’s Theorem proved?

  • When written in Euler’s form the proof of de Moivre’s theorem is easy to see:
    • Using the index law of brackets: open parentheses r straight e to the power of straight i theta end exponent close parentheses to the power of n equals blank r to the power of n straight e to the power of straight i n theta end exponent
  • However Euler’s form cannot be used to prove de Moivre’s Theorem when it is in modulus-argument (polar) form
  • Proof by induction can be used to prove de Moivre’s Theorem for positive integers:
    • To prove de Moivre’s Theorem for all positive integers, n
    • left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of n equals r to the power of n left parenthesis cos invisible function application n theta plus isin invisible function application n theta right parenthesis
  • STEP 1: Prove it is true for n = 1
    • left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of 1 equals r to the power of 1 left parenthesis cos invisible function application 1 theta plus isin invisible function application 1 theta right parenthesis equals blank r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis blank
    • So de Moivre’s Theorem is true for n = 1
  • STEP 2: Assume it is true for n = k
    • left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of k equals r to the power of k left parenthesis cos invisible function application k theta plus isin invisible function application k theta right parenthesis blank
  • STEP 3: Show it is true for n = k + 1
    • left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of k plus blank 1 end exponent equals left parenthesis left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of k right parenthesis left parenthesis left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of 1 right parenthesis blank
    • According to the assumption this is equal to
      • left parenthesis r to the power of k left parenthesis cos invisible function application k theta plus isin invisible function application k theta right parenthesis right parenthesis blank left parenthesis r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right parenthesis
    • Using laws of indices and multiplying out the brackets:
      • equals blank r to the power of k plus 1 end exponent left square bracket cos invisible function application k theta cos invisible function application theta plus icos space k theta space sin invisible function application theta plus isin invisible function application k theta cos invisible function application theta plus straight i squared sin invisible function application k theta sin invisible function application theta right square bracket
    • Letting i2 = -1 and collecting the real and imaginary parts gives:
      • equals blank r to the power of k plus 1 end exponent left square bracket cos invisible function application k theta cos invisible function application theta minus sin invisible function application k theta sin invisible function application theta plus straight i left parenthesis cos space k theta space sin invisible function application theta plus sin invisible function application k theta cos invisible function application theta right parenthesis right square bracket
    • Recognising that the real part is equivalent to cos(kθ  + θ ) and the imaginary part is equivalent to sin(kθ  + θ ) gives
      • open parentheses r blank cis blank straight theta close parentheses to the power of k plus 1 end exponent equals r to the power of k plus 1 end exponent left square bracket cos invisible function application open parentheses k plus 1 close parentheses theta plus isin invisible function application invisible function application open parentheses k plus 1 close parentheses theta right square bracket blank
    • So de Moivre’s Theorem is true for n = k + 1
  • STEP 4: Write a conclusion to complete the proof
    • The statement is true for n = 1, and if it is true for n = k it is also true for n = k + 1
    • Therefore, by the principle of mathematical induction, the result is true for all positive integers, n
  • De Moivre’s Theorem works for all real values of n
    • However you could only be asked to prove it is true for positive integers

Exam Tip

  • Learning the standard proof for de Moivre's theorem will also help you to memorise the steps for proof by induction, another important topic for your AA HL exam 

Worked Example

Show, using proof by mathematical induction, that for a complex number z = r cisθ  and for all positive integers, n,

z to the power of n space equals space left square bracket r blank left parenthesis cos invisible function application theta plus isin invisible function application theta right parenthesis right square bracket to the power of n blank equals r to the power of n left parenthesis cos invisible function application n theta plus isin invisible function application n theta right parenthesis

1-9-3-ib-aa-hl-proof-of-de-moivres-theorem-we-solution



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
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                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
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                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
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                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
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                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        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.


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