Revision NotesExam QuestionsFlashcardsPast PapersMock Exams
IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraFurther Complex NumbersConversion between Forms of Complex Numbers

Conversion between Forms of Complex Numbers (DP IB Applications & Interpretation (AI)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HL

Conversion of forms

How do I convert out of Cartesian form?

  • To convert z equals x plus straight i y from Cartesian form to modulus-argument (polar) form or exponential (Euler) form, you first need to find

    • the modulus 

      • r equals open vertical bar z close vertical bar equals square root of x squared plus y squared end root

    • and  the argument

      • theta equals arg invisible function application z

    • then use

      • z equals r open parentheses cos space theta plus isin space theta close parentheses equals r straight e to the power of straight i theta end exponent equals r space cis space theta

Examiner Tips and Tricks

The relationship z equals r open parentheses cos space theta plus isin space theta close parentheses equals r straight e to the power of straight i theta end exponent equals r space cis space theta is given in the formula booklet.

How do I convert into Cartesian form?

  • To convert from modulus-argument (polar) form or exponential (Euler) form to Cartesian form

    • you need to work out the values of

      • cos space theta

      • sin space theta

      • which may be exact

    • then expand z equals r open parentheses cos space theta plus isin space theta close parentheses into the components z equals x plus straight i y

Examiner Tips and Tricks

It is not possible to go directly from exponential (Euler) to Cartesian form without having to first convert to modulus-argument (polar) form.

Examiner Tips and Tricks

Your GDC may be able to convert between complex number forms, as some have the option to 'convert to polar' or 'convert to rectangular' (Cartesian).

Worked Example

Two complex numbers are given by z subscript 1 equals 2 plus 2 straight i and z subscript 2 equals 3 straight e to the power of fraction numerator 2 pi over denominator 3 end fraction straight i end exponent.

(a) Write z subscript 1 in the form r straight e to the power of straight i theta end exponent where r greater than 0 and 0 less or equal than theta less than 2 pi.

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-a

(b) Write z subscript 2 in Cartesian form.

1-9-2-ib-aa-hl-forms-of-cn-we-solution-3-b

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself
    Previous:Exponential (Euler's) FormNext:Frequency & Phase of Trig Functions
    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.