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IB DP Maths: AA HL復習筆記1.9.2 Forms of Complex Numbers

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月11日 下午4:19

How do I write a complex number in modulus-argument (polar) form?

How do I multiply complex numbers in modulus-argument (polar) form?

  • The main benefit of writing complex numbers in modulus-argument (polar) form is that they multiply and divide very easily
  • To?multiply?two complex numbers in modulus-argument (polar) form we?multiply their moduli?and?add their arguments
  • The rules of?multiplying the moduli?and?adding the arguments?can also be applied when…
  • The rules for multiplication can be proved algebraically by multiplying?z1?=?r1?cis (θ1) by?z2?=?r2?cis (θ2), expanding the brackets and using compound angle formulae

How do I divide complex numbers in modulus-argument (polar) form?

  • To?divide?two complex numbers in modulus-argument (polar) form, we?divide their moduli?and?subtract their arguments
  • The rules for division can be proved algebraically by dividing?z1?=?r1?cis (θ1) by?z2?=?r2?cis (θ2) using?complex division?and the compound angle formulae

Exam Tip

Worked Example

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

 

 

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

Exponential (Euler's) Form

How do we write a complex number in Euler's (exponential) form?

    • This shows a clear link between exponential functions and trigonometric functions
    • This is given in the formula booklet under 'Modulus-argument (polar) form and exponential (Euler) form'
  • The argument is normally given in the range 0 ≤ θ < 2π
    • However in exponential form other arguments can be used and the same convention of adding or subtracting 2π can be applied

How do we multiply and divide complex numbers in Euler's form?

    • Euler's form allows for quick and easy multiplication and division of complex numbers

What are some common numbers in exponential form?

Exam Tip

  • Euler's form allows for easy manipulation of complex numbers, in an exam it is often worth the time converting a complex number into Euler's form if further calculations need to be carried out
    • Familiarise yourself with which calculations are easier in which form, for example multiplication and division are easiest in Euler's form but adding and subtracting are easiest in Cartesian form

Worked Example

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

Conversion of Forms

Converting from Cartesian form to modulus-argument (polar) form or exponential (Euler's) form.

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

Converting from modulus-argument (polar) form or exponential (Euler's) form to Cartesian form.

  • To convert from modulus-argument (polar) form to Cartesian form
    • Write?z?=?r?(cosθ+ isinθ)as?z?=?r?cosθ+ (r?sinθ?)i
    • Find the values of the trigonometric ratios?r?sinθand?r?cosθ
      • You may need to use your knowledge of trig exact values
    • Rewrite as?z = a + bi where
      • a = r?cosθ and?b?=?r?sinθ
  • To convert from exponential (Euler’s) form to Cartesian form first rewrite?z?=?r?eiθin the form?z?=?r?cosθ+ (r?sinθ)i and then follow the steps above

Exam Tip

  • When converting from Cartesian form into Polar or Euler's form, always leave your modulus and argument as an exact value
    • Rounding values too early may result in inaccuracies later on

Worked Example

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

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

 

 

 

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