IB DP Maths: AA HL复习笔记1.9.2 Forms of Complex Numbers
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
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?
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- 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
Conversion of Forms
Converting from Cartesian form to modulus-argument (polar) form or exponential (Euler's) form.
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- 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
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