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

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?

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?e^iθin the form?z?=?r?cosθ+ (r?sinθ)i and then follow the steps above