Forms of Complex Numbers (DP IB Analysis & Approaches (AA)): Revision Note

The Cartesian form of a complex number, , is written in terms of its real part, , and its imaginary part, 

If we let and , then it is possible to write a complex number in terms of its modulus, , and its argument, , called the modulus-argument (polar) form, given by...

• 

• This is often written as z = r cis θ

• This is given in the formula book under Modulus-argument (polar) form and exponential (Euler) form

It is usual to give arguments in the range   or 

• Negative arguments should be shown clearly

• e.g.

  • without simplifying   to either  or 

The complex conjugate of r cis θ is r cis (-θ )

If a complex number is given in the form , then it is not in modulus-argument (polar) form due to the minus sign

• It can be converted by considering transformations of trigonometric functions

  • and 

• So   

To convert from modulus-argument (polar) form back to Cartesian form, evaluate the real and imaginary parts

• E.g.  becomes 

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

So if z₁ = r₁ cis (θ₁) and z₂ = r₂ cis (θ₂)

• z₁ z₂ = r₁r₂ cis (θ₁ + θ₂)

Sometimes the new argument, , does not lie in the range (or   if this is being used)

• An out-of-range argument can be adjusted by either adding or subtracting 

• E.g. If and   then   

• This is currently not in the range 

• Subtracting from  to give , a new argument is formed

  •  This lies in the correct range and represents the same angle on an Argand diagram

The rules of multiplying the moduli and adding the arguments can also be applied when…

• …multiplying three complex numbers together, , or more

• …finding powers of a complex number (e.g.  can be written as )

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

To divide two complex numbers in modulus-argument (polar) form, we divide their moduli and subtract their arguments

So if z₁ = r₁ cis (θ₁) and z₂ = r₂ cis (θ₂) then 

Sometimes the new argument, , can lie out of the range  (or the range  if this is being used)

• You can add or subtract  to bring out-of-range arguments back in range

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

A complex number can be written in Euler's form as  

• This relates to the modulus-argument (polar) form as 

• 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

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

If and then 

• 

  • Multiply the moduli and add the arguments

• 

  • Divide the moduli and subtract the arguments

Using these rules makes multiplying and dividing more than two complex numbers much easier than in Cartesian form

When a complex number is written in Euler's form it is easy to raise that complex number to a power

• If ,    and  

As  and  you can write:

Using the same idea you can write:

• 

• where k is any integer

As  and  you can write:

• 

• Or more commonly written as 

  • This is known as Euler's identity and is considered by some mathematicians as the most beautiful equation

As and  you can write:

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

and  

To convert from modulus-argument (polar) form to Cartesian form

• You may need to use your knowledge of trig exact values

• a = r cosθ and b = r sinθ

• Write z = r (cosθ + isinθ ) as z = r cosθ + (r sinθ )i

• Find the values of the trigonometric ratios r sinθ and r cosθ

• Rewrite as z = a + bi where

To convert from exponential (Euler’s) form to Cartesian form first rewrite z = r e^iθ  in the form z = r cosθ + (r sinθ)i and then follow the steps above

Your GDC may also be able to convert complex numbers between the various forms

• TI calculators, for example, have 'Convert to Polar' and 'Convert to Rectangular' (i.e. Cartesian) as options in the 'Complex Number Tools' menu

• Make sure you are familiar with your GDC and what it can (and cannot) do with complex numbers