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Mod-arg & Polar forms
rcisθ and eiθ
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Exercises
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rcisθ and eiθ
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
No exercises available for this concept.
The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:
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By noticing a right angled triangle, we can say that
When a>0:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:
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When a<0:
When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have
This can be seen on the complex diagram by remembering that bi lies on the yi axis:
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Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:
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It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that
And we use the shorthand cisθ=cosθ+isinθ:
There is one more way to express complex numbers:
For example, we can write z=−1=1⋅cis(π)=eiπ. This leads to the classic result
We call this Euler's form (or sometimes exponential form) because of the presence of Euler's number, e.
The main advantage of Euler's form is that it makes multiplying complex numbers much easier:
In words, when we multiply two complex numbers the arguments add and the moduli multiply.
Similarly for division:
In words, when we divide one complex number from another, we subtract the arguments and divide the moduli.
In polar form this becomes:
If z=rcisθ=reiθ, then the conjugate z∗ is