Polar Form
Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.
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Polar form of complex numbers
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:
It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that
And we use the shorthand cisθ=cosθ+isinθ:
Euler's form
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.
Multiplying in polar / Euler form
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:
Complex conjugate in polar form
If z=rcisθ=reiθ, then the conjugate z∗ is
Distance in the complex plane
The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.
This is a reflection of the familiar distance formula
except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.
Angle between complex numbers in the plane
The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:
Nice work completing Polar Form, here's a quick recap of what we covered:
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