Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.
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:
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
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.
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:
Exercises checked off
Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.
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:
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
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.
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:
Exercises checked off