How to convert into polar form?
z = 1 - i
w = 1 - √3i
The polar form expresses a complex number as:
z = |z| exp(i theta)
theta is the angle with the positive real axis.
For z = 1 - i, we have:
|z| = sqrt(2)
Then it's easy to see that you can take theta = -pi/4:
exp(-i pi/4) = cos(pi/4) - i sin(pi/4) =
1/sqrt(2) (1-i).
Then in case of
w = 1 - √3i
you have:
|w| = sqrt(4) = 2
And you easly see that theta = -pi/3
You can compute theta directly by taking arctan of imaginary part divided by the real part, but you may then need to add pi to this. If you multiply z by -1, theta changes by plus or minus pi, while the ratio stays the same.
In polar form, the first is (1/sqrt(2),-pi/4)
The second is (2,-pi/3)
To convert a complex number into polar form, follow these steps:
1. Express the complex number in the form a + bi, where a and b are real numbers.
For z = 1 - i, a = 1 and b = -1.
For w = 1 - √3i, a = 1 and b = -√3.
2. Calculate the magnitude (absolute value) of the complex number using the formula:
|z| = √(a^2 + b^2)
For z: |z| = √(1^2 + (-1)^2)
|z| = √(1 + 1)
|z| = √2
For w: |w| = √(1^2 + (-√3)^2)
|w| = √(1 + 3)
|w| = 2
3. Calculate the argument (angle) of the complex number using the formula:
θ = arctan(b/a)
For z: θ = arctan(-1/1)
θ = arctan(-1)
θ ≈ -45° or -π/4 radians
For w: θ = arctan(-√3/1)
θ = arctan(-√3)
θ ≈ -60° or -π/3 radians
4. Write the complex number in polar form as |z| * (cos(θ) + i*sin(θ)).
Complex number in polar form:
z = √2 * (cos(-45°) + i*sin(-45°))
Complex number in polar form:
w = 2 * (cos(-60°) + i*sin(-60°))
Note: The angle (θ) represents the counterclockwise rotation from the positive real axis to the vector representing the complex number in the complex plane.