Write the complex number z = −6 in polar form
-6 = -6+0i = (-6,0), so polar is (6,π)
To express the complex number z = -6 in polar form, we need to find its magnitude (r) and argument (θ).
Step 1: Calculate the magnitude (r):
The magnitude (r) of a complex number is given by the absolute value of the complex number. In this case, |z| = |-6| = 6.
Step 2: Calculate the argument (θ):
The argument (θ) of a complex number can be determined using trigonometry. Since the real part of z is negative and the imaginary part is 0, the argument is 180 degrees (or π radians).
Step 3: Write the polar form:
Now we can express the complex number z = -6 in polar form as z = 6∠180° (or z = 6∠π radians).
To write a complex number in polar form, we need to express it in terms of its magnitude (also known as the modulus) and argument (also known as the angle).
Let's start by finding the magnitude, which is the distance of the complex number from the origin (0,0) in the complex plane. The magnitude is given by |z| = sqrt(Re(z)^2 + Im(z)^2), where Re(z) is the real part of z, and Im(z) is the imaginary part of z.
For the complex number z = -6, the real part is -6, and the imaginary part is 0. Therefore, |z| = sqrt((-6)^2 + 0^2) = sqrt(36) = 6.
Next, we need to find the argument, which represents the angle between the positive real axis and the line connecting the origin and the complex number in the complex plane. We can use the arctan function to find the argument. The argument (θ) is given by θ = atan2(Im(z), Re(z)), where atan2 is a function that takes both imaginary and real parts as inputs.
For the complex number z = -6, the real part is -6 and the imaginary part is 0. Therefore, θ = atan2(0, -6) = 0 degrees (or in radians, 0).
Now we have the magnitude (|z|) and the argument (θ). We can write the complex number z in polar form as z = |z| * (cos(θ) + i*sin(θ)).
In this case, |z| = 6 and θ = 0. Therefore, z = 6 * (cos(0) + i*sin(0)).
Simplifying further, we know that cos(0) = 1 and sin(0) = 0, so the polar form of z is z = 6 * (1 + 0i).
Therefore, z = 6.