Optional Practice Problem: Complex Numbers
Consider the complex number x = 1+1j (where j represents the imaginary unit, \sqrt{-1}).
Compute the following values. Each of the following can be represented in rectangular form (as a + bj) or in polar form (as re^{j\theta}). Write each of the following values in rectangular and polar form by specifying appropriate values of a, b, r, and \theta to represent that number.
You can enter your answers as python expressions, which may include the sqrt function and the constant pi.
Computing x^{0}
If x=1+1j, what is the value of x^{0}, in rectangular and polar coordinates?
Computing x^{1}
If x=1+1j, what is the value of x^{1}, in rectangular and polar coordinates?
Computing x^{2}
If x=1+1j, what is the value of x^{2}, in rectangular and polar coordinates?
Computing x^{3}
If x=1+1j, what is the value of x^{3}, in rectangular and polar coordinates?
Computing x^{4}
If x=1+1j, what is the value of x^{4}, in rectangular and polar coordinates?
Computing x^{63}
If x=1+1j, what is the value of x^{63}, in rectangular and polar coordinates?
General Trends
What general pattern do you notice in the rectangular forms of these numbers? How is the angle of the response changing with $n$? How is the magnitude of the response changing with $n$?
Show/Hide
In polar form, we can represent a number as $re^{j\theta}$, where $r$ is the magnitude of the number, and $\theta$ is its angle in the plane.In this case, we had $1+1j$, which has a magnitude of $\sqrt{2}$ and an angle of $\frac{\pi}{4}$ radians. The polar form of this number, therefore, is: $$\sqrt{2} e^{j\frac{\pi}{4}}$$
When we raise this to the power $n$, we have: $$x^n = \left(\sqrt{2} e^{j\frac{\pi}{4}}\right)^n = \left(\sqrt{2}\right)^n \left(e^{j\frac{n\pi}{4}}\right)$$
This is a new complex number whose magnitude is $(\sqrt{2})^n$, and whose angle is $n\pi / 4$.
When $n=2$, we have magnitude $(\sqrt{2})^2 = 2$ and angle $2\pi / 4 = \pi / 2$.
When $n=3$, we have magnitude $(\sqrt{2})^3 = 2\sqrt{2}$ and angle $3\pi / 4$.
When $n=4$, we have magnitude $(\sqrt{2})^4 = 4$ and angle $4\pi / 4 = \pi$.
Each timestep, the magnitude is changing by a factor of $\sqrt{2}$ and the angle is increasing by $\pi / 4$.
Try sketching the locations of $(1+1j)^k$ for $k=0,1,2,\ldots,6,7,8$ on the complex plane. How do the patterns from above manifest in your graph?
Show/Hide
