Question

Express the following complex numbers in the $x+iy$ form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others- -try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. $$4e^{-8i\pi /3}$$

Short answer

-2 - 2\sqrt{3} i

Step-by-step solution

Understand the problem

The task is to express the given complex number, which is in polar form, in the Cartesian form (as a + bi).

Identify the given complex number

The given complex number is $4e^{-8i\pi /3}$. Here, the modulus is 4 and the argument is $-\frac{8\pi }{3}$.

Simplify the argument

Notice that the argument $-\frac{8\pi }{3}$ is not in the standard interval $[0,2\pi )$. Simplify it as follows: $$-\frac{8\pi }{3}=-2\pi -\frac{2\pi }{3}$$. Since $-2\pi$ is an integer multiple of $2\pi$, it can be disregarded. Therefore, the argument simplifies to $-\frac{2\pi }{3}$.

Convert to Cartesian form

To convert from polar form to Cartesian form, use the following formulas: \ Re(z) = r \cos(\theta) \ and \ Im(z) = r \sin(\theta). So, we need to calculate: $$x=4\cos (-\frac{2\pi }{3})$$$$y=4\sin (-\frac{2\pi }{3})$$.

Calculate the cosine and sine

Recall that $\cos (-\frac{2\pi }{3})=-\frac{1}{2}$ and $\sin (-\frac{2\pi }{3})=-\frac{\sqrt{3}}{2}$. Using these values: $$x=4(-\frac{1}{2})=-2$$$$y=4(-\frac{\sqrt{3}}{2})=-2\sqrt{3}$$

Write the final answer

Combine the values of x and y to express the complex number in Cartesian form: $$4e^{-8i\pi /3}=-2-2\sqrt{3}i$$.

Key concepts

polar to Cartesian conversion

The polar to Cartesian conversion allows us to express complex numbers in a more conventional form. In polar form, a complex number is represented by its modulus and argument as $r{e}^{i\theta }$. Our goal is to convert this into Cartesian form, which looks like $a+bi$, where $a$ is the real part and $b$ is the imaginary part.
Here are the steps to convert from polar to Cartesian form:

• Identify the modulus $r$ (magnitude) and the argument $\theta$ (angle).
• Use the formulas:
  $Re(z)=r\cos (\theta )$ and $Im(z)=r\sin (\theta )$.
• Calculate the real part $a$ and the imaginary part $b$.
• Combine $a$ and $b$ to get the Cartesian form $a+bi$.

For example, converting $4e^{-8i\pi /3}$:
The modulus $r$ is 4 and the argument $-8\pi /3$ simplifies to $-2\pi /3$.
Using the trigonometric functions, we find the real part: $4\cos (-2\pi /3)=-2$ and the imaginary part: $4\sin (-2\pi /3)=-2\sqrt{3}$.
Thus, the complex number in Cartesian form is $-2-2\sqrt{3}i$.

complex number visualization

Visualizing complex numbers can make it much easier to grasp their properties. Complex numbers are often presented in a 2D plane, known as the complex plane, where the x-axis represents real numbers, and the y-axis represents imaginary numbers.
When you plot a complex number, you can think of it as a point or a vector originating from the origin (0,0) and pointing towards $(a,b)$, where $a$ is the real part and $b$ is the imaginary part.

• The length of the vector (or the distance from the origin) is the modulus $r$.
• The angle the vector makes with the positive x-axis is the argument $\theta$.

This visual approach is extremely useful in understanding operations with complex numbers, like addition and multiplication, as these operations can be seen as transformations in the complex plane. For instance, the multiplication by a complex number rotates and scales another complex number's position in the plane.

argument simplification

Arguments of complex numbers are angles and can sometimes be outside the typical range of $[0,2\pi )$. Such arguments need to be simplified to fall within this principal range. When you encounter an argument that is not within $[0,2\pi )$, you should add or subtract integer multiples of $2\pi$ to bring it into this range.
Here's how to do it:

• If $\theta$ is negative, keep adding $2\pi$ until it falls within the interval $[0,2\pi )$.
• If $\theta$ is greater than $2\pi$, keep subtracting $2\pi$ until it is within the desired range.

In our example, the argument $-8\pi /3$ is simplified as follows:
$$-8\pi /3=-2\pi -2\pi /3$$.
Since $-2\pi$ is an integer multiple of $2\pi$, it can be disregarded, giving us the simplified argument $-2\pi /3$. This is now within the interval $[-\pi ,0)$ which is often used in place of $[0,2\pi )$ for easier calculations.

trigonometric identities

Trigonometric identities are essential tools in converting complex numbers and simplifying arguments. They relate the angles and lengths in a right triangle and help us express the Cartesian coordinates of a point given its polar form.
Two key trigonometric functions used in polar to Cartesian conversion are:

• Cosine $\cos (\theta )$: represents the horizontal coordinate of a point on the unit circle for a given angle $\theta$.
• Sine $\sin (\theta )$: represents the vertical coordinate of a point on the unit circle for a given angle $\theta$.

Recall these important identities:

• $\cos (-\theta )=\cos (\theta )$
• $\sin (-\theta )=-\sin (\theta )$

In our example: $\cos (-2\pi /3)=-1/2$ and $\sin (-2\pi /3)=-\sqrt{3}/2$.
By knowing these identities and values, we were able to convert the complex number $4e^{-8i\pi /3}$ into its Cartesian form $-2-2\sqrt{3}i$.