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. $$\frac{(1+i{)}^{48}}{(\sqrt{3}-i{)}^{25}}$$

Short answer

The answer is \frac{\sqrt{3}}{4} + i \frac{1}{4} \.

Step-by-step solution

Simplify the Numerator

First, simplify $1+i$ raised to the power of 48. Since $1+i$ can be represented in polar form as \sqrt{2} e^{i \frac{\pi}{4}}\, use De Moivre's theorem: $$(1+i{)}^{48}=(\sqrt{2}{e}^{i\frac{\pi }{4}}{)}^{48}=2^{24}{e}^{i\frac{48\pi }{4}}=2^{24}{e}^{i12\pi }$$ Since \ e^{i 12\pi} = 1\, we get: $$(1+i{)}^{48}=2^{24}$$

Simplify the Denominator

Next, simplify \ (\sqrt{3} - i)^{25} \ using polar representation. \ (\sqrt{3} - i) \ has a magnitude of 2 and an angle of \ -\frac{\pi}{6} \, hence: $$(\sqrt{3}-i{)}^{25}=(2e^{-i\frac{\pi }{6}}{)}^{25}=2^{25}{e}^{-i\frac{25\pi }{6}}$$ Since \ 25\pi/6 = 4\pi + \frac{\pi}{6} \, $$e^{-i\frac{25\pi }{6}}=e^{-i(4\pi +\frac{\pi }{6})}=e^{-i\frac{\pi }{6}}$$ Hence, $$(\sqrt{3}-i{)}^{25}=2^{25}{e}^{-i\frac{\pi }{6}}$$

Compute the Quotient

Now, combine the results from steps 1 and 2: $$\frac{(1+i{)}^{48}}{(\sqrt{3}-i{)}^{25}}=\frac{2^{24}}{2^{25}{e}^{-i\frac{\pi }{6}}}=\frac{1}{2e^{-i\frac{\pi }{6}}}=\frac{e^{i\frac{\pi }{6}}}{2}$$

Express in Cartesian Form

Convert \ \frac{e^{i \frac{\pi}{6}}}{2} \ to Cartesian coordinates: \ e^{i \frac{\pi}{6}} = \cos{\frac{\pi}{6}} + i\sin{\frac{\pi}{6}} \ which translates to \ cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \ and \ sin(\frac{\pi}{6}) = \frac{1}{2} \, thus: $$\frac{e^{i\frac{\pi }{6}}}{2}=\frac{\cos \frac{\pi }{6}+isin\frac{\pi }{6}}{2}=\frac{\frac{\sqrt{3}}{2}+i\frac{1}{2}}{2}=\frac{\sqrt{3}}{4}+i\frac{1}{4}$$

Key concepts

Polar Form of Complex Numbers

Polar form is a way to express complex numbers using the magnitude (or modulus) and angle (or argument). It is particularly helpful in simplifying the multiplication and division of complex numbers.
Instead of writing a complex number as $z=x+iy$, we write it as $z=r{e}^{i\theta }$, where $r$ is the magnitude and $\theta$ is the angle.
The magnitude $r$ is calculated as $r=\sqrt{x^2+y^2}$ and the angle $\theta$ can be found using $\theta ={\tan }^{-1}\frac{y}{x}$.
When the complex number $z$ is raised to a power or multiplied/divided with another complex number, polar form makes the calculations much simpler.

De Moivre's Theorem

De Moivre's theorem is a powerful tool in complex number theory. It provides a simple way to compute powers and roots of complex numbers expressed in polar form.
The theorem states that for any complex number $z=r{e}^{i\theta }$ and any integer $n$, the $n$-th power of $z$ is given by:
$$z^n=(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }$$
This converts the multiplication of complex numbers into the addition of their angles and the multiplication of their magnitudes. This makes the process as easy as multiplying and adding real numbers.
In our example, we used De Moivre's theorem to simplify $(1+i{)}^{48}$ and $(\sqrt{3}-i{)}^{25}$. By expressing them in polar form first, we then applied De Moivre's theorem to handle the exponents easily.

Cartesian Coordinates

Cartesian coordinates are the standard way to represent complex numbers, as $z=x+iy$. Here, $x$ is the real part, and $y$ is the imaginary part.
To convert from polar form back to Cartesian coordinates, we use the formula:
$r{e}^{i\theta }=r(\text{cos}\theta +i\text{sin}\theta )$
This means we multiply the magnitude $r$ by cosine of the angle to get the real part $x$, and by sine of the angle to get the imaginary part $y$.
In our solution, we converted $\text{Extra close brace or missing open brace}$ back to Cartesian coordinates and got $\frac{}{4}+i\frac{1}{4}$. This step involves evaluating the cosine and sine of the angle, then dividing by the magnitude.