The complex plane

In mathematics, many fascinating things can be modeled using the complex plane. The complex plane has a real axis (i.e. all numbers in set $\mathbb{R}$) and an imaginary axis (respectively {$ bi : b \in \mathbb{R}$}). With these two axes, you can have numbers like:

• $3i+2$
• $-5i$
• $8$
• $-2i+5$

As the complex plane can contain more information than a single axis could, it has many applications.

The Cool Trick

We can use the equation below where $x$ is the angle measurement in radians to get the coordinates.

$$ f(x) = e^{xi} $$

The function will return an imaginary number, with the X-axis value in the real part, and the Y-axis value in the imaginary part respectively.

Trying It Out

$$ x = \frac{\pi}{3} $$

$$ f(\frac{\pi}{3}) = e^{\frac{\pi}{3} i} $$

$$ = 0.5000000000 + 0.8660254038i $$

Now that works, and we get our desired result. As $\frac{\pi}{3}$ is equivalent to $60^{\circ}$, a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle should prove sufficient in making sure everything is fine. We know that $\sqrt{3}$ is across from $60^{\circ}$, and $1$ is across from $30^{\circ}$. However, the hypotenuse is $2$, rather than being $1$ (as the radius of the unit circle is $1$). So, we divide all sides by $2$. Now, the value for the X-axis is $\frac{1}{2}$, and the value for the Y-axis is $\frac{\sqrt{3}}{2}$, perfectly matching the earlier $0.5000000000 + 0.8660254038i$.

Conclusion

I found that trick to be pretty cool. I first found out about it through a 3Blue1Brown video. You should watch it as it’s explained better there (in the context of Fourier series). Also, I hope for posts to be longer and for the typesetting to be better in the future. Thanks for reading!