Looking Deeper at $\sqrt{-1}$
A lyrical poem by DAoM student Shane Murray
It's impossible, at first glance
To solve this type of problem
We don't have a fighting chance
There are just so many of them
I mean Argand plane? Cartesian plane?
What I thought is that all planes were the same!
Thinking about it differently might not keep me sane
Looking back, I'm the one to blame
For not noticing that this can't be true
First Argand planes have imaginary numbers on the vertical axis
Cartesian planes are different as we already knew
They have imaginary numbers instead, that's what the fact is
Turns out all along I had the wrong attitude
Figure out complex solutions wasn't that bad
All I needed was the measurements of argument and magnitude
No algebra needed at all, I wasn't even mad
Add the arguments and multiply the mags
This opened my eyes to finding solutions using geometry
I'll leave all the algebra to the old hags
Once I started thinking this way I was home free
Solutions to the complex equations could be found on the plane
I'm surprised math like this hasn't gone viral
The solutions could be summed up in the shape it became
Upon closer look the solutions came out to be a spiral
One of the best parts of geometry is the patterns it entails
$z^2$, $z^3$, $z^4$ have 2, 3, and 4 complex numbers as solutions
Even graphing these show the pattern never fails
Geometry, in fact, eliminates the confusions
Look back on the completion of this work
I realized that my original view was just a misunderstanding
My professor wasn't trying to seem like a jerk
Having opened our minds it was nothing more than smooth sailing
