I have to write a paper and give a presentation on the topic

File: 56492658.jpg (22KB, 377x264px) Image search: [Google]

22KB, 377x264px

>>9170752
>he hasn't been paying attention in class
ah, i remember high school, good times

>>9170752
Smoothness.

>>9170765
Explain please.

>>9170752

I am kinda in the middle of being thaught complex numbers as well. So far as I understand it complex numbers are a way to write two dimensional numbers, where the real numbers sit on a real axis and the imaginary numbers sit on the imaginary axis. Due to this complex numbers can be thought of as vectors (or rightangled triangles) where the length of the two catheti are given by the size of the real and imaginary number which make up the complex number. The hypotenuse (or the length of the vector) in that regard is called the modulus, whereas the angle between the modulus and the x-axis is called the argument.

There exist two main ways of writing complex numbers (representations), the rectangular form (i.e. Z=3+5*i) and the polar form (i.e. z=IzI*e^(i*arg(z)). The polar form can be derived through trigonometry as well as Euler's formula (https://en.wikipedia.org/wiki/Euler%27s_formula). Complex numbers on polar form can easily be converted into rectangular form and vice-versa through trigonometry.

Complex numbers are interesting because they can be used to describe a wide range of phenomena (i.e. electrical impedance, Schrödinger's formula, etc.) which are useful within physics and engineering.

>>9170752
>why the fuck should anyone care?
That's a pretty interesting question. Why should anyone give a shit about mathematics at all? Any pragmatic arguments fail because so many mathematicians claim to study mathematics for its own sake. Then it just seems like mathematicians are in their own world with their own rules, and you should care about a piece of mathematics because it relates to other pieces of mathematics, and that's "interesting," in whatever way you define "interesting," and, apparently, "interesting" things are worthwhile to explore.

>>9170890
I'm asking for "cool" things about complex numbers that can make my audience oo and awe.

>>9170898
R I E M A N N

M A P P I N G

T H E O R E M

File: 1501979289983.png (160KB, 695x900px) Image search: [Google]

160KB, 695x900px

>>9170752
So basically I'm just going to dump various videos and playlists which mention complex numbers, see if you like anything in the videos.

Welch Lab's "Imaginary Numbers are Real", title should imply what it's about.
https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

3Blue1Brown's videos makes mention of the complex numbers, you can probably find some interesting properties of Imaginary Numbers through these videos. I'll leave a couple of videos which bring up the role of Imaginary numbers in these topics at least once.

First one: "Some light quantum mechanics", brings up how Complex numbers can be used to describe waves.
https://youtu.be/MzRCDLre1b4

Second One: "Pi hiding in prime regularities" and "All possible pythagorean triples, visualized", these videos bring up Guassian Primes and Integers, could be interesting if you would like valuable uses of Complex Numbers in a manner that isn't too complex (heh) in the manner of concepts mentioned for your peers (which I assume are in about high school or maybe undergrads)
https://youtu.be/QJYmyhnaaek
https://youtu.be/NaL_Cb42WyY

Third One: "Visualizing the Riemann Zeta Function and Analytic Continuation", it has a small part which explains what it means for a real number (don't get onto my case for mentioning real numbers, bergerfags) to be raised to an Imaginary power
https://youtu.be/sD0NjbwqlYw

There are also some cool videos on Complex/Imaginary Numbers on Numberphile, I'll post one of their videos on it and leave it to you to dig deeper.

"Complex Numbers are Awesome" by Numberphile2, the title should imply what it is about.
https://youtu.be/-IJuqR6nz_Q

I'm sure there are more videos which take a closer look at the properties of Imaginary Numbers but those are just the videos that I've seen recently that have mentioned them in some fashion or another. Good luck and also don't post semi-homework posts like these /sci/, the rules on the sticky clearly state they belong on >>>/wsr/

>>9170898
In this video, Dr. Koch manages to point out a connection between every meme math concept... using complex numbers:
https://www.youtube.com/watch?v=gv0TFo-G8Ik

>>9171012
>>9171025
>>9171014
Thanks senpai.

File: pictured, two i.png (1MB, 1132x926px) Image search: [Google]

1MB, 1132x926px

>"today we're going to talk about something very beautiful."
>a man appears, reddened and doused in anxious sweat
>"I use the complex numbers every day"
>"we represent [math]\sqrt{-1}[/math] by i"
>he tries to open his marker but can't grip due to sweat
>shakily, he begins speaking through a grimace
>"a mathematician always has to use latex"
>he puts on a latex glove and writes down the letter i on a lunch bag
>"as you can see"
>he chuckles between gasps
>"I have two eyes"
>he blushes while coyly looking at the camera
>camera zooms onto one of his eyes
>his eye bulges like the moon
>"without the number i"
>his eye noticeably moistens
>it's unclear if it's from sweat or tears
>camera zooms back out
>he moans, sobbing
>"i wouldn't be able to see"
>he pulls a slice of white bread out of the lunch bag
>the rest of the video is him wiping his tears with the bread and then eating it

File: 1505376826335.gif (494KB, 245x240px) Image search: [Google]

494KB, 245x240px

>>9171046
Into the screencap archive you go next to mr. klein bottle and 5 fingers. Shame there aren't more Numberposts, I'm too lazy and horrible at writing to make them myself

>>9171046
top chuckle

>>9170752
Well, the obvious example would be
e^(iθ) = cos(θ) + isin(θ).
And this result often helps you to get general solutions to specific types of differential equations.

i is also used in Schrodinger's equation

>>9171325
^This is my post. I meant a general solution in a sinusoidal form. Sorry.

How about sucking my dick

>>9170898
the stereographic projection is kind of cool and easy to explain
the theorem that circles on [math]S^2[/math] stereographically project circles in [math]\mathbb{C}_\infty[/math] is kind of unintuitive and pretty neat, but also easy to demonstrate geometrically

>>9171046
Kek. Glad to see a new one of these popped up.

>>9170898
Unless your class consists of a bunch of geeky nerds, that's probably not going to happen.