The Square Root Of Negative One

In which we look at the numerous results of defining 'i' to be the square root of negative one.

00:00 Intro
00:12 The Imaginary Unit
00:47 Multiplying "i" By Itself
01:34 Imaginary And Complex Numbers
03:57 Addition
04:54 Multiplication
06:10 Square And Square Root
07:02 Euler's Identity
10:23 Review

[ankittiwari]

I appreciate your work deeply. I know how hard it is to make these animations. Thanks for doing it.

[rohan]

Imagine working countless hours and getting 600 you're so underrated bro


Nice video...<3

[BenKarcher]

Absolutely amazing video! I found this channel because of the video explaining quantum computers and decided to watch through the other videos in order to develop more intuition for the math in that video. I absolutely loved the intuition at the beginning of the video with the animations of rotating and scaling numbers. The only part I didn't like was introducing exp(z) via the Taylor series. While that is obviously a mathematically correct definition it didn't give any intuition for the properties described. After putting the words e defined by d/dx exp(x) = exp(x) in the video you have all the tools necessary to introduce what I find by far the best intuition for the described property: Take the function exp(i*y) at some point. Incrementing the function by some small step dy increment the result by i*exp(i*y)dy (yes I know infinitesimals arent formally defined like this but I find the intuition is great). This means that any output of this function the derivative is that point rotated by i (90 degrees). Or in other words, the derivative of the function is orthogonal to the line from zero to the output at every point which is one way to define a circle.

Again I loved the rest of the video and am not trying to detract from it, just trying to add a different geometric intuition because it always annoyed me in school when things like this were introduced as a Taylor series because while I could see that it was correct I didn't know why it was correct.

[erikziak]

This channel reminds me of both Viascience and 3Blue1Brown. Keep it up. Great work! Subscribed and liked. I rarely "like" videos, I do not know if the YT algorithm takes into account how often I "like" a video, but if it does, this channel should get a boost. This is really highest quality content.

[joshgroves]

It was so cool how phi was counting up to and landed on pi as the lead up for your explanation of Euler's Identity! Really awesome video!

[oh.]

i found your channel from your recent video about bits and abstraction and I’m really glad I did (but i did see the kurzgesagt one lol). I’m really interested in maths and computer science and physics and your channel has the best explanations ive ever heard in my life- really appreciate your hard work on this channel, you’re in for a lot of recognition soon if you keep going for sure😊

[asseater]

I watched this after watching your quantum computing video and what you do here is fantastic. I literally said "That's so cool!" out loud at the end lol
Great visualization, great sound design and great explanation of course!

[vitaszernys]

Great as always. Had the privilege of finding your channel and your videos are amazing. Thank you so much for contributing so beautifully to our lives.

[jeesimplified]

good work, I'll suggest your channel to my students.

[andreacarpi]

This is the very best video on the topic I have ever seen! Huge congrats to the author

[vittorioful]

Wonderful!!!! Great video and very clear explanation . Keep on going!!!! ⭐⭐⭐

[alphalunamare]

I really enjoyed this visual representation but I will admit that I stopped and reran the video, at certain points, umpteen times to clearly understand. It is like learning a new language's grammar. I think if I spent another hour that I could explain it to a high school student .. that is amazing! :-)

[jaocbj]

The clicking sounds at 4:40 are so satisfying!

Cool video

[sciencefordreamers]

Awesome value, thank you for your diligent and original work man!

[cristhianvictorrojasmarque]

Este video es una obra de arte, en verdad la forma en la que explcas y como lo complementas con las animaciones es simplemente maravillosa, gracias por este video buen hombre

[ericksuzart]

I love your content, please continue and you eventualy will get the recognition that you deserves. You make me think that complex numbers are the natural extension of the concept of real numbers not an arbritary thing. Thanks so much for this!

[antonioponce]

the way this taught me so much in so loved your demonstrations definitely made it all "click"

[Hector-bjls]

I feel like I'm learning maths backwards.

So, I got a job where I had to do a lot of Linear Algebra. I learned about vectors, matrices, and quaternions. Eventually I got pretty decent at it, but I never really understood where the formulas came from.

Then, I stumbled onto Geometric Algebra. It felt like I finally understood how some of these formula worked. It takes a different route. Vectors, Bivectors, Trivectors, etcvectors... But, it was really interesting learning about the "geometric numbers". I.e. x*x=-1, x*x=1, x*x=0.

Now, I'm learning about complex numbers. Which is basically just 2D GA. They used Euler's Identity in some of their examples, and I'd already seen it when investigating the dot product.

It's interesting coming at these things from so many different angles. Everyone is talking about the same thing, but in different ways.

[mikafoxx]

Imaginary numbers being associated with phase makes them very applicable to the real world.
Almost every radio reciever and transmitter these days use quadrature sampling, so they can sample with a 90 degree and regular phase, so that you can essentially sample above and below zero frequency. You can directly mix your target center frequency instead of needing to offset it in a way to negate the mirror images. With the imaginary component, you can then derive the difference between positive and negative frequency. It takes two dacs, one for the phase shifted "imaginary" component, and one for the real component. Both are mixed with the target frequency before getting sampled.

[jakuzzy_san]

Thank you verry much for helping me understand intiuitively my math lessons ! Great work !