At 9:27 shouldn't the blue cos have an offset of 90°? like this it's two sin waves but not one sin and one cos.

The level of care put into these videos is matched only by Spongebob tucking pickles under a cheese blanket and reading them a bedtime story.

Tibees is the person we all need. She's so relaxing and calm

Tibees is an imaginary friend for many people, while being real.

I remember finding out about imaginary numbers and being like "OK, cool... but why?" This explains it really well.

I m an high school student in Turkey. Our educational system is all about rote learning and the curriculum is too overloaded. This kind of videos ( logically explaining..) really helpful. Thanks a lot

4:43, why is Tibees in the back? Tibees: This is the negative real world This is one of the greatest pieces of cinematography I've ever seen. Style and content perfectly fuse to produce meaning! This youtube video needs to be archived for future generations.

I was taught imaginary numbers in a 3 year engineering degree, and also have watched a load of KZfaq videos including the whole of the "Imaginary numbers are real" series and this is the first time I feel like I've got a grasp of why imaginary numbers work and are useful. It was your line about "imaginary numbers are good at describing circular motion" that really made things click. Als the 45° example. Thanks very much 🙏

That really started to make sense! It's been over 30 years since I learned about imaginary numbers and only now are they starting to make any sense. More "Joy of Mathmatics", please and thank you.

That cat’s cuteness is immeasurable! It’s just like trying to use a ruler to mesure the sqrt(-1)

If you were my Math's teacher, I would be PhD mathematics today.

That one dislike is from tibees... the cat accidentally clicked on the mouse

ive studied imaginary numbers before but im pretty glad to say that I've gained a new perspective on this topic!! Thanks Toby!

Watched this twice because of her voice, it's like she makes my brain feel my heart beat

I remember learning about imaginary numbers at high school, and thinking "what a waste of time". And then a few years later I was training as an electronics technician (late '70s), and learning that I had to use imginary numbers for waveguide engineering calculations. Not so imaginary now, I thought!

Traveling back and forth between the real world and the imaginary world feels so wonderful! Thank you so much for providing us with these precious passports, Toby!

Never knew math could be so peaceful and relaxing, thank you

Your Bob Ross style videos are always a treat.

YESSSSSS. Joy of mathematics is back!

Your voice is very calm and soothing.It’s captivating and helps to make learning easy

This is so enveloping and hugging. There is something just magical about the way you go about it all. I wish I had a teacher like you as a child. I would have excelled much further and would have felt supported instead of stressed by the mathematics & physics that I bonded with so well, whilst not bonding well at all with the humans that taught them to me - which in combination with a difficult household environment, led to me not advancing with math despite that huge potential teachers kept telling me I had. I think what I'm trying to say, is thank you, and I believe you are wonderfully magical and pure. Do you teach children? Have you considered it? Some of the more talented ones are talented because it's their escape, and they may need a guide through the door to imagination. Again, I wish I had you as a teacher growing up ♥ No one else understood my attempts at four dimensional humor, either. You totally would have.

Hello Tibees, (1+i) is not on the circle, so there's a sqrt(2) factor involved as well as a rotation. Our intuition can easily be carried away by (1+epsilon i) which is indeed a infinitesimal rotation :-) That said, very nice video as always. I love your work and your way of presenting things, which is always clever, beautiful and elegant. Thanks for being who you are and best regards.

This is my new favorite quote regarding imaginary numbers: 6:42 "Despite the name, they are not imaginary at all. Some people just have trouble seeing them."

I can’t express how excited I am that this is back

You taught me more in 10 minutes than my Physics professor did in a semester

Thanks Toby.

Such a great video! Thanks! I've never read _Alice Through the Looking Glass_ but as you were drawing and placing yourself inside and outside the 'imaginary' world, and the 'negatives' of those worlds, I thought of that story and was thinking it could totally be an allegory about imaginary and negative numbers. I wonder if Lewis Carroll had that in mind?

I left the topics on imaginary numbers in my text book years back, untouched. Now they are courting me again. What a way to relax my mind, listening effortlessly to Tibees' genius teaching style. She's like Vivaldi piece, listening to her.

these videos are just, truly magical

That was unbelievably good. I've filled notebooks on imaginary numbers and operations with complex numbers, but that was excellent.

I just love listening to your smooth soothing voice. I was sitting by the kitchen window looking out to my parent’s patio still holding my school bag and watching your program on my iPad. I woke up with my iPad on the floor and I had to rewatch you. I would love having you as a prof...it’ll make me actually want to attend classes. Love you so much, Toby. ...Diane.

What can be more adorable than this. I teach maths to small grade students in exact the same way ma’am does. but we gotta appreciate her drawing skills too! Thanks!

I love this series so much!! It's very relaxing

Loving how she uses chalk as a medium.

I'm a 71 year old retired electronic engineer and I have to say that's the first time I've really understood why imaginary numbers are good for describing electromagnetic waves - thank you!

We learn so much from you, much more than just mathematics: A way of being in the world. A pleasant way to teach. Art. Math. Kindness

this is one of the best videos on the internet

So relaxing explanation your accent is southing too 😍

"But mostly we will be learning how to escape from reality..." . Glad you got down to business quick, lol. Why I love this channel.

I was told in first grade that you couldn't subtract a greater number from a smaller number. When I got home I was feeling a bit transgressive, so got hold of my father's calculator, and gave such a subtraction a try. When I reported back the results the teacher just said the calculator must have been mistaken. It was a pretty discouraging experience. In retrospect, I think I was searching for patterns so I could infer new knowledge from principles instead of spending time on tedious rote memorization, as I was very lazy as a child (and still is). It took 15 years for me to realise that this was the correct way all along, unfortunately being too lazy to do the rote memorization I ended up being quite bad at math.

As imaginary as my relationship with Tibee.

As soon as you revealed the map, everything you said came full circle

I really enjoy the picture. I especially like that you included the roots of the trees in there, and in a way that they show that they draw the nutrients from the ground.

I learnt about imaginary numbers 16-17 years ago. I used them many times, I know how to calculate them. For all these years I didn't know about this angle rotation like from your example. This story about a forest and changing path by 45 degrees and how we can calculate this using imaginary numbers... IT'S BRILLIANT! Now I understand why I need it 😂

interesting to think about what other numbers we don't yet know about... thanks for making maths so enjoyable Tibees :)

Hooray! Absolutely fantastical and enlightening vid. Thanks.

"We're sorry. You have reached an imaginary number. Please hang up, rotate your phone 90 degrees, and dial again."

One of the best maths series on KZfaq. Running the concepts along the bottom helped it feel even more true to the original.

This is really mind opening! Lovely presentation and in a fun way, honestly great job!

3am, I should get into another imaginary world but can't stop watching this video. :)

Thanks. I WAS going to play a video game but now I want to learn some new maths.

9:10 brilliant way of understanding waves from basic complex numbers

Such a relaxing math lesson from such a wonderful teacher 😊

My math teachers were never good at coloring or drawing stuff but yes they draw decent enough shapes for us to understand

This has to be the most soothing voice ever recorded by a human❤️❤️❤️

The change of camera angles made everything so clear!!!

An imaginary friend needs an imaginary friend to say "oh yes. we're both real"?

Saludos desde Puerto Rico. Me encanta la forma como has explicado este tema. Es fantástico como integras el arte y la narración con las matemáticas .

Thanks Toby!

"But mostly, we will be learning to escape from reality. Some people do this through drawing, painting, reading or playing video games." Wow, I do all of these!

Me after watching Tibees videos: I should top this year in Mathematics. But reality is often disappointing

good morning Toby

Favorite series from your channel dear Tibees! Thank you so much!

I've never met a mathematician/physicist who was also an artist. I just watched a video of yours from 3 years ago where you talked about being a "nerd" and not "partying". I'm not a scientist at all, but you're an awesome, lovely person.I would give anything to buy you a pint and wax philosophical/physics with you all day. Cheers.

Toby this manner is really nice like i just enjoy these kind of videos and don't forget your cat is damn cute 😄

This was so sweet! Just finished class on advanced complex geometry.

I’ve been watching your videos from quite some time now. You’ve been fantastic in explaining each and every concept and your review videos of different examinations are also great to watch. Best of luck for your future endeavours!✨

Very nice to see you again! It's been a very visual explanation of complex numbers. I have never try to understood as rotation while we are multipling by i number. Now I have a more clear concept of a complex number. Thank you!

Hi from Turkey 🇹🇷

i^4 be like: snap back to reality

Opening Scene is very apt....the joy of mathematics...always made me sleep like a cat...

I just realized that this is from 2 years ago, and here are you, sending hale and cheer (in an imaginary way?) across time, across the digital landscape.

08:08 I would get lost and die from starvation lol

is this really how a genius mind functions? wish I had that thinking

This was the saddest story I’ve heard all month. Not only did I have to go through the emotions as you worked things out between your negative friends and your imaginary friends, but I was also reminded of how stupid I am in real life.

Very creative teaching concept. If I was a fellow brainiack I'm sure I would understand all this right away. But your a heart throb. Love your hair.

Thank's from Brazil...

Escaping from reality with Toby ❤️

First the parabolas video and now this... I think Toby is trying to discreetly teach us how to direct artillery fire on a map.

Your channel deserves way more subscribers.

Hi, I'm from Ecuador 😃🇪🇨

Thanks for making my day better

The best video on imaginary numbers in KZfaq

You have such a soothing voice... if I ever should have sleep problems I will listen to your video.

Biggest fan from India, ma'am 🙃

No matter what she want to explain , she always remain calm . 😎

I don't think I've ever quite grasped the concept of imaginary numbers until this video. Pretty amazing

Multiplying by 1+i does not just rotate by 45 degrees, it also stretches by a factor of sqrt(2).

Thing make us to luv her more is She combine relaxation with mathematics😍😍😍🤩

Don't stop watching halfway or you'll be trapped in Imaginary Land.

I never thought in my life that I could watch such explanation, very phenomenal

Very relaxing and informative. I'm guessing most of your viewers already know about complex numbers, but this was enjoyable to watch!

Meow

1 D might be concentration or focus. 2 D might be point of contact between two surfaces or vision 3 D might be state of being or physical matter that present in vacuum space or sky that is infinite in scale.

I just taught my 15 year old the concept of i last month. You do it a lot better than I did. I'm going to have him watch this video. Thank you for your awesome videos.

Interesting. I will be watching for this. It’s SO critical that you’re reminding people that these are probabilities, not fundamental truths.

06:30 The crux of the matter. Lady I am 36 years old and have for the first time understood why we use imaginary numbers. Thank you. I have seen too many videos on this and understood nothing. You did the impossible for me. Respect from India 🇮🇳 🙏

Complex numbers concept was just too imaginary for me to understand until now. Thank you for making it possible for me to understand

You had me at “I’m going to run some things across the bottom of the screen.”

Hi Tibee! Im your imaginary friend from Brazil. Im glad to watch your content, its creative and interesting. And of course, your kind way to teach science is absolutely lovely.