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.

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.

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 🙏

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

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

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 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

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

YESSSSSS. Joy of mathematics is back!

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

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.

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!✨

Your Bob Ross style videos are always a treat.

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

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

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

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

I was waiting for this so badly... This type of video is what makes this channel so unique, thank you toby!

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

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

these videos are just, truly magical

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.

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!

Hooray! Absolutely fantastical and enlightening vid. Thanks.

Thanks Toby.

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

God Toby, this was simply brilliant. Truly amazingly well done. I've never seen anyone describe complex numbers with such a childlike joy and simplicity.

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.

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

I just wanna say your videos always (well, not exactly always, but) helped me stay afloat since I discovered this channel back in somewhere 2018 (August or September, I guess). Thank you. What can be better than such form of escapism...

The calmess is intoxicating ! Makes me want to learn math on my own then following litearly the school circulum .. thanks teebs for making me a better person 😄

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?

Such a relaxing math lesson from such a wonderful teacher 😊

I just love how you explain math. I really wish I had a teacher like you when I was younger. You're the best!

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.

this is one of the best videos on the internet

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 .

Absolutely LOVE this type of videos!

Honestly the best thing I’ve seen all day Thank you a lot, I used to love mathematics but since entering med school I haven’t had time to study it and this was vey nice to watch thanks again

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

So relaxing explanation your accent is southing too 😍

Thank you again Toby, great insights in your teaching work !

This is the same subject I've been studying for online classes and you just made it so much easier!

Loving how she uses chalk as a medium.

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

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!

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

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

Thanks for the informative video. It is great to see you outdoors again.

I like the method you use, Tibees! Good job!👍

As imaginary as my relationship with Tibee.

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

Well explained in a simple way.... and OH the tingles... your voice is so ASMR to me 😌

your voice and wordz gives my mind shiverz

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

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

I like your way of teaching and your calmness

I appreciate all Tibees explanations on any math subjects. Tibees has a very clear point of view; very precise reasoning; and a math problem solving technique that I call "UNIQUE AMAZING".

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

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

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

This is an awesome video. Well done ,well explained and calming.

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

Thank's from Brazil...

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

This is so far more fascenating then anything I've learned in school! This really expands my horizon and is thererfore useful in two dimensions!

good morning Toby

Hi, I'm from Ecuador 😃🇪🇨

I love your channel, and these videos are so unique. I applaud you and all your brilliant work in communicating physics and math :)

this is so relaxing . . . 😴 love it so much. such a fun drawing. hi ignacio!

i^4 be like: snap back to reality

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

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

Thanks Toby!

Hi from Turkey 🇹🇷

Thanks for making my day better

I enjoy watching your vids even if I don't understand anything 😂. Love seeing how far you've come ❤

love the absolute values in the end. they really spread some positivity! :) (and bring you back to reality)

Escaping from reality with Toby ❤️

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

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

Concept finally cleared... thank you so much 🙃❤

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

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

thank you, you made a very clever way to show this set of concepts.

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."

Biggest fan from India, ma'am 🙃

Meow

What a fascinating mix of media! I liked that a lot!

Your channel deserves way more subscribers.