Imaginary Numbers Explained Bob Ross Style 

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

I was taught imaginary numbers in a 3 year engineering degree, and also have watched a load of RU-vid 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.

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.

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

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

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

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!

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

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!

Your Bob Ross style videos are always a treat.

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.

YESSSSSS. Joy of mathematics is back!

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

Thanks Toby.

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!✨

One of the best maths series on RU-vid. 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

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

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

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."

these videos are just, truly magical

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

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 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

Hooray! Absolutely fantastical and enlightening vid. Thanks.

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

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

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?

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

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!

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...

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.

Such a relaxing math lesson from such a wonderful teacher 😊

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.

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 😄

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 one of the best videos on the internet

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

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.

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

Loving how she uses chalk as a medium.

Absolutely LOVE this type of videos!

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

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

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 .

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!

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

So relaxing explanation your accent is southing too 😍

As imaginary as my relationship with Tibee.

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

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

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

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

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!

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".

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

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

I like your way of teaching and your calmness

your voice and wordz gives my mind shiverz

good morning Toby

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

Thanks Toby!

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

Hi from Turkey 🇹🇷

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

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!

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.

i^4 be like: snap back to reality

Thank's from Brazil...

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

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

Hi, I'm from Ecuador 😃🇪🇨

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

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

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

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

Thanks for making my day better

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

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

Escaping from reality with Toby ❤️

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

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

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

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.

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

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

Biggest fan from India, ma'am 🙃

Meow

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

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