10 - What are Imaginary Numbers? 

The moment this gentleman made me understand the imaginary numbers i have literally got tears in my eyes. YOU ARE A GREAT HUMAN BEING SIR! GOD BLESS YOU!

This is the first time I am seeing a math professor combining with physics professor in one man.. Awesome....

I am a pretty old dude to be studying math, i came back to pre-algebra and saw 'i', which i vaguely remembered. after seeing all of the concepts in this video (sin, cos, etc) it is super clear to me now why they taught me 'i' in high school. Amazing explanation, for me. Though judging by the comments, I can see some guys in early high school that are being shown 'i' for the first time, without any idea about trig or calculus, this could be improved someway to make this imaginary completely understandable to grade 9 students. Excellent work! Any high school teacher introducing students to 'i' should require this video as homework.

"There's a lot more depth from what's on the board" I think is a reference to how many many boards he has

If only I could give this video a million likes. The explanation of imaginary numbers and how to use them was so helpful. I feel confident that I’m going to do well on my test today ;)

I wish I had got a teacher/professor like you in my school/college I would have not struggled in my studies. You are a blessing. GOD bless you.

What a great teacher.. I feel instead of attending school, kids should just watch his video. You are great!

I get it. I get it. Thank you! Excellent, effective, educative presentation style!

Truly, you explained/presented the complex world of math in its simplest manner! Math made easy! Thank you for all of your tutorial videos!!!

been out of school for decades and have literally no use for this. BEST EXPLANATION EVER. I came here after watching a stick figure animation where he was fighting math and (i) kept coming up and I had no idea what that was. after almost giving up watching other terrible videos and guides, tried your video and I feel extremely enlightened and happy. back in my day, these would all be "undefined". I remember that word. almost wish I was back in school to solve these and commit them to muscle memory. ty!

Wow perfect teaching, love how you show advanced math showing it’s use. Every class should do that

The Greatest.. Teacher of all times.. Love the class..

I loved this lecture! So clear and informative... Thank you!

I love the great depth you go-to for all your videos

You brought significant clarity to this content for me. Thanks sir. ❤

I appreciate all of your content. Thank you. Wish you would have been my professor for college!

Thank you so much. Wasn't understanding the concept of imaginary numbers on Khan Academy, but you cleared it up thoroughly.

Damn, i liked your content right here. keep up the good work man, good job.

Very very interesting and inspiring 💐 Your lessons are far more thrilling than any thriller 😊 Thanks a lot ❤

I was looking into number systems and ended up wanting more information on imaginary numbers in number systems, I didn't find my answer here but the clarity of the information and its relation was greater than any post-secondary school lessons I took.

Thank you so much,sir. I am not a maths teacher and I have difficulty in explaining about it. Your explanation greatly helps me.

Thank you for this video. After watching this video, it cleared a lot of confusion surrounding the topic of imaginary numbers. Afterwards, I was able to apply my knowledge and practice what I have learnt. Thank you once again. It's always a great feeling when you finally understand a topic in mathematics.

Wonderful presentation and explanation of concepts!!!

This video got you a sub. It's very hard to find videos or resources that elucidate concepts like imaginary numbers well, but this is one of those. Thank you!

Very helpful for reviewing things, and also interesting and engaging. Great video!

This lesson was really good. I like how he combined various topics in math.

you are without a doubt the best math guy on youtube

Sir you are powerful than others and really inspire me for curiosity of unknown staff. I have easily understood about “i” concept and application just as other introductions you presented. I am so joyful from my heart. Thank you Sir❤

Really good explanation. Thank you🙏

You're the best. Thanks.

Perfect explanation, thank you!

This was incredibly helpful, thank you so much. I'm starting a Calculus class in two weeks after not having taken a math course in over 10 years! Nervous but excited, wish me luck.

You and Khan Academy are my two favorite resources for this kind of information. Thanks for everything!

This was a great explanation. Good history involved to. I haven't done this stuff in over 30 years. A friend that I work with showed me his math homework because he knows I'm great with math, but it's basic I can't remember certain stuff. This triggered a lot of memories. Such as Sin X=1/Cos X and Cos X= 1/Sin X and Tan X = Sin X / Cos X and Cotangent X=Cos X/Sin X Which allows me to do a lot. It's slowly coming back. Thank you.

It's nearly been a half a decade. Still I am enjoying the subject as if it was yesterday. ty.

Thanks for the explanation, it was very clear and useful

Thank you, your lectures have been very helpful for me. You make math seem beautiful even to a non mathematian like me.

I came here because I needed a good introductory video for my students. This is indeed a good one. However, I wish the math teacher would start with the "why" and not the "how ". My students can already solve quadratic and cubic equations when I explain imaginary numbers. I usually let them solve either of the two and, ideally, one from a real-world example. Most importantly, I pick one that yields a solution with the square root of a negative number but has a real number solution and can be solved with some guessing. When they guessed the right solution, I then showed them how applying the concept of "i" would get them to the solution they had guessed previously. I had students who could already do all the stuff shown in this video, but they had yet to learn why they were doing it or what the problem was in the first place. Showing them the "why" first provides purpose and reason and leads them down the path of understanding there once was a problem that someone cleverly solved.

Its been a long time since I have had to use any of this. I enjoy your explanations and simplification of the subject. I would have done much better in cal with you as an instructor.

Fantastic video! Keep up the good work sir!

You are a great teacher!

Imaginary numbers have always been a bit of a mystery to me. I wish that I had teachers like this when I was in school. The use of the imaginary # I see is easy to understand after seeing this video. 👍👍

Throughout high school, I was in "high honors" mathematics, so that by my senior year I was learning "advanced caculus" in preparation for the advanced placement exam. Which of the two exams I took, I do not remember, but I do remember this. The very existance of "imaginary numbers" was not revealed to me and my fellow traveling "high honors" math classmates until a month or so before we took the test (the year was 1975). I was so confused from misunderstanding imaginary numbers, I ended up scoring only a 3 out of 5 on the exam, even though, so far as I can remember, the test I took required NO understanding of imaginary numbers at all! I was devastated---prior to my teacher introducing "i" on the blackboard, I was among the top two, no more than three, students in my class! I was so ashamed, my life went in a completely different direction in college----I never took another math or science course again in my life. I wish I hadn't, because now my love of "Layman Physics", takes me only so far. I can read and understand books like "A Brief History of Time" by Hawking, or better yet, "Time Reborn" by Lee Smolin, but don't ask me to derive Schrodingers Equation, the Uncertainty Principle, or even E=Mc2. Nevertheless, I greatly appreciate the above tutorial. At least now I don't have to feel stupid if I were to run into a Physicist and ask him or her; 'when an observer causes the wave function to collapse, can it collapse to an imaginary or complex number, or is it always a "real" number'! (parenthetic pun intended)

Thanks for this video, i like all your videos math and science

Thank you for your service

Thank you so much sir, may God bless you a long life, love from India

You are awesome, thank you so much!

Great teacher!!!

God bless you and your work

Wish I had this guy for my math classes

Thank you're the best

This is very great.

thanks great stuff

Thank you, mr.

This is my first time I watch a full video talking about math

Thank you so much.

What gift, to learn from such teachers. Thank you seems so so lame. Soon I shall be able to buy your help, each month. I thank you for your service to us.

I suggest that you put everything related to the radicals lessons in one group so that we can follow the lessons in order to make it easier for us. as well as exponents

Thank you🌹

This is my third imaginary numbers vid and the first one where I’m understanding it thank you ✨✨

All of this for free, i lovveeee you

Thank you very much

Thank you sir

THANK YOU...SIR...!!!

Powerful stuff.

Thankyou sir!

After knowing and been through all this, I find the pesistance to force sense out of an impossibility, insane.

Is there a playlist for complex numbers chapter? Otherwise where could i find part 12 of this series?

Where can I get the last part ? Have you made any video about the sin(x)?

I love this guy

Very interesting

Great explanation! Though things like "ignoring the sign" creates a lot of cogintive dissonance in my brain. I see it this way: If sqrt(a) = the number that multiply with itself to get a (which makes geometrical sense since the sqareroot then is the 'root'/line of the area of the square). Then using this rule: (a*b)^2 = a^2 * b ^ 2. So (2I)^2 = 2^2 * I^2 = 4*(-1) = -4, which fullfils the definition of a squareroot, and takes the sign into consideration.

Thanks. From Bangladesh.

love it❤️

Brilliant - 30 years after I first learned about imaginary numbers, I finally get it.

Now's eye's smart. Thank ya

Wish my imaginary ‘70 Dodge Charger R/T became real

This is helpful. I am happy I understand now the imaginary numbers.

This teacher’s style is a 1,000 times better than the teacher I have.

I like the future look at why things are important.

Imaginary numbers are like the fourth dimension. Hard to imagine, but necessary to explain things.

The reason you have to invent imaginary numbers to do these calculations is because all numbers are technically positive. When you have a negative number, it's in reference to wherever your starting point is before you start subtracting. All numbers are positive in reference to an absolute zero though. The celsius and kelvin temperature scales would be a good illustration of this.

Nice sir

GRACIAS MIL POR LA EXPOSICION...SALUDOS DESDE KITU-ECUADOR

Good explanation

Does the power rule apply to imaginary numbers? @29:46 you wrote i^3=i(-1). Couldn’t you use the power rule and say i^3=3i^2=3(-1)=-3?

thank you

I was wondering my whole life why was i sqr = -1 thank u so much

Question: if we raise i to 0 and Sr of -1 to 0 does that mean 1=1

Who invented negative numbers? The TAX MAN

can someone explain me what is 1/4th power of -ve1 because 1/2nd power of - ve 1 is called or defined as i........ so the question is what's the 1/4th power of - ve 1 or 1/8th power of the same

23:00

gets interesting at 31:56

1.5

This is a too-long description of an obscure topic with no demonstrated purpose. You need to revise your lecture to put some tangible examples in the first few minutes. This type of lesson is why math is so hard to learn, math teachers cannot make the concepts and equasions relate to anything in the student's knowledge of other things. -- How do I know this? I have a BS and an MS in engineering and I remember listening to lectures like this one and not really understanding anything until it was applied to mechanical and electrica problems. The expression F(t)=e^i·omega·t meant nothing at the start of the hour and still nothing after an hour of mathmatical gymnastics trying to explain it By the way, that is alternating current. -- Example One: In calculating the lift of a wing, there are certain portions of the wing where the math may result in imaginary numbers, so early engineers discarded these numbers because they did not make sense. They crashed a few planes too. Then they discovered these parts of the wing were not contributing to the wing's overall lift. It was imaginary lift. -- Example Two: In an alternating-current circuit with resistors the voltage and current are in phase and volts times amps results in watts, real power. If there are inductors and capacitors in the circuit there is a non-resistive element that is drawing current without dissipating power. Consider an unloaded motor running at normal speed, 120 volts and 7 amps; volts times amps appear to be 840 watts, or a little more than a horse power. But where is that power going? It only takes about 50 watts to make the unloaded motor run, and you can feel that by the motor getting warm and feeling the air blowing around the windings. That is where you introduce imaginary numbers and the concept of phase angle and power factor; and current flow without producing any useful work. A good motor has a power factor better than 95%, as in the phase angle between voltage and current is such that the cosine is better than 0.95.Side note: Yes, 0.95 for an engineer is the same as .95 for a math person, but the problem is when the leading decimal point is mistaken for a fly spec and ignored. Think about it this way, does it weigh zero-point-one pound or does it weigh flyspec-one pound? One-tenth or one? -- End of rant.

Ha ha…thank you for the “secret sauce” knowledge!

why do you put " [ " after equal sign?

Where was this guy when I was in advanced math classes?

x=1/4 x1/2

This is the “doorway” between exists and non-existence? Like thoughts being a most primitive form of existence?