The Real World Uses of Imaginary Numbers
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- čas přidán 10. 09. 2018
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This video covers how imaginary numbers are used to solve real worlds problems in math, science, and engineering as well as a derivation of the most beautiful equation in math. The main topics include signals, controls, quantum mechanics, electromagnetism, and circuits but imaginary numbers do have more applications.
#complexNumbers
#mostbeautifulequation
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One thing I was going to mention in the video which some people have said in the comments is that 'imaginary' isn't a great word for the square root of negative 1. Even Gauss, one of the greatest mathematicians of all time, has a famous quote in which he says...
"“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.”
why would you use euler's formula for phase shift if you can use sum to product formula?
Please answer this- I'm an undergraduate. Going to university next year actually. I was just wondering after getting Bachelor's in EE can u study Master's and do a part time or full time job at the same time. ??????
@ Fugitive
Yes however your master's may take more time since you may not be taking the full load of classes a "normal" full time grad student would (instead of finishing in 2 years maybe 3). It also depends on you and how well you could time manage and balance responsibilities.
MajorPrep I mean the software draw the math function pro😂
@Figitive Part time working, yes. But full time working, your master will probably take a very long time!
Good video, this is a topic my audience have asked me to make so I'm glad you did it instead :P
xD
Hello tibbee. You watch his videos too!
Thank you Toby! I've been a fan of your channel since I came across your 'astrophysics exam unboxing' video a few months back. Would be interesting to see this topic from a physics perspective though. Keep up the good work!
@@zachstar toby
LOVE YOU TIBEES! DAY 1 SUBSCRIBER!!!!!!
Can I use imaginary numbers to count my friends and money?
yes offcourse.....all those artists and people that you love but never met them before and all that money that you dont have...you cant count them and you will get an IMAGINARY RESULT ))))))
my friend count : 10i
money : $1 M . i
No, because “imaginary numbers” really exist, unlike your money and friends
@@AndresFirte True. LOL
you cannot use negative numbers to count your friends either ;)
"this is a graph of sinx, if you haven't seen this before now you have" 😂😂😂 this made me laugh
I haven't seen that before
Just kidding I'm a Mathematician ,well at least I will be.
Just Why? And why did 214 people agree w u wtf
he should have done "this is a graph of sinx, if you haven't seen this before you should probably click away now" 😂
A lot of people’s difficulty in math is that they are trying to memorize an equation and variables without truly understanding what it means and how it can be applied
Math is logic disipline that’s taught as a series of memorization course in grade school. In reality you could teach a kindergartener Calculus given that you explain the logic carefully enough
that’s how the education system (at least in America and I’m pretty sure most of the world) works. Most students don’t really care about anything learnt in classes that involves calculus, take it from a current sophomore in high school. So students learn it to memorize since it’s easier than to fully understand it and its applications
It doesn't help that when a concept is first introduced the teachers explain it using formal math lingo which there is no way the students will understand, at least not at first. So basically the least resistance path to getting a good grade is to memorize a bunch of equations, and learn the mechanics of the exercises by doing a lot of them. This way a student can get a very high grade without ever having a proper understanding of the concepts behind what they are learning.
Of all disciplines taught at school I think Math is the most broken out of them all imo in how poorly it's being taught
@@miguelpereira9859 facts
@@miguelpereira9859 I'm glad to see this mentioned. Most people learn by rote memorization, because it's made needlessly complicated by things like asinine vocabulary, and in truth most people don't have a concrete way to make it applicable to their daily lives, and they never will. Mathematics as taught isn't "just logic bro" as some claim, it's a game of trying to decipher the hidden message of the esoteric math cult. It's hard to answer something when the question itself needs to be explained. The established syntax is why a lot of people struggle with math, and that's the facts. Sadly if anything were ever formally done about it, I'm sure things would end up significantly worse, if other school initiatives are anything to go by. I think it's something teachers should just keep in mind.
The grave mistake lies in calling them "imaginary numbers".
Quahntasy - Animating Universe Always preferred saying complex numbers instead
Complex numbers have a real and imaginary part, with i being the imaginary part.
@@azizalaliq8 Actually i doesn't belong to the imaginary part. The imaginary part is what you multiply i with.
Lateral Numbers
And second wa you were born
I remember telling friends that aren't engineer majors the value of square root of -1. It's definitely your best video on mathematics.
Yess, pls more!
Once you start understanding it Math is so interesting and cool how all these things apply in the real world
Actually, math and science have only made me more depressed. Sure I see more connections, but I also have a lot more questions.
@@thewalkingjoke3843 so... You haven't started understanding it?
@@James-jh3sz Thing is you can't understand these things right away. I studied complex numbers and Advanced calculas in high school. I am in 3rd year of my college degree and now I understand almost everything.
@@alokbaluni8760 I agree, but the person i responded to seems to be the type of person to overestimate themselves which is why I was saying that.
@@alokbaluni8760 You can’t understand Advanced Calculus without first understanding all of Elementary Calculus and Linear Algebra.
"The Real World Uses of Imaginary Numbers"
High School Students: Impossible...
@@randylejeune Complex*
This is infinitely better than that _Stop Making These Mistakes_ video
Good job 👍🏻
hello vector
Engineering students: *Are happy*
Imaginary numbers: Hold my beer
SMH. could’ve said “hold my roots”
@@shayorshayorshayor *hold my root beer
Engineering students don't even know what "being happy" means
@@jimmyh2137 why is that?
@@jimmyh2137 haha you said big funni stereotype I must like now
Here's an idea for a video you might want to consider. I think it was in my sophomore or junior year in college when a professor went through the laborious process of adding two electrical impedences using traditional math (lots of calculus if I recall). If I remember correctly, it took him something like 30 minutes. Then he did the same thing using complex numbers, getting the exact same answer, and it took him maybe two minutes. That's when I was sold on complex numbers.
I wish I had kept those notes of him going through that long process. I can't even find it done that old way online. If you could make a vid showing how those calculations were done prior to the use of complex numbers, and then with complex numbers, I think that would be a major eye opener, and make people really appreciate the benefit of using complex numbers.
Math would have been better in school if they explained how we could apply what we learned to the real world and would probably motivate more students
Andy Dufresne well in my country there is 2 type of maths. Emath and amath. E math is elementary math (applicable to the real world) and additional maths.
I feel like explaining math would have the reverse affect for a lot of people. So many are simple minded and if explanations dont come naturally they give things a bad stigma and choose to neglect the importance of things.
For people like myself and some of the others in the comment section im sure what you said makes sense. When I am perplexed by an idea I stop at nothing to fulfill my need to understand things.
I personally was a little frustrated in school because like you i got to points in school where i had eurika moments in regards to specific concepts that I feel i couldve been explained in middle or high school and still understood.
So the part thats like you i had wished they had just explained things in ways earlier on to build understanding.
But instead the school systems focus on solely applications of math so people can just simply understand how to evaluate problems and feel good about getting to a definite answer. With or without an in depth explanation. Some would rather not have that in depth explanation because they will realize the extremes of what they are getting into and stop.
@@woooshwooosh2867 from SG
People who don't *want* to understand mathematics will never understand mathematics.
The trouble is, we've made it socially acceptable to be *proud* of not understanding, and not wanting to understand, mathematics.
@Yani Zhou Because ypu're spending thousands of dollars for this exact reason? And because children mostly don't care that much about school.
Nobody:
Literally nobody:
Fourier transforms: time to make some engineering students cry
Just finished calc. This hurts so much.
As someone who gets really frustrated and has a hard time with maths when I can't apply it to the real world this type of video is pure gold and I am very thankful for them! :)
Mathematicians: e = 2.718....; pi = 3.14159...
Physicists: e = 3 = pi = 3
Engineers: e was somewhere between 0 and 150 meh
@@wetfart420 I know how to find more digits of pi. That is, I theoretically know any finite number of digits of pi (given enough time)
@@wetfart420 I know all numbers of pi
I know the last 2 digits of pi
I know all of Pi in a base-Pi number system
@@jordanwoods728 that's smart
Studying Engineering but never had e^ix visualised graphically with all the 3 maclaurin's series', cheers for the great explanation it was really interesting
One of the best videos I've ever seen about this topic, helped me to understand it better, thanks!
I'm just now getting into engineering and plan to go into electrical engineering after completing the industrial electrician course at my college which touched on these topics in their basic forms but never really explained more than just measurement and basic understanding of how circuits work. This seems like a lot of fun honestly and I can't wait to get into this more down the road. Very inspiring. Love the videos
I am in AC circuit analysis right now and I have been looking for a proper explanation as to why we use complex numbers and you by far have been the most helpful. I hate just following math equations without knowing why I'm doing them. Thank you for the great work you are doing.
Wow, I have tried watching your videos, a number (8?) of them, and you are … Brilliant and amazing. I am an IT professional, never pursued math, but use it every day in what I do, and you make me look like an idiot. I don't understand a lot of what you are talking about, but it is like a picture being painted in front of me by Rembrandt, slowly unveiling the picture of the universe. I appreciate what you and all the other super-nerds do. Thanks for posting.
8:08 the real part should pe approximately 1.609 not 1.069
Thanks✌
This makes me so much more excited for Physics and Further Maths at A-level... and beyond.
Very well laid out and explained video, thanks 🙂
well just a heads up
AS level physics is boring and AS maths is easy af dk about Further maths
Why does this video only have 17k views? This is explained really well...
Because the average person doesn't care about 'i'
People are scared of mathematics.
Because it's terribly explained. A maths student and an aspiring tutor opinion.
Sorry...
Too less explanation, presenting new ideas as if I'm supposed to be familiar with them in order to follow along, but also be surprised when seeing they make use of i.
My moral of the video is "i is useful in maths describing how the universe works, although I have no idea how really"
Becous it only shows the real part.
@@nitsanbh but you will know if you study it?
"Can't measure the 'i' of anything"
Well, that's not exactly true, it depends on what the 'i' of something is. It could be representative of a geometric aspect (where i is one of the quaternions), in a complex number representing the interdependent population growths of a predator and prey species the real part would represent one species and the imaginary would represent the other. There's more examples in quantum physics but don't let the imaginary label of i mislead you, it's better to think of complex numbers (real + imaginary) as multidimensional numbers
Actually he is right. No measurement gives you an imaginary number. We are talking about measurements in an experiment. A quantity can be complex but you can measure the magnitude or real part only.
You can't measure an imaginary quantity in quantum physics otherwise it wouldn't be Hermitian
In MRI machines, the measurements we get from machine are the imaginary and real parts of the 2D Fourier transform of the image from a slice of the patient.
Also, in optics, if you throw parallel rays throw a convergent lens, the light at a focal distance from the lens shows the Fourier transform (the magnitude, I guess)
These things blew my mind.
This video came up in my recommendations today and I’ve been binge watching your channel since. Love your work
I have a class presentation on imaginary numbers, your video is a huge help! I am still in school and reading articles on the net about the use of complex numbers in AC circuits was of a little help, your video was amazing as I was able to analyze the use of imaginary numbers. thank you!
Sees someone use degrees instead of radians
TRIGERRED
Radians are fucking annoying, I'm much more used to more basic units.
Sees someone spell a word incorrectly
TRIGGERED
Triggered by what? Degrees are equally, if not more, helpful and convenient to use compared to radians.
It doesn't matter. When I use reading word synthesis, I use degrees because I cut 2pi out of my mind. That way I make less error. When I am analysing the stability,observability,Linearity, Controllability and a lot of other things I prefer to go in rad/sec. They make the math in both of these cases simpler.
Degrees have much more real world application than radians. Radians are only correct on the unit circle technically. Unless you always make whatever radius you have one unit but personally if my circle has a radius of 3.5m then that's not one unit.
Amazing! You explained the application of imaginary numbers in such a way that I now understand their importance and significance. I believe that using i to mean imaginary is a unfortunate choice and a misnomer since it turns people off since it is not real, but from your explanation, I could now remember that the i could stand for intermediate instead. As they don't directly appear in reality or have direct analogs in reality, but they are convenient "intermediate" shorthands when transitioning from one state to another.
Had to watch it twice in order to START getting it. Good stuff! Thanks for all the effort you put into making this video!
You're the best. I've been looking for more channels related to my major & you make everything so clear. :) Thanks!
I have not learnt any of this.
I don't know why CZcams recommended me this.
My brain hurts right now.
I still don't know why I'm watching this.
Do you study maths?
Carlos Dominguez Nope, just the joys of CZcams recommendations algorithm.
Why are you replying to a year old comment?
@@livethefuture2492 I don't know, just I did it
@@livethefuture2492 Two years later (and a year from your reply) and I understand the concepts in this video - still nothing of how it’s used.
Quite interesting that I got this recommended to me yet again, I completely forgot about this comment.
Thanks for making this. I got all the way through Chemical Engineering without using complex analysis in engineering classes (only in Math classes), so when I heard about complex numbers having "real applications", I believed it but wasn't sure of specifics. We did use Fourier and laplace transforms so I guess it was technically there all along.
I looked this up independently of your other, more comedic videos, and was happy to see you!
All your videos are really good, but this one was even better man, thanks!
Whew, now I have a backup to calculate -1.
@@epsi what...
Electrical and electronics engineers know how important imaginary numbers are.
Kalumba mutembo agree
Agreeeeeed
No, I'm in second year (bachelor degree) and i don't know it yet
this was actually very well done, the visual representation is great. I wish i seen this while i was taking my deterministic signals class. I struggled in the beginning with simple things like this, which made the class feel impossible at the end.
Your videos are really helping me getting a better idea of how concepts link together. Thank you!
It's my senior year in high school, this made me understand lots of things that were skipped in class, like why is (e^ix) is equal to (cosx + isinx)
Thank you !
Well explained!!!
Excellent stuff!! I think people would love to see technical explanation videos like this about how math is used in various other majors.
Just thank you. I have always wanted to know this it seems for my whole life. I appreciate it so much and you make it as simple to understand as I can imagine.
Amazing video as always! You are making youtube a better place! At the 8:20 I did not believe you though and I found a small mistake, your real part should be 1.609 instead of 1.069 and then everything works fine
My favorite application of imaginary numbers, by far, is the link that electrical circuits have with magnetic ones through capacitors and inductors
This was such an awesome and informative video... as a senior in high school I've learned all of these formulas but never the real proofs behind them - definitely sharing it with my math class tomorrow.
I’m an Aero engineer at CPP currently and I’m in both a Controls class and an Advanced Engineering Math class. Both of these deal heavily with complex analysis. Thanks for the insight and the clarification on this topic!
I prefer e^(i*pi)+1=0 - because it contains both 1 and 0, the neutral elements of multiplication and addition.
Man!Please be technical,I would love to watch and follow through like hours of your lectures.Good Job!!!
Watching the graph change dynamically as you write out the sequences is an amazing learning reference, brilliant
Thank you so much.
I did not understand a word.
definitely scares the excitement out of me as well as stir the possibility of growth at the same time. I can tell you love this world you live in and it is so helpful to hear your heart speaking thorough all those formulas.
I am retired and wanting to go back to math where i left off in pre-algebra in the 70's. It is overwhelming at times, and i literally do not know where to start, so i am using pre-algebra in Alecks to try to find my place. you know.. just Jump right in.
then i run into invisible numbers and think i must be in a kindergarten disney world...is this real? a search to understand practical use of invisible things leads me to you and i get a tiny glimpse of its purpose. it feels like it is there to find balance.. like a doing the problem over backwards to check your answer. i am an artist. when i looked at the xy graph and saw a mirrored number under it that was invisible.. i thought, oh, a reflection of the above, like looking into a pool of water...
thanks again
Very good video.
Such a good explanation that makes sense! I have seen explanations using geometry or derivatives and they just don't make sense, but using the Maclauren series was a much better explanation
Thank you for showing the technical side! I learn better this way!
Thank you so much for the incredible insight of where the square of -1 came from. I've been searching the internet that explains the nature of it to no avail, until I watched this video. I now know it's just a mathematical necessity which united and equated the two quantities in Euler's equation. This is the deepest insight behind the mysterious "complex" number! And thank you again.
Who else doesn't understand most of what's he's saying but still watches the video because it seems interesting?
Only me?
It’s explained using Elementary Calculus.
number e complexia
www.desmos.com, as a mathteacher, I use it almost everyday.
good content , love the animation of numbers, and the content of the videos is solid , keep up the good work!
Thank you for sharing
Your video is absolutely incredible
It is so much clearer to me now💝
So in high school, when a teacher is asked _"why do we need to know this?"_ and they respond _"You just need to know it"_ they leave a lottttt to explain which would help..
There's two possible reasons why your teacher gave that response:
1. Your teacher never learned the applications of complex numbers
2. The background knowledge you'd need to know to understand their applications, is far beyond the scope of the class. For instance, you'd need to know the background of electric circuits, and what capacitors and inductors even are, to show you how to apply complex numbers in this particular real world application.
Ah, imaginary numbers - my old arch nemesis
I've loved imaginary numbers since I was told about them by my dad when I was 6, maybe 5.
You have no idea how much you helped me. It's all so clearer right now!
Thank you. It really helped. It was just what I was searching for. Thanks alot.
At 8:28, shouldn't it be 1.481e^(i * 43.796°)? Because with the expression you gave you'd get the real component as 1.609, not 1.069
Yes
Yes. I got the same answer as you got too!
Actually, at 8:11 he made a mistake. 0.966 + 0.643 = 1.609, so the next slide should have said 1.609 + 1.025i. But then 1.908e^(32.5 deg)i is correct.
Not first, second, or third, but..
i-th.
(i^4)st
No
pi-th
So, you didn't watch the video, but only imagined it?
Flick Penrose yup
Thanks for this great video. Subbed!
I’ve been excited about this video since you mentioned it in your last one
thanks, as a mech eng student, i didn't know why they use the word impedance instead of just resistance
Resistance refers only to the opposition of a component to conduce electric current, impedance refers to the factor that relates a voltage with it's current or vice-verse (Basically Ohm law but instead of the R, it's a Z which stands for impedance). Resistance is actually an impedance, but just for the case of a resistor, as capacitors and coils have a different expression for their impedance that relates to it's capacitance and inductance respectively.
@Justin Chan
The impedance of a circuit has two components;
1) Resistance : how the circuit resist the current by heating up
2) Reactance : how your circuit augment or diminish vibration of current
Those two together will determine how your AC circuit reacts.
For exemple, an Electric motor in a circuit will make your circuit vibrate, which will result in less usable power in your circuit. In order to reduce reactance, you will add an capacitor to your circuit to tamper vibration.
At 8:35 it should be 1.609 not 1.069
Totally right, my mistake
How did you determine that?
Owen you just compute cos(15)+cos(50) in degrees
Or add .966 to .643 like a normal person
Fantastic, the only video I've found that helps convey the usefulness of imaginary numbers, and I've seen many.
one of the only channels i watch on normal speed. awesome stuff
has anyone observed the effect of balls on an isochronous curve.
I just want to know that, what does it physically really mean if, i is multiplied to any sort of scalar or vector quantity in physics eg., Mass(i), velocity(i) etc. Does it really makes any sense?I will be glad, if a knowledgeable person like you will suggest me anything about it, and draw your views forth to me.Thank you great and enlightening video anyways.👍👍
Multiplying by i rotates a vector (or a phasor as I really learned it) 90 degrees in the complex plane actually. From what I talked about in the video you can use eulers formula to see that i is the same as e^90i. so multiplying by some other exponential will result in addition of exponents and by adding 90 to the angle but keeping the magnitude the same you end up with the same vector rotated 90 degrees.
Multiplying any constant with i just tells that that quantity is just lies in the imaginary axis of the 3d complex graph. Moreover real things which has tendency to holds it's shape and size, under all circumstances, doesn't come near to i. Complex numbers definitely helps us to make the math of something changing or the math of changed quantity w.r.t. some other Changing quantity easier for us to understand.
My guy you rocked it! Beautiful job
i can stand and clap for you! really well done !.. i always try to teach like this.
I didnt understand anything.
He is just basically showing you that imaginary numbers is a very useful tool to deal with complicated physics equations. Thanks to euler's identity, we can use imaginary numbers to manipulate real functions (physics laws), i.e. treat them as the real part of complex functions (imaginary numbers as inputs and outputs) because lots of useful calculations are easier in complex domain.
This series in the following link is much much better than this video, though it doesn't go into engineering applications of them.
czcams.com/video/T647CGsuOVU/video.html
For better explanation of Fourier Transform, which is a big deal in electrical engineering, see this:
czcams.com/video/spUNpyF58BY/video.html
Sonay Yalım Sonay Yalım i was watching dash cam australian when youtube recommended me a video about imaginary number[the first video link, ”imaginary numbers are real”]!!! so, awesome. the i stumbled on to this: czcams.com/video/IUTGFQpKaPU/video.html
to flex on your imaginary friends!
Nice job, especially with the graphing as evidence. I learned this stuff 20 years ago and forgot most of it, but this video jogged a lot of memories.
Really great video mate.... Loved it.
-cries every time j is used instead of i-
cries every time i is used instead of j
Found the engineer
@@Mot-dh5sx Cries every time i or j is used instead of k. quaternions and their imaginary, joke and kooky numbers
Reality Is multidimensional like at least 12 dimensions....you can prove fermat's last theorem in like 1 page Using higher dimensions, rather than the accepted 200 pages of 1 dimensional thinking.......!!!!!! Thinking in higher dimensions simplifies everything.
I would like to see that prove
Emil Müller :v well I can email it to you if you like. You can get in touch also in Skype , my ID is " haniffdin " . You can get my smartphone number from there also and use other chat apps, sms or iMessage etc.....
lol, good one
@@hanniffydinn6019 You lost such a good opportunity to say "I'd send you the proof, but the margins of this comment section are too small"
Such a waste of potential...
Naimad Granted, bit of cliche though I thought. Nobody has enough brain cells to even ask, never mind the joke !
This is the perfect explanation for what I'm learning in Circuits one at university right now as an Electrical Engineer. Thanks for summing up a confusing lecture
Thank so much for kind effort to teach us in such an interesting way.
At 8:10 there is a small error. e^i50° + e^i15° = 1.609 + 1.025i and not 1.069 as proudly shown in the video
8:06 *1.609, not 1.069
This is really nice. Previously I have not seen the derivation of Euler's equation from the Maclaurin series. Big fun.
man if you continue publishing stuff on the mathematics of circuits you will have a very happy subscriber.
The world is complicated
You should know, as an electrical engineer, that we use j and not i for the sqrt(-1) :p
agreed.
You didn’t make it to the end of the video :P
Stunning demonstration ! And beautiful as well
I can't believe how everything became so clear in just 1 video. Thanks a lot 👍
7:47
this is why i hate engineers
Assume e=2, gravitational acceleration=10 and pi=3. Margin error will be reasonable. /S
@@ankushmenat that's wrong
The engineering fundamental theorem:
e = π = 3
I would have thought their j-imaginary numbers would be the the spawn of your hatred instead
I sign pie.
You’re a phenomenal teacher. I had a laughing fit @ the e^ipi mclauren series proof and I don’t know how to explain that is the highest compliment I can and have ever paid anything. It tickled my brain.
Well done. I will definitely use this video when introducing complex numbers to my students.
Tesla knew all of this more than 100 years ago? Woah
He was a genius
GIT GUD Yeah, but not only that. Intuition that he had (as he said) had been given to him. If you haven’t check “My inventions” by Nikolai Tesla as well as his interviews.
@@aimilioner I will thanks tesla is the reason I want to be an engineer
tesla is a god compared to you a mere man
lazer gorrillas im aware of that
Thank u so much make this kind of video more and more it very helpful
great video. please continue this series especially in maths
This is such a good recap for me. Also this is well explained. Thanks!