I really love how happy he is the whole vedio, shows how he loves the subject... I really was smiling the whole time seeing him explain while enjoying what he is doing ...
i'm a japanese student studying math you looked very excited and i was maked interested by your speaking i have thought cos(x) equals or be lower 1 but i learned cos(x) can be 2,by your speaking it's very exciting thank you
Or we can say solve cos(x) = 2, where x belongs to the set of complex numbers. That is why I always try to tell my students to define the variables you are working with just as if you are writing code. :)
@@DoctrinaMathVideos I mean, you'd have to define that 'z' is in the complex world as well, doesn't really change anything apart from him using the variable 'z' over 'x'.
Another way to do this is to use one of my favorite identities, which is: cos(ix) = cosh(x) So you really just need to solve: cosh(x) = 2 and then attach an i to your answer! So the answer is x = arccosh(2)*i
@@sniperwolf50 you can get the rest the usual way when you have trigonometric equations: by recognizing that cosx = cos(x +2pi*m) and that cosx = cos(-x). these identities are where the other solutions got absorbed, but you can add them back in by being more general.
Arg! Reminders of 35+ years ago at University doing electrical engineering - impedance, capacitance, alternating current, power factors, EXCEPT we used "j" instead of "i" (since "i" is current in electricity) - so glad I went into software engineer 🙂
this cosine or sine getting larger than 1 has real applications in wave mechanics For instance: w(k)=asin(bk)where a and b are just some constant real numbers, if the wave number k becomes a complex number while the angular frequency is still a real number, then in wave mechanics we say that spatial attenuation has occured
Complex numbers are not only used in electrical engineering but they are also used in quantum physics in calculating the wave function in the Schrödinger equation and many more. Even the father of complex numbers AKA Quertonions are used in quantum physics.
It was a fun video to watch until the last part: -“it happens in our imagination” Me: haha :) -“but it is used on electrical engineering” Me, studying EE: Ouch :(
cos() and cosh() are even functions, so x=2piM+/-i*ln(2+/-sqrt(3) You can get the other answer by multiplying by e^-ix in the beginning and picking y=e^-ix.
@@iabervon Because 2+√3 = 1/(2-√3) -ln(2-√3) = ln((2-√3)^(-1)) = ln(2+√3) 1/(a - √b) = a + √b whenever a² - b = 1 This means ln(a+√b) = -ln(a-√b) if a² - b = 1
*Or not entire. Since we know it is not constant, if it was bounded it would have to be not entire. First time I learnt about Liouville's theorem it really blew my mind! Edit: Picard's little theorem takes that even further (at most 1 point can be omitted from the image) but I didn't learn about that at the time.
If you don't know Euler's identity, to solve cos(z)=2, set z=x+y*i, and then expand cos(x+y*i) using standard trigonometric identities, you then use cos(x*i)=cosh(x), sin(x*i)=-sin(x) and equate real and imaginary parts. The identities can be proven using power series.
Given that cos(x) is a symmetric function, I am surprised that the answer is not symmetric as well, i.e. I would have expected the simpler result x = [(+ or -)(some number) times i] + (2.pi.m), but this is apparently not the case!
For the rest of the integers (conjecture not yet proven); cos(-i*ln(n+sqrt(n^2 - 1)) = n which, after a fashion kinda implies that: f(x)=cos(-i*ln(x+sqrt(x^2 - 1)) = x if you solve this in wolfram alpha, it does appear to behave like the identity function, but would require more rigorous proof. An interesting rabbit-hole, nonetheless.
after 35 years, i'm loving watching these vids and relearning my math ! before watching, this was also beyond my limit of understanding. lol, you Dr Peyam !!
writing z for exp(ix) one gets z + 1/z = 4 i e. z^2 - 4*z + 1 = 0 i.e (z -2)^2 = 3 i.e z = 2 + √3 and z= 2 - √3 Hereby either x =-i* ln(2+√3) and x =i* ln(2+√3)
@@mastershooter64 Im new at this, so sorry any ignorance, but two questions, why to vizualice in n dimensions if the function u are working with,its dimension is,i dont know n-1 ?, and two are there any negative dimensions ?
I love your enthusiasm, I’m on holiday from school right now and not the greatest at maths, but I just watched your whole video and even finished some steps for you completely voluntarily at 6:30am your great at this 👍👍👍
Casi todo cuando se expande al dominio de los complejos tiene solucion, disculpa por qué al final colocaste 2πMi, fue como para generalizar la ecuación? no entendi esa parte
Even though I know that in complex variable it is possible, cos(z)=2, I thought there was a mistake in your proof. I like your videos. Congratulations!
And now let's return to the ordinary world, where the cos is the ratio of the leg to the hepatinuse, and the fact that the leg cannot be larger than the hypatenuse automatically makes it less than one in modulus.
@@drpeyam Thanks for the great content, by the way. I don't understand the vast majority of the topics you address, but your energy and personality really makes all of it enjoyable to watch. There are a lot of math channels out there, but your enthusiasm is so infectious that it's just fun, even with subjects that are far beyond the scope of anything I'm currently trying to study or learn (or capable of really intuiting). There's definitely multiple attributes that make a great teacher, but that's one of them! Keep doing what you do.
I'm a student in Japan, and I was wondering if the range of cos from -1 to 1 is the range of real numbers, and if I use imaginary numbers, it can also be 2? I'm sorry. My English is somewhat poor.
Good job 👌💙 I love it. Professor, I have a question. I want to learn more mathematics and be expert in it. My ultimate is to be a mathematician and holding extensive knowledge in the subject. What is your advice for me and what books should I read?
I really love how happy he is the whole vedio, shows how he loves the subject...
I really was smiling the whole time seeing him explain while enjoying what he is doing ...
ikr! cocaine can work wonders
@@ANTEUEX is someone loving you?
i'm a japanese student studying math
you looked very excited and i was maked interested by your speaking
i have thought cos(x) equals or be lower 1 but i learned cos(x) can be 2,by your speaking
it's very exciting
thank you
Thanks so much!!!
if you are in a calculus class then do not listen to this video
Onii-chan~ Arigato for telling me cos(x) can be 2 A-Ahh~!
@@bodbodii8654 cringe
@@bodbodii8654 xd wouldnt be sensei ir Sempai?
Solve cos(x)=2, *disgusted face*
Solve cos(z)=2, *pleased face*
Or we can say solve cos(x) = 2, where x belongs to the set of complex numbers. That is why I always try to tell my students to define the variables you are working with just as if you are writing code. :)
@@DoctrinaMathVideos I mean, you'd have to define that 'z' is in the complex world as well, doesn't really change anything apart from him using the variable 'z' over 'x'.
@@valizeth4073 it is a convention that is elegant
@@valizeth4073 You missed my point. Stating your variables is important and assumptions should not be overlooked.
@@DoctrinaMathVideos cos(x)=2, x∈C
Another way to do this is to use one of my favorite identities, which is:
cos(ix) = cosh(x)
So you really just need to solve:
cosh(x) = 2
and then attach an i to your answer!
So the answer is x = arccosh(2)*i
woah amazing
...which means arccosh(2)=ln(2+-sqrt(3))... 😁
This answer is right, but incomplete. By this method, you can only get the purely imaginary solutions
@@sniperwolf50 what even? He just missed an extra factor of 2πni which appears when taking the "arcosh", and that'd make it completely correct.
@@sniperwolf50 you can get the rest the usual way when you have trigonometric equations: by recognizing that cosx = cos(x +2pi*m) and that cosx = cos(-x). these identities are where the other solutions got absorbed, but you can add them back in by being more general.
The final solution for x has a 2-Peyam suffix. So in addition to having infinite values of cosine, you also get infinite Peyams. How cool is that?
Общеизвестно, что "в военное время косинус может достигать значения 4."
Российско-советская военно-научная шутка.
Объясни мне лучше пожалуйста
🤣🤣
I'm sorry, what does it mean? I need to know now
@@alessandrocarbotti9241 It is well known that in "wartime, the cosine can reach the value of 4" Russian army joke
For a cheap shortcut: if cos(x)=2, then sin(x)=+/-i sqrt(3). So cos(x)+i sin(x) = 2+/-sqrt(3)=e^i x.
amazing
Very interesting! Amazing!
man's speedrunning math
Arg! Reminders of 35+ years ago at University doing electrical engineering - impedance, capacitance, alternating current, power factors, EXCEPT we used "j" instead of "i" (since "i" is current in electricity) - so glad I went into software engineer 🙂
this cosine or sine getting larger than 1 has real applications in wave mechanics
For instance: w(k)=asin(bk)where a and b are just some constant real numbers, if the wave number k becomes a complex number while the angular frequency is still a real number, then in wave mechanics we say that spatial attenuation has occured
thank you so much sir, not only for solving such an interesting equation , but also for the vibe you gave off.
Complex numbers are not only used in electrical engineering but they are also used in quantum physics in calculating the wave function in the Schrödinger equation and many more.
Even the father of complex numbers AKA Quertonions are used in quantum physics.
It was a fun video to watch until the last part:
-“it happens in our imagination”
Me: haha :)
-“but it is used on electrical engineering”
Me, studying EE: Ouch :(
書き順のクセが強すぎて全然頭に入ってこない
Your videos are great!! Love the unique concepts 🔥
Peyam has the talent to be big on UA-cam.
@@henrymarkson3758 that's for sure!! but the number of idiots on UA-cam needs to decrease for Dr. Peyam to rise up!
Awww thank you 🥰
Its not unique, i saw blackpenredpen do this the same way like two years ago with sin(z) = 2
Great way to teach…. Easy language and easy way to approach mathematical problem, great job sir!
i am really thankful to you for sharing your knowledge in such a great and unique way .
I'm from Morocco and I love math very much I watch all your videos. We want big videos at a time. Thank you, sir
cos() and cosh() are even functions, so x=2piM+/-i*ln(2+/-sqrt(3)
You can get the other answer by multiplying by e^-ix in the beginning and picking y=e^-ix.
You don't need both ± signs, for the surprising reason that ln(2+√3)=-ln(2-√3).
@@iabervon
Because 2+√3 = 1/(2-√3)
-ln(2-√3) = ln((2-√3)^(-1)) = ln(2+√3)
1/(a - √b) = a + √b whenever a² - b = 1
This means ln(a+√b) = -ln(a-√b) if a² - b = 1
@@iabervon oh yeah forgot about that. It's not strictly true of all numbers of that form, but 2+sqrt(3)=1/(2-sqrt(3))
The dark side of the math is a pathway to many abilities some consider to be unnatural ;)
Ex: cos(x)=2
It's very important that cosine be unbounded in the complex world because then it would be constant! 😱
Great remark!!
*Or not entire. Since we know it is not constant, if it was bounded it would have to be not entire.
First time I learnt about Liouville's theorem it really blew my mind!
Edit:
Picard's little theorem takes that even further (at most 1 point can be omitted from the image) but I didn't learn about that at the time.
If you don't know Euler's identity, to solve cos(z)=2, set z=x+y*i, and then expand cos(x+y*i) using standard trigonometric identities, you then use cos(x*i)=cosh(x), sin(x*i)=-sin(x) and equate real and imaginary parts.
The identities can be proven using power series.
Who the hell learns hyperbolic trig functions before Euler identity
No need of minus sign at the start of the answer since the arguments of ln(...) are reciprocals.
x = ln(2 ± √3) + 2nπ
Smart boi
i really enjoy your way of teaching , fast solve and slow at the right place .
Given that cos(x) is a symmetric function, I am surprised that the answer is not symmetric as well, i.e. I would have expected the simpler result x = [(+ or -)(some number) times i] + (2.pi.m), but this is apparently not the case!
Oh but actually it’s symmetric, -ln(2-sqrt(3)) = ln(2+sqrt(3)) even though it doesn’t look that way at first sight
This guy looks so much happier than i am while doing maths...this is the end goal
For the rest of the integers (conjecture not yet proven); cos(-i*ln(n+sqrt(n^2 - 1)) = n which, after a fashion kinda implies that:
f(x)=cos(-i*ln(x+sqrt(x^2 - 1)) = x
if you solve this in wolfram alpha, it does appear to behave like the identity function, but would require more rigorous proof.
An interesting rabbit-hole, nonetheless.
I love his excitation, it just feels like he proofed the whole humanity wrong.
after 35 years, i'm loving watching these vids and relearning my math !
before watching, this was also beyond my limit of understanding. lol, you Dr Peyam !!
writing z for exp(ix)
one gets z + 1/z = 4
i e. z^2 - 4*z + 1 = 0
i.e (z -2)^2 = 3
i.e z = 2 + √3 and z= 2 - √3
Hereby either
x =-i* ln(2+√3) and x =i* ln(2+√3)
Great video Dr Peyam. I take pleasure watching the video and get some inspiration for mine. Thanks
3:58 "Plus 2 Peyam"
There are two of them now?
I wish I could visualize n-dimensions so that I can visualize the graphs of complex functions
also yes, in the words of the great bprp, use chen lu
Could you make ur complex function 2D so you can work with a third?
@@mmek10 what?
@@mastershooter64 did you mean, vizualize above the third dimension ?
@@mmek10 I meant visualizing every dimension, hence the "n"
@@mastershooter64 Im new at this, so sorry any ignorance, but two questions, why to vizualice in n dimensions if the function u are working with,its dimension is,i dont know n-1 ?, and two are there any negative dimensions ?
Interesting result. Complex numbers are powerful.
Thanks for the fun video today Dr. Peyam!
cosの定義を忘れて「無理ゲーーー！」って思いながら見るのが楽しいです！
Normal, we all mean from the same problem, from a mistake, we learn
What about quaternions, octonions or any hypercomplex number? Can you put these in the cosine function?
Wtf are those?
“your calculus teacher says you can but actually you can.” genius
Very interesting! Thanks for the video
"Imaginary worlds exist only in the imagination of the imaginer" - Frank Zappa from album Joe's Garage
I had signals theory during my studies but it was rather about Fourier transform. Then on circuit theory I had Laplace transform and complex numbers.
Every thing possible in math after complex no. Arrive.. These are though in our school. Studying in Indian institution of technology currently..😀
cos x = 2
e^ix = 2+-3^1/2
e^ix = cos x + i*sin x = 2+-3^1/2 +0i =>
=> cos x = 2+-3^1/2, i*sin x = i0
cos x = 2 => cos x = 2+-3^1/2 !?
Great channel. Thanks for all the videos.
درستها بصف ثالث كلية الهندسة و كنت محتار بوقتها بطريقة الحل و ما مصدق بقيمة جيب او جيب تمام اكبر من الواحد 😊😊😊😊
Что значит пмі?
Absolute legend
日本の数学系UA-camrもこれくらい明るくかったらいいのに
I love your enthusiasm, I’m on holiday from school right now and not the greatest at maths, but I just watched your whole video and even finished some steps for you completely voluntarily at 6:30am your great at this 👍👍👍
This dude looks like a younger version of ElectroBoom, apart from that, really liked the video !
LOL, I mean we’re both Persian
That was satisfying to watch!
Thank you!!!
Casi todo cuando se expande al dominio de los complejos tiene solucion, disculpa por qué al final colocaste 2πMi, fue como para generalizar la ecuación? no entendi esa parte
There's a basic problem, when we write a complex number x+iy, x doesn't take the place of y but that answer is suggesting that.
Addition is commutative, the order doesn't matter.
Commutative? Do you mean that x+iy=y+ix?
@@lazaremoanang3116 No, x + iy = yi + x, which is how he wrote the answer at the end.
So when I say that x becomes y when that equation is solved, you don't understand what I mean.
I wish I could add multiples of yourself in the real world so I can have you teach math at my university 😞
ily king 🤧❤️
Bro how u focus so much plz guide me too
You are a medical aspirant
Why you want to know abt mathematics?
Perfect thanks a lot !!
but cos(x) E (-1;1) in R
but you show cos(x) in IR
0:02 "your calculus teacher said you can, but actually you can"
*said you can’t
Fun fact everyone looking for real solutions until a Dr pi m comes with complex solutions
That's a smart move! I really wish our school teach us this
question: cosinus range is between -1 and 1?
Yes if x is real, but here x is complex
Math is the true universal language!
I never want to go to the math world with imaginary number...
우리는 이것을 통해 식당에 가면 코스는 2인분부터 주문가능하단것을 알 수 있습니다.
Even though I know that in complex variable it is possible, cos(z)=2, I thought there was a mistake in your proof. I like your videos. Congratulations!
sir ,can you please explain flint hill series ?
What’s that?
Me: looks at thumbnail
Complex Universe: *playing Beethoven symphony*
How to solve cos(x)=2?
Easy, just use arccos and then you get x = arccos(2).
Sure 😂
And now let's return to the ordinary world, where the cos is the ratio of the leg to the hepatinuse, and the fact that the leg cannot be larger than the hypatenuse automatically makes it less than one in modulus.
If cosine has infinite values in the complex plane, where does it ‘curve over’ as in where its gradient is 0
also ln(2±√3) = ± ln(2+√3)
i = the integral btwn zero and z, Z = z/w boson
w = zero or i W/Z = d(S)/d(t) S=x, y, z or Infinity
Why (x)? = (x)(y)(z) (?) x (f) = S
Dr . PI*M brilliant :-)
"That's the limit of my knowledge--no pun intended"
amazing
Hahaha
@@drpeyam Thanks for the great content, by the way. I don't understand the vast majority of the topics you address, but your energy and personality really makes all of it enjoyable to watch. There are a lot of math channels out there, but your enthusiasm is so infectious that it's just fun, even with subjects that are far beyond the scope of anything I'm currently trying to study or learn (or capable of really intuiting). There's definitely multiple attributes that make a great teacher, but that's one of them! Keep doing what you do.
What is M in 2piMi? Thanks! Very interesting.
Any integer
@@drpeyam Thanks!
You guy look so friendly !!
Impossible is an over used word.
Really loved it👍🤗
なんで急におすすめで出てきたんだw
x = - shirt cos 2 + k2π
x = shirt cos 2 + k2π
I'm a student in Japan, and I was wondering if the range of cos from -1 to 1 is the range of real numbers, and if I use imaginary numbers, it can also be 2?
I'm sorry. My English is somewhat poor.
Exactly
@@drpeyam Thank you. It was a very interesting lecture.
You’re welcome :)
in the calculator cos (i(2+√3))=1.00026 not 2 there is a fault
The calculation is correct
Good job 👌💙 I love
it.
Professor, I have a question. I want to learn more mathematics and be expert in it. My ultimate is to be a mathematician and holding extensive knowledge in the subject. What is your advice for me and what books should I read?
But the range of cos is 1
I finished writing my last high-school exams yesterday and youtube recommends this to me💔
I didn’t understand what exactly is the M on the e
I don't understand about e^ix + e^-ix ..
알고리즘이 여기로 끌었다
ElectroBoom's long lost brother
cos(x) = 2
x = acos(2)
QED
I'm a huge fan of the tee-shirt, and given your occasional references to blackpenredpen, I know at least one possible source for the joke.
At 0:14 you said something wrong about x, listen closely.
Edit:- i was wrong, sorry,(mistook range as domain)
I was referring to cos being between -1 and 1
@@drpeyam oh i am so sorry, i thought that you were talking about the domain of cos x
I can't understand
exp(ix) = exp(ln(2+-root3)+2piMi)
i'd never seen "Mi"
what is it ??
Any integer
The premiere ends:
Mathematics meme creators:
Directed by @Dr Peyam.
Amazing!
If you had asked me to solve cos(z)=2, I would have guessed it right in the beginning. Somehow x stands for a real number...
x＝cos-1(2) ????
Thanks a lot.
Can't believe i'm watching this for fun.
Thank you