This is a nice video of the very basic implications of Euler's identity, and I must agree that it is the most beautiful formula in the world. The reasons you put however I don't think is why it's beautiful, the graph of the Euler's identity as can be seen on the Wiki page better explores the beauty of this equation I think
Oh good god. It's from a piano piece by Beethoven, popularly known as the "Moonlight Sonata." Do people really have that association with it? Jesus . . . What have we come to.
you didn't explain why this exponential function (considering pi as the argument) can be represented as a sum of sin and cos. The beautiful for me, relies in the expansion of the exponential as 2 maclauring series, one of them multiplied by i (imaginary) resulting in e^(ix).
For ppl who know the meaning of this function, my advise is to look at the meaning of cos and sin and study some maths of complex functions and understand the relationship between a complex number and it's parametric representation. If u understand that than u should understand this reasoning.
@trudbol Euler's formula [e^(x i) = cos x + i sin x] is actually quite useful when working with complex numbers. Euler's identity [e^(pi i) = -1] on the other hand is just a beautiful mathematical relationship with no real use.
My god people, how can you rage over the fact that he didn't do it the "hard" way (it's NOT hard, it's just more work). This video was to show of Eulers Identity. Go poke your parents if you feel that you're getting too little attention. And remember the KISS rule, keep it seriously simple.
@gabisosa79 Simple geometric interpretation. Draw a circle of radius one in the complex plane (closely resembles the x, y plane we know and love from school). Angle, theta, can be measured in radians as well as degrees. 2pi radians equals 360 degrees. Pi radians is 180 degrees (= -1 on unit circle). At any point on the unit circle you can look down and see the real part (x). You can look over and see the imaginary part (y). The real part is cox theta and the imagi9nary part is sin theta.
What you showed is only impressive at a very trivial superficial level, it only impressives people who are not accustom to seeing natural occuring constants like e, pi. And imaginary number "i" is simply the SqRoot of -1 ... a mathematical symbol... what's interesting there is that i allows an extension to the real numbers and new fields such as complex algebra.
big deal... this is not significant, just b/c you can evaluate Euler's Eq for value of pi.
A more wonderful mathematical result is deriving Euler's Eq using a power series expansions on those functions (e, sin, cos)... why don't you try that!
after watching a few videos on this, i still didn't quite get how cos(pi) + isin(pi) = -1
but then i watched this and it cleared it up. which means that i got to take up an entire ap calc class showing my teacher and the class the proof of this, but i did a really long version of the proof haha. thanks man, clear and simple
The primary ASCII code shown at Wilton Windmill crop circle on May 22nd 2010 contained 96 binary digits, as 12 ASCII characters of 8 binary digits each. And it gave a close approximation to Euler’s Identity from advanced mathematics ( /watch?v=MqzNojwSPzE ) , Yet it also contained 9 anomalous binary digits. Those 9 anomalous digits equal 011010001 or 011-010-001 in octal base-eight meaning “3-2-1”. (a countdown to 2012) THE EPS /watch?v=N6QMxgWCtls
Hidden in the Euler’s Identity is the ‘Story of CREATION’ and this is what makes it most beautiful. It is the mathematical representation of Creation! Can there be anything more beautiful?
@seriallk This is the first movement of Beethoven's piano sonata no. 14, commercially known as 'The Moonlight Sonata'. A very profound piece. It manifests complex harmony and modulation in a simple but beautiful way; an appropriate choice for this video.
@edward6000 math is a universal language....if you dont speak/understand it...u are the idiot my dear.....i take it you didnt get past pre-algebra? its ok...there is still time to educate yourself...knowledge is power! Blessings, love , and light......
pollardrho06, ignore all the gain-sayers. All we mathemagicians know that you're just demonstrating the beauty of maths. If someone cannot see this beauty, there is not point trying to explain it. May I also say that you have very nice handwriting? :-)
how is it clever, dont you think euler who actually invented the identity exp(i*x)=cosx+iSinx realised that if you substitute pi in then you get said identity. Im sure he knew his trig functions. Doing somthing thats already known by every single mathematician in the world is not "clever", what i was saying is if this formula is to be demonstrated completley one must use the taylors series expansion
... seriously? I'm an undergraduate mathematics and physics student, I've known since high school. "but that of e^x... with a clever substitution." By skipping that, I guess you don't know how to derive Euler's formula... you have to substitute x=i(theta) is what I meant, sorry for the discrepancy.
It is an elegant equation, but I don't understand the 'mystery.' I mean, these constants are defined to mean these things. It's as if I said, "We'll let 'apples' = 1, 'bananas' = 2, and 'oranges' = 3. Now isn't it amazing that apples + bananas = oranges?!"
It's because it relates so much with 1 statement. It like magic how it works out. Apparently you can't see that, and comparing eulers formula with apples in oranges shows that you don't fully understand the equation.
That it relates so much in one statementwas the elegance I mentioned. I was talking about the "oh wow, isn't that amazing how that works out" sentiment that makes me yawn. If you understand what each of those constants mean and how they relate to each other, then it's obvious.
Uh, my bad that I'm only in high school. I do plan on majoring in math though, but right now, my understanding of math if very elementary compared to what you have.
@ schlynn, that's wonderful that you want to pursue greater studies in mathematics! It is a very rewarding subject. If your skills are up to it, I recommend taking an AP calculus class while in high school. That'll fast-track you into the really fun stuff in college.
You talk like if the definition e^ix=cosx+isinx is like this just because. It is defined like this because it makes sense (the links between the proprieties of the exponential and the trigonometric functions).
Actually, you can prove that e^iPi + 1 = 0 without the definition of e^ix, for example with Taylor series expansion.
I know, I have seen the proof you're talking about. But, you are extremely blind if you don't see the brilliance of the equation. It's called "the most beautiful equation in mathematics" or "the most elegant equation" for a reason.
You guys are so cute. I love how a pack of rabid wanna-be math nerds jump on me because they misunderstood my analogy. Sorry for seemingly "trolling" and sounding arrogant, but I've been doing calculus - recreationaly - since 1990.
I can assure you I am not 'blind to the beauty of mathematics.'
"You talk like if the definition e^ix=cosx+isinx is like this just because."
Um.. no. :-))
Think about it. pi is the ratio of circumference to diameter of a circle. e^ix is the equation of a circle in the complex plane, so of course at pi radians around that circle you're going to get to a cosine of -1
That's what I meant by it working because how the constants are defined.
Did you miss the part where I get paid to teach math to people?
I'm pretty sure I don't misunderstand.
I don't have enough room here to explain this, but here's the short version:
e^ix is the equation of a circle in the complex plane in polar form. At x = pi radians, the sine of that angle is zero so the imaginary part drops out leaving the cosine of -1.
That's really all there is to it.
Yes it is beautiful - I never said it wasn't - all I said was that it isn't mysterious.
thanks verymuch you have given me the answer and i now understand a lot better! Even ifit wasnt essentiall to know this you have just improved myunderstaning of this equation. this make you a teacher! thank you!! f.martin
ok im fiftteen so please dont judge me for asking this but what is 'i'. probably a stupid question but i would greatly appreciate an answer. thanking you in advance f.martin,
I doesn't need to have applications, it's beauty is cause enough to study it. It shows how with the introduction of complex numbers we are able to unify fundamental concepts which previously had appeared distinct, the exponential and trigonometric functions, calculus and geometry, the constants e and Pi. I find it hard to believe how you can not be amazed when you see the constants of mathematics elegantly linked together in euler's identity.
One thing that bothers me about complex numbers is that if you raise a number to an irrational power there are infinite solutions for it. Can't that be fixed? Could (or should) we introduce another kind of number?
I'm a mechanical engineer. This equation is used in most sciences that I use: Vibrations, thermodynamics, continuum mechanics, electrical systems etc etc. It's especially useful in electronics tho.
Although, I must say that this video does not show the full beauty off the proof of this identity based off the relatively simple series expansions of e^x, cos(x), and sin(x).
moshpot93 ur such an idiot. you dont understand how intelligent people think. math is the single most important field in the history of the world. if you cant comprehend that youur fucked
It's beautiful metaphorically not literally. I mean it isn't supposed to replace a woman or anything c'mon. It's just that it forms really well and is pretty amazing how it comes together.
Its e (which is (1+1/n)^n, where n is very close to infinity, but not quite there. i is the square root of -1, and pi is just pi. So, its e to the power of i times pi
( + , x )
xsyaz 3 weeks ago
nice try - but when u chainging x to pi - u must change all x to pi or/and ALL PI TO X !
blank4mexx 1 month ago
This is a nice video of the very basic implications of Euler's identity, and I must agree that it is the most beautiful formula in the world. The reasons you put however I don't think is why it's beautiful, the graph of the Euler's identity as can be seen on the Wiki page better explores the beauty of this equation I think
branbones69 2 months ago
Youtube won't work and I'm trying to learn. WHAT?!?!?!
clairebeanbagchair 2 months ago
2girls 1cup!??? nice background music
DEEEKAYify 2 months ago
This has been flagged as spam show
@DEEEKAYify "2girls 1cup!??? nice background music"
Oh good god. It's from a piano piece by Beethoven, popularly known as the "Moonlight Sonata." Do people really have that association with it? Jesus . . . What have we come to.
cufflink44 2 months ago
so what was the point of the contains bit ?
KillerXoX 3 months ago
shouldn't u draw the argand diagram for better illustration?
DiabloMercy 3 months ago
you didn't explain why this exponential function (considering pi as the argument) can be represented as a sum of sin and cos. The beautiful for me, relies in the expansion of the exponential as 2 maclauring series, one of them multiplied by i (imaginary) resulting in e^(ix).
aikeda86 3 months ago
For ppl who know the meaning of this function, my advise is to look at the meaning of cos and sin and study some maths of complex functions and understand the relationship between a complex number and it's parametric representation. If u understand that than u should understand this reasoning.
dekippiesip 3 months ago
I want to understand this!!!
eretrece 5 months ago
@eretrece The formula itself is actually "just" beautiful, it isn't really useful.
Its beauty lies in its extremely simple form and the fact it connects 5 of the most important numbers of mathematics:
- zero, the only number you can add without changing anything,
- one, the only number you can multiply by withut changing anything,
- pi, important in trigonometry,
- exponential, important in calculus,
- and the square root of minus 1, a key element in complex number theory.
Hope this helps. ;-)
trudbol 4 months ago 2
@trudbol Euler's formula [e^(x i) = cos x + i sin x] is actually quite useful when working with complex numbers. Euler's identity [e^(pi i) = -1] on the other hand is just a beautiful mathematical relationship with no real use.
(Sorry for nitpicking, but I couldn't resist it)
someonep93 2 months ago
@someonep93 Yes, that's pretty much what I said: it's beautiful and useless. ;-)
trudbol 2 months ago
@trudbol What I was saying was that you, in reality, were talking about Euler's identity, not Euler's formula.
Euler's identity: Useless yet beautiful
Euler's formula: Useful and (in my opinion) pretty beautiful as well
someonep93 2 months ago
son mamadas!!!
monserrathpeza 6 months ago
This a nice idea how to create a simple video and attract 54k to see it!!
Photon98 6 months ago
Look up "pi is (still) wrong" by vihart. She'll convince you of something more beautiful.
michalchik 7 months ago
The version using Tau is better. e^(i*Tau) = 1 = 1 + 0 if you really want zero in there.
Tau is equal to 2*Pi and is the ratio of a circles radius to it's circumference (obviously).
Sweet handwriting, by the way.
gloconnor 7 months ago 2
I understand it but. . . why the need to draw boobs? Seriously who draws boobs!!
mrvirus888 8 months ago
My god people, how can you rage over the fact that he didn't do it the "hard" way (it's NOT hard, it's just more work). This video was to show of Eulers Identity. Go poke your parents if you feel that you're getting too little attention. And remember the KISS rule, keep it seriously simple.
perejlm 8 months ago
this formula also contains the concept of equality
whoisthisguy002 9 months ago
What this mean in easy words, please. :)
gabisosa79 9 months ago
@gabisosa79 Simple geometric interpretation. Draw a circle of radius one in the complex plane (closely resembles the x, y plane we know and love from school). Angle, theta, can be measured in radians as well as degrees. 2pi radians equals 360 degrees. Pi radians is 180 degrees (= -1 on unit circle). At any point on the unit circle you can look down and see the real part (x). You can look over and see the imaginary part (y). The real part is cox theta and the imagi9nary part is sin theta.
camgere 9 months ago
Whatever, nerd.
kkriegg 9 months ago
what does this shit mean?
r3playretro 9 months ago
Nice hand writing hahah
smallfrye182 9 months ago
do you have an idiots' guide to understanding this?
78rikkers 9 months ago
i dont get it how is this beautiful it is just a formula
Kastralis 10 months ago
dude u saved my pre-calc grade so thank you
jlail572 10 months ago
L.Euler,(1.707-1.783),merece la música de Bach y no la de Beethoven.
Cierto,es la ecuación mas bella de todos los tiempos.Tres números irracionales con el 1 el 0!
0.Elemento neutro de la suma.
1.Elemento neutro de la multiplicación.
pi.La constante circular
e.La base de los logaritmos naturales
i.La unidad imaginaria.La raíz cuadrada de -1.
¿Cabe mas belleza?
En ausencia del hombre está la inmutabilidad de la verdad matemática.
paradoxicus 10 months ago
ugh derive it or don't... you just plugged in x=pi here, what a waste of time
wownov83 10 months ago
What you showed is only impressive at a very trivial superficial level, it only impressives people who are not accustom to seeing natural occuring constants like e, pi. And imaginary number "i" is simply the SqRoot of -1 ... a mathematical symbol... what's interesting there is that i allows an extension to the real numbers and new fields such as complex algebra.
cesarjom 10 months ago
big deal... this is not significant, just b/c you can evaluate Euler's Eq for value of pi.
A more wonderful mathematical result is deriving Euler's Eq using a power series expansions on those functions (e, sin, cos)... why don't you try that!
cesarjom 10 months ago
That has to be the most amazing handwriting ever! Well the cursive anyways.
arhodes18 10 months ago 9
@arhodes18 : Thank you :-)
pollardrho06 10 months ago
sonata #14 in c#minor, 1st movement. great song choice
FaithBane 10 months ago
@FaithBane thanks for the name of the music
bp56789 10 months ago
@bp56789 Oh, you didn't know? In that case, you're very welcome.
FaithBane 10 months ago
Comment removed
bp56789 10 months ago
Meraviglioso davvero
Xplessoable2 11 months ago
@1992R THANK YOU!!!
Euler himself would be proud
messakg123 11 months ago
Proof of first line?
(e^ix=cosx+isinx)
messakg123 11 months ago
At a glance Euler's Formula makes no sense, but in proofs, there is nothing more understandable.
DevoutSkeptic 11 months ago
ohhhhh,I hate math!!!!!!!!!!!!!!!!!!!!
DOPSIF 11 months ago
after watching a few videos on this, i still didn't quite get how cos(pi) + isin(pi) = -1
but then i watched this and it cleared it up. which means that i got to take up an entire ap calc class showing my teacher and the class the proof of this, but i did a really long version of the proof haha. thanks man, clear and simple
MrGreenknight11 11 months ago
"Beautiful!" the author writes there.
Yes, it is a simple equation of great mathematical beauty,
which is used today in everything from x-ray crystallography
to engineering to DNA. ...please read my next comments...
ArchangelWaiting 1 year ago
...And the 'crop artists' showed it to us
in a field of oilseed rape in southern England on May 22, 2010.
Whoever did this ... If we cannot now appreciate
the great beauty of their high intelligence,
and their yearnings to make contact with other advanced scientists
or mathematicians on Earth (for whom it was clearly meant),
then our local, often-struggling human race here will be the worse for it. ...please read my next comments...
ArchangelWaiting 1 year ago
The primary ASCII code shown at Wilton Windmill crop circle on May 22nd 2010 contained 96 binary digits, as 12 ASCII characters of 8 binary digits each. And it gave a close approximation to Euler’s Identity from advanced mathematics ( /watch?v=MqzNojwSPzE ) , Yet it also contained 9 anomalous binary digits. Those 9 anomalous digits equal 011010001 or 011-010-001 in octal base-eight meaning “3-2-1”. (a countdown to 2012) THE EPS /watch?v=N6QMxgWCtls
ArchangelWaiting 1 year ago
MATHEMATICIANS
Please see this link at cropconnector website .... /2010/wilton/comments.html
help solve the riddle
ASCII code contains Euler's Formula
ArchangelWaiting 1 year ago
Comment removed
ArchangelWaiting 1 year ago
Well maybe riemann hypothesis, but we just posed the question there
sitwosaints 1 year ago
Lavabugs right, about the best thing we have done so far, as a species
sitwosaints 1 year ago
Moonlight sonata, Really?
siniceblood2 1 year ago
I have a Question how did cos(pi) +i sin(pi) become -1+i(0)
TheDEATHMYSTERY 1 year ago
@TheDEATHMYSTERY cos(pi)=1 and i*sin(pi)=0
leonicc 1 year ago
@TheDEATHMYSTERY It's simple trig using radian measures of angles on the unit circle pi radians=180 degrees
PwnTheN00bz 1 year ago
This has been flagged as spam show
blah blah blah
ArtisanTony 1 year ago
Our jewel.
Lavabug 1 year ago
nice
cmatthew91 1 year ago
nice <3 !
TheKliverman 1 year ago 3
@TheKliverman : Thanks!
pollardrho06 1 year ago
Hidden in the Euler’s Identity is the ‘Story of CREATION’ and this is what makes it most beautiful. It is the mathematical representation of Creation! Can there be anything more beautiful?
azk2020 1 year ago
@azk2020 : Very well put..
pollardrho06 1 year ago
@azk2020
have you ever stopped to consider what it is that we are creating on this earth - a place where most suffer and lack the basic necessities
consider a mathematically proven solution in equal money - google it
cameronvcope 1 year ago
@azk2020 Well put, math and science is the only valid religion out there.
vkon15 11 months ago
@azk2020 How do you mean the story of "CREATION", do you mean the religious creation?
aneedfortheory 5 months ago
@azk2020
that rhetorical question, takes the form of irony?
dariuszskitek 1 month ago
Forgot that the identity also has a power operation (^) :)
samurai3502 1 year ago 2
@samurai3502 : Right, thank you..
pollardrho06 1 year ago
@samurai3502 but then again the power operation is simply multiplication.
baefe15 10 months ago
@samurai3502 Bear in mind, that's just multiplication a certain number of times. And (s)he mentioned multiplication.
wreynolds1995 9 months ago
@samurai3502 but Power is basically multiplication nth time.
mohnish0999 5 months ago
Unfortunately, according to Gauss, I'll never be a first class mathematician :(.
I can memorize it but I don't really understand it.
Barrettm95 1 year ago
I got this equation tattooed on my left forearm yesterday, and to my surprise people are not amazed at all.
sirisnin 1 year ago
@sirisnin I've always said, "if I ever get a tattoo, it will be of Euler's formula"
However, I going to wait until I'm more comfortable understanding complex trig before I do.:)
stampmaille 1 year ago 14
@stampmaille
LOL!
gregg4 9 months ago
mind=blown
ThemTheyHeShe 1 year ago
What O_O
wendyXpeggyXpancho 1 year ago
thank you, you just saved my ass :)
babyelephant121 1 year ago
I know how to proof Euler's formula , using Taylor's expanding for sin and cosin functions
TheKenthope 1 year ago
@TheKenthope
thats not a real proof right like going the long way home on the bus around all the houses
donnyab 1 year ago
Aww.. I wish I was this good at maths ... I always get the idea behind the math, but just am too slow to perform the maths consistently...
derealmoister 1 year ago
Taking rather a long time to show a very trivial derivation >.<
Suziessoliloquy 1 year ago
OK, can some-one explaiun to me what this is? what applications does this have and can I have an analogy.
I do not claim to understand this and I would love to please can I have a laymen term please.
PureZOOKS 1 year ago
@PureZOOKS
i have no idea o.o he lost me at 1:28
XXEMOsxrxhottXX 1 year ago
wats this music called?
seriallk 1 year ago
@seriallk This is the first movement of Beethoven's piano sonata no. 14, commercially known as 'The Moonlight Sonata'. A very profound piece. It manifests complex harmony and modulation in a simple but beautiful way; an appropriate choice for this video.
xxxxcensoredxxx 1 year ago
lol simplistic though the explenation is.. it is correct XD
MorrisWales 1 year ago
Thank you for the beautiful explanation!
gentlehorseman 1 year ago
@gentlehorseman Glad you think this way...
pollardrho06 1 year ago
beautiful?!?!?!? Are you joking? This is marvelous
Caporacolo 1 year ago 4
@Caporacolo Indeed...
pollardrho06 1 year ago
mathmaticians are self important idiots
edward6000 1 year ago
@edward6000 Hmm, learn how to spell first "mathmaticians!!"
pollardrho06 1 year ago 9
@pollardrho06 :))
cristiantimofi 1 year ago
@edward6000
Math is the universal language, and it's what makes the world work.
itsthebrod 1 year ago
@edward6000 math is a universal language....if you dont speak/understand it...u are the idiot my dear.....i take it you didnt get past pre-algebra? its ok...there is still time to educate yourself...knowledge is power! Blessings, love , and light......
islandgal75 1 year ago
@edward6000 Mathematicians and scolars fill the gap of ignorance and stupidity left by useless failed people like you, how do you like it now?
Umby64 1 year ago
As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Arthur Cayley
orcodrilo 1 year ago
@orcodrilo Indeed.
pollardrho06 1 year ago
ok the ecuation most beautiful ???????
RUDO1990 1 year ago
pollardrho06, ignore all the gain-sayers. All we mathemagicians know that you're just demonstrating the beauty of maths. If someone cannot see this beauty, there is not point trying to explain it. May I also say that you have very nice handwriting? :-)
DeadChickenHat 1 year ago
Thank you. Appreciate it and I totally agree with what you have written.
pollardrho06 1 year ago
la ecuacion matematica mas hermosa de todos los tiempos o_0 no tiene nada de hermosa
RUDO1990 1 year ago
Translate to English please? Thanks.
pollardrho06 1 year ago
lol you can't derrive that expression starting from what your supposed to be proving, you need to use a taylors series expansion
paulio2293 1 year ago
@paulio2293 I'm not proving it... Just plugging in x = pi. Does it say 'proof' anywhere?
pollardrho06 1 year ago
no babes
paulio2293 1 year ago
He wasn't proving anything... and not just a taylor series expansion, but that of e^x, cos(x), and sin(x) with a clever substitution.
Arycke 1 year ago
how is it clever, dont you think euler who actually invented the identity exp(i*x)=cosx+iSinx realised that if you substitute pi in then you get said identity. Im sure he knew his trig functions. Doing somthing thats already known by every single mathematician in the world is not "clever", what i was saying is if this formula is to be demonstrated completley one must use the taylors series expansion
paulio2293 1 year ago
... seriously? I'm an undergraduate mathematics and physics student, I've known since high school. "but that of e^x... with a clever substitution." By skipping that, I guess you don't know how to derive Euler's formula... you have to substitute x=i(theta) is what I meant, sorry for the discrepancy.
Arycke 1 year ago
Into the Taylor series expansion that is.
Arycke 1 year ago
Thank you Arycke.
pollardrho06 1 year ago
This is so true!
bukottmernok 1 year ago
Your handwriting is something in which to aspire.
Thanks.
joeybenn 1 year ago
Thank you so much for the appreciation and watching.
Regards.
pollardrho06 1 year ago
You have really nice handwriting!
Anyways, great video. The way that three seemingly unrelated constants fit together like that is just amazing to me.
OrphanEater1 2 years ago
Thanks for the appreciation and watching. This is just so beautiful... I just can't stop getting fascinated by it... :-)
pollardrho06 1 year ago
how about some explanation?
ibizaboyz 2 years ago
It is an elegant equation, but I don't understand the 'mystery.' I mean, these constants are defined to mean these things. It's as if I said, "We'll let 'apples' = 1, 'bananas' = 2, and 'oranges' = 3. Now isn't it amazing that apples + bananas = oranges?!"
Not really...
mdiem 2 years ago
It's because it relates so much with 1 statement. It like magic how it works out. Apparently you can't see that, and comparing eulers formula with apples in oranges shows that you don't fully understand the equation.
schlynn 2 years ago
That's funny. I understand the equation fully. I have degrees in physics, mathematics and engineering.
And I tutor mathematics on the side.
I get the math. That's why it *isn't* mysterious to me.
If you can't see the comparison to apples and oranges, then *you* don't understand where the constants come from.
;-)
mdiem 2 years ago
That it relates so much in one statementwas the elegance I mentioned. I was talking about the "oh wow, isn't that amazing how that works out" sentiment that makes me yawn. If you understand what each of those constants mean and how they relate to each other, then it's obvious.
It's very clean and neat, but *not* mysterious.
mdiem 2 years ago
Uh, my bad that I'm only in high school. I do plan on majoring in math though, but right now, my understanding of math if very elementary compared to what you have.
schlynn 2 years ago
@ schlynn, that's wonderful that you want to pursue greater studies in mathematics! It is a very rewarding subject. If your skills are up to it, I recommend taking an AP calculus class while in high school. That'll fast-track you into the really fun stuff in college.
;-)
mdiem 2 years ago
You talk like if the definition e^ix=cosx+isinx is like this just because. It is defined like this because it makes sense (the links between the proprieties of the exponential and the trigonometric functions).
Actually, you can prove that e^iPi + 1 = 0 without the definition of e^ix, for example with Taylor series expansion.
dupontproducoes 2 years ago
I know, I have seen the proof you're talking about. But, you are extremely blind if you don't see the brilliance of the equation. It's called "the most beautiful equation in mathematics" or "the most elegant equation" for a reason.
schlynn 2 years ago
@schlynn I see its beauty, of course. It's my favourite equation. My comment was to mdiem, sorry if I didn't make that clear.
dupontproducoes 2 years ago
You guys are so cute. I love how a pack of rabid wanna-be math nerds jump on me because they misunderstood my analogy. Sorry for seemingly "trolling" and sounding arrogant, but I've been doing calculus - recreationaly - since 1990.
I can assure you I am not 'blind to the beauty of mathematics.'
No hard feelings, seriously.
Just settle down.
LOL!
mdiem 2 years ago
"You talk like if the definition e^ix=cosx+isinx is like this just because."
Um.. no. :-))
Think about it. pi is the ratio of circumference to diameter of a circle. e^ix is the equation of a circle in the complex plane, so of course at pi radians around that circle you're going to get to a cosine of -1
That's what I meant by it working because how the constants are defined.
mdiem 2 years ago
maybe you've misunderstood...
e is the constant such that when you take the derivative of e^x, you get e^x.
π is the constant defined as the ratio of the circumference of a circle to the diameter of that circle.
i is √(-1)
approximately 2.718... raised to the power of approximately 3.14159 times an imaginary constant.... = -1
They weren't defined in terms of each other, and yet this somehow works. That's what makes it so darn cool!
100percentrekkie 2 years ago 2
LOL. ;-)
Did you miss the part where I get paid to teach math to people?
I'm pretty sure I don't misunderstand.
I don't have enough room here to explain this, but here's the short version:
e^ix is the equation of a circle in the complex plane in polar form. At x = pi radians, the sine of that angle is zero so the imaginary part drops out leaving the cosine of -1.
That's really all there is to it.
Yes it is beautiful - I never said it wasn't - all I said was that it isn't mysterious.
mdiem 2 years ago
thanks again oall ofthe replys i have had! u lot are AWSUM!!!!!!!!!!!!!!!! XD much love f.martin
upatree2009 2 years ago
thanks verymuch you have given me the answer and i now understand a lot better! Even ifit wasnt essentiall to know this you have just improved myunderstaning of this equation. this make you a teacher! thank you!! f.martin
upatree2009 2 years ago
Well explained, 5 stars for u! :)
MrNightLifeLover 2 years ago
ok im fiftteen so please dont judge me for asking this but what is 'i'. probably a stupid question but i would greatly appreciate an answer. thanking you in advance f.martin,
upatree2009 2 years ago
its the imaginary number, you can't obtain an answer from an even square of a negative number
Sqrt(-1) = i
it is used, e.g., Sqrt(-9) = Sqrt(9) * Sqrt(-1) = 9i
keine031 2 years ago
correction (not to be a douche)
3i
xxhorseriderxxx 2 years ago
crap, you're right, well I hope upatree got the concept
keine031 2 years ago
Thanks for elaborating...
pollardrho06 2 years ago
i = √-1
It's an imaginary number, as you cannot find the square root of a minus number: No number multiplied by itself can yield a negative result. :]
MentisIncorruptus 2 years ago
To author, if you are proving the Euler's formula you must use either complex graph, either Taylor's formula... Here, your writtings mean nothing
TheCentralasian 2 years ago
This video isn't showing a proof, just consequences of the formula..
mdiem 2 years ago
You have nice handwriting
gerberfletch 2 years ago
I don't understand why this formula is so great, what is it used to calculate?
Yu2Kal 2 years ago
I doesn't need to have applications, it's beauty is cause enough to study it. It shows how with the introduction of complex numbers we are able to unify fundamental concepts which previously had appeared distinct, the exponential and trigonometric functions, calculus and geometry, the constants e and Pi. I find it hard to believe how you can not be amazed when you see the constants of mathematics elegantly linked together in euler's identity.
wowsa0 2 years ago 16
One thing that bothers me about complex numbers is that if you raise a number to an irrational power there are infinite solutions for it. Can't that be fixed? Could (or should) we introduce another kind of number?
TaiFerret 2 years ago
@wowsa0 : Well said!!
pollardrho06 1 year ago
it's amazing because it links together some of the most important numbers and constants in mathematics - 1,0,pi,e and i
jakstrike1 2 years ago
It's very useful in solving higher order ODEs and PDEs.
Refer to one-dimension heat and wave equations.
plazmaz 2 years ago
I'm a mechanical engineer. This equation is used in most sciences that I use: Vibrations, thermodynamics, continuum mechanics, electrical systems etc etc. It's especially useful in electronics tho.
s27448632 2 years ago 2
Great video Pollard! I loved it. Check out my channel. I proved it the other way by using the series expansion of e^x.
Best regards,
Mr. RiemannCalculator
P.S I created another account because I had to delete my first one.
MrRiemannCalculator 2 years ago
Although, I must say that this video does not show the full beauty off the proof of this identity based off the relatively simple series expansions of e^x, cos(x), and sin(x).
prussianpanther 2 years ago
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prussianpanther 2 years ago
moshpot93 ur such an idiot. you dont understand how intelligent people think. math is the single most important field in the history of the world. if you cant comprehend that youur fucked
Johnsintomath 2 years ago
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moshpot93 2 years ago
Yea, sure, try telling people that when you're like, 30, and you're homeless because of your idiosity
zapper469 2 years ago
Comment removed
moshpot93 2 years ago
It's beautiful metaphorically not literally. I mean it isn't supposed to replace a woman or anything c'mon. It's just that it forms really well and is pretty amazing how it comes together.
MrRiemannCalculator 2 years ago
erm thanks 4 like putting on the video but could you make one explaining what it is???
tgtbatc 2 years ago
Its e (which is (1+1/n)^n, where n is very close to infinity, but not quite there. i is the square root of -1, and pi is just pi. So, its e to the power of i times pi
zapper469 2 years ago
You forgot exponentiation. :p
demolishmentor7 2 years ago
Yes... thanks...
pollardrho06 2 years ago
Nice video though. Demonstrates the simplicity of one of the most beautiful equations.
demolishmentor7 2 years ago
True... this is the beauty of mathematics... the preciseness and compactness of expressing things... Thanks for watching and commenting...
pollardrho06 2 years ago
Its + and =
Not + and x
Surely best fomula ever
cms271828 2 years ago
the x is the multiplication operator...
Addition and multiplication are the two most important operators in all of mathematics...
The equality is there of course too...
pollardrho06 2 years ago
the most beautiful relation in science even beats Einstiens maa Enegy Equivalence....(Ahem...its E=m(c^2))
vaibhav1803 2 years ago
True.
pollardrho06 2 years ago
Why (+ , x)..? What does the "+" mean?
UntakenNick 2 years ago
Plus (Addition) and * (Multiplication) are the two most fundamental operations in mathematics. Thanks for watching.
pollardrho06 2 years ago
Thanks.. I had a teacher who would write it that same way but I always thought that it was just because he's nuts..
UntakenNick 2 years ago
Euler rules!!!!!!
Thanks for putting this vid :D
charcoalfilter1101 2 years ago
Glad you like it. Thanks for watching and commenting...
pollardrho06 2 years ago
it would seem more beautiful to non mathematicians if you show the infinate series of sin x, cos x, and e^ix.
stockshunter 2 years ago
now use it to derive the double angle formula XD
zgr9 2 years ago
Indeed a formula to show the magic and majesty of Mathematics. Thanks for the proof.
MoroccoFriendofUSA 3 years ago
Greatest mathematician of all time, unfortunately SO MANY people ignore the contributions that mathematicians have made to this world.
gincampana 3 years ago
Euler was an amazing mathematician, thanks to show the world the most beautiful equation
elyayon 3 years ago