Compliments young man. I'm a teacher with many years in the job so I know a little of which I speak - you are a loss to the profession. Please continue to make this kind of video and I'll continue to point my colleagues in the direction of this sort of high-quality teaching resource. At the very least you can take some satisfaction in knowing that many people will benefit from your hard work.
your point about being 'pulled away' from the real growth by the imaginary growth was such a great explanation! I appreciate the intuitive approach you have taken with this lesson, it makes understanding Euler's identity that much more beautiful!
It's great to see a REAL discussion of this. Most other videos simply show that e^ix = cosx + isinx. Euler's formula is actually a DEFINITION and a convenient convention because the trig version behaves the way one would expect the e^ix version "should" behave (regarding differentiation, exponential behavior, infinite series expansion). While the numbers i and pi may seem superficially unrelated, i represents a rotation in the complex plane and pi represents a rotation (in radians) around the unit circle (with polar coordinates). So i and pi are basically just both units of rotation that are being compared in differents contexts. i and pi / 2 (the argument of i when i is expressed in polar coords) have essentially the same "meaning" (just in different contexts), thus showing that Euler''s formula is not so terribly mysterious, but rather quite intuitive and due to Euler's DEFINITION which simply relates complex plane rotation to trigonometric rotation. Great video! The only video that really gets to the meat of the problem and makes things clear and less "mysterious". That's the talent of a great teacher.
Wait wait wait, i'm profoundly unimpressed by your explanation, I think i'm not getting it. I see how you describe this complex plane rotation with pi and the number i, that's very clear to me, but how does the number e (a number seemingly unrelated to all this) fit so incredibly nicely in this polar coords? You say Euler's formula is actually a "definition", a "convention" but this makes no sense. The number e was discovered by Bernoulli when he was trying to solve a compound interest financial problem. When you take the number e and you elevate it with this polar imaginary coordinates then suddenly all the irrational infinite decimals of pi and all the irrational infinite decimals of the number e suddenly vanish. Where is the explanation of this?
lautaa33 Perhaps you're secretly wanting pi and e to be super mysterious. Pi is simply the ratio of the circumference of a circle to its diameter. en.wikipedia.org/wiki/Circumference There's nothing super mysterious about a circle, one of the simplest geometric shapes. e is also not super special. It appears in many contexts in mathematics, both in interest theory and in calculus. This is a long discussion, but there are many simple definitions of e. en.wikipedia.org/wiki/E_(mathematical_constant) More importantly, any complex number z = x + iy, can also be defined as z = re^(i * theta), where r is the distance from the origin (in the complex plane), i.e. square_root ( x^2 + y^2) and theta is the angle in radians. As such, an angle for theta of exactly pi (or 2 pi or 3 pi etc.) simply means the number is completely real, i.e. it has no imaginary part. That's it. Nothing more. If theta is 1/2 * pi (or 3/2 * pi or 5/2 * pi etc.), it simply means the number is completely imaginary, i.e. it has no real part. Is this helpful? This is not to deny that the universe itself is a great mystery and, as such, every detail of it is mysterious in some way. But, from this perspective, the number 2 is also very mysterious. In fact, e^(i * 2 pi) + 1 = 2. You can make anything mysterious if you are creative enough. ;-)
Douglas Chorpita The number 2 is perfectly senible. The only mysterious thing about e^(i * 2 pi) + 1 = 2 is where did all the irrational decimals go. I understand where pi and e come from. They come from seemingly unrelated places. I'm not trying to make those numbers mysterious. Is the relantionship of those numbers who are mysterious. You say "z = re^(i * theta)" but how do you even get to this (I don't distrust that it's true, I'm just saying, why does an irrational number that came from finance relate so perfectly to the polar coords of a complex circle)
lautaa33 e, like the number 0, was always “there”. It was first noticed by humans in the context of interest theory. But that doesn’t mean the number is an "interest theory kind of number". You can define e^x and e^ix in terms of sin x, cos x, sinh x, cosh x and i. And you can define sin x, cos x, sinh x, cosh x in terms of e^x, e^ix and i. So these functions are inherently and deeply related to each other. Where does pi come in? The period of sin x and cos x is 2 pi. Why? Because they represent the rotation about the unit circle, x^2 + y^2 = 1, whose circumference is 2 pi. e^ix represents the phase (the angle) of a complex number. Here e^i pi = -1 and e^i 2 pi = 1, because e^ix is a periodic function that, just like sin x and cos x, exactly repeats itself after every 2 pi radians. en.wikipedia.org/wiki/Periodic_function It might be worth thinking about the unit hyperbola x^2 - y^2 = 1 to understand how sin x, cos x, sinh x, cosh x, e^x and e^ix all fit together nicely. en.wikipedia.org/wiki/Unit_hyperbola It might be worth thinking about the Taylor series expansions for all these functions to understand how they are all built of the same kinds of polynomial "components". en.wikipedia.org/wiki/Taylor_series It might be worth thinking about differential equations. The derivative of e^x is also e^x. The second, third and fourth derivative of e^x is also e^x. The fourth derivative of sin x is also sin x. The fourth derivative of cos x is also cos x. The fourth derivative e^ix is also e^ix. Such functions that can be expressed in terms of their first (or higher order) derivatives are called Pfaffian functions. en.wikipedia.org/wiki/Pfaffian_function sin x, cos x, e^x and e^ix are all such functions. It is worth mentioning that it's not useful to think in terms of decimal places. Decimal places are just an arbitrary convention that arises from the fact that humans have ten fingers. The number 10 (the number one greater than 9) which is the basis of our most widely accepted number system is just a human convention. If you know computers, you are familiar with base 16 numbers like 3a5c. You could also use base e numbers if you wanted to. In this case 10 would be equal to e and 100 would be e^2. Don't worry about decimals. Numbers don't need to be based on how many fingers you have. Does that help you now?
Well thank you very much for taking the time to put all that together, but unfortunately I already knew all of it (except Pfaffian functions, but I knew that they both have ciclic derivatives) Although I'm sure I don't have the most profound understanding of all those concepts, I did just pass my Calculus final exam from engineering so I'm fresh in the demonstration that Taylor expansions provides for Euler's identity. Once you establish the connection between the number e and the sine/cosine functions, then the connection to pi is trivial, I know. In fact, Euler's identity is just an specific case of The Formula of Euler, which relates cosine and sine functions to the number e. But that's my mistery, why is the number e, when considered in the complex realm of numbers, somehow related to the sine and cosine, which are concepts related to circunferences? The number e has no business with circunferences, but when you plug the square root of -1 in the equation, then suddenly 2 irrational numbers vanish. Your dismissing of the decimals profoundly misses the point of the inherent characteristics of an irrational number. Irrational numbers are incommensurable. The reason why this numbers impressed so much the greeks, was not because they had infinite decimals in the decimal base, but because they just can't be expressed, not even as a fraction of integers. Your trick of putting the irrational number as a base is just passing the problem to the base. The fact that pi and e, 2 seemingly unrelated irrational numbers can become a goddamn integer when you plug a square root of -1 is outstanding. The fact that they are both Pfaffian functions doesn't help me to understand how the complex realm can provide a connection between the number e and the trigonometric functions.
Dude you are a life saver! You know what would be even way cooler? If you could get a list of all high school math topics and create analogies for each topic. You'd be literally bringing joy to the world's math education. You'll change the world.
I've been to a lot of math lectures with many experienced professors. This young man has done an amazing presentation. The best 10 minutes on an academic topic I have ever witnessed.
Wow, what a fantastic explanation. I haven't fully internalized this formula yet, but this was an excellent 10 minutes spent in helping me get there. Well done, and thank you so much!
i ve been following your blog for a while and very glad to see a screen cast :) Its much easier and fun to watch a video before starting on text. I have come across Benjamins quote on Eulers equation, and took his quote for granted and never made an attempt to understand the equation. Once you have explained all the components of the equation, it doesn't seem to be a mystery anymore ! Excellent work :) subscribing right away .
Got here after noticing that today's Google doodle is celebrating Euler's birthday. I'm not a mathematician, but easily understood your explanation. Very clear and concise. Great job!
I love this channel so much. They should teach this way in schools. I'm hoping to become a lecturer some day and I will try to use your methods as much as possible. These thorough explanations give a much better understanding of the concept and I believe the extra time put into understanding the math intuitively will be saved time on how much students need to revise and will probably be a whole lot enjoyable to learn.
so sin/cos have two properties (may call them x and y or whatever) of which periodically the one looses and the other gains, the one gives the other takes, they hand it over ad infinituum. the same is the case for the definition of imaginary space. you have two properties again, the 'imaginary' dimension of i and the dimension of 'real' numbers. that is just a helpful tool to be able to have something in between of 1 and -1, it is the root of -1 which otherwise couldnt be defined, so if you take i*i you get -1, and if you take i*i*i = -i you are left in the imaginary space between and if you take i*i*i*i you have 1 so you circle around between the two properties again. thats a beautiful insight. thank you for sharing :)
Absolutely Excellent! The idea of i being a rotator is very seldom, if ever, discussed. In fact I contend it is a unit of precession and Euler's identity is simply the compact form of Hamilton's quaternions, that is, e to the i pi=i squared=j squared=k squared=ijk=-1.
Helpful. One thing I wonder about is how much of the startling nature of Euler's Identity is due to just the convention of how mathematicians have defined raising something to the "i" power. In other words, does the identity represent some amazing discovery of an inherent property of numbers or is it more of just an artifact of how we define certain mathematical notations (multiplying a number by itself an imaginary number of times) which have no real-world grounding.
Thank you bro; I finally grasped the concept. My intuition tells me that Euler's Identity holds the key that will unlock the emergence of consciousness(biological)
@anzatzi Thanks for the note! Yes, there's a few ways to tackle e -- as you say, e^i has rotation built in. I might be confused about your comment -- I think growth following the path of a circle is basically a rotation right? However, there might be another way to state it which gets the point across more clearly too. I'd definitely like to make some more videos from the book :)
@Padenormous Yeah, I don't see it as much as a "definition" and more as showing another way to get to i. It's a bit like saying 4/2 = 2... yes, you used 2 in the explanation of itself, but the idea is to say "You can see 2 on its own, or as half of 4." Similarly, you can see i as a single rotation on its own (just straight "i"), or the result of taking "1" and rotating it upwards 90 degrees (e^i * pi/2).
@consultjohan I used ScreenFlow on Mac -- it makes it easy to both capture and combine your screen and webcam. Yeah, I like having a way to show gestures, etc. vs. just your voice.
@lankyjuggler Thanks, hope you enjoy it! As a chemist you may enjoy the articles on ln and e, as they seem to appear in a ton of chem formulas :). It took a while before I could start to see them intuitively.
This is WONDERFUL. Thank you for expressing in intuitive analogies the functional relationships underlying mathematical symbols. Please continue this work! You're doing great. (Are you on Patreon? If not, you might wish to look it up!)
@supergopi Thanks -- yes, that quote really struck a nerve with me because I had resigned myself to not understanding the formula when I first encoutered it also. Appreciate the support :)
From 5:00 to 5:19 , when Kalid talks about 1 unit -- is that same as 1 unit of time? I'm not clear how he is representing 1 unit of time on the circle? At 5:19 , he says "we travel 1 unit" ? What's the unit here ? is it a time unit or distance unit? thanx
@EclecticSceptic Hi, sorry it didn't work for you! Analogies are personal -- for me, I see the equation as relating linear motion (sine + cosine) to rotation (move out, and rotate up).
Great presentation! Props! but i'm confused. Pythagorus turned counting and measuring distances and angles to form geometry, algebra was developed which made it easier to work with equations as units change. Euler led to polar notation (radius of a circle, and angles) which is really useful when things spin or move in a circle. trigonometry develops to make this easier. calculus makes it possible to calculate rates of change. all of this derives from Euclid's 5 axioms.. albert learned all these, but as a physicist, using maths to try to understand some things about the real physical world, realized that Newtons methods didnt work. but working with the maths, discovered equations that energy and mass and time and the speed of light were cleary in a very distinct relationship. he put gravity into the equations, and he resisted believing the implications. but the maths worked! and I may very well be wrong, but my understanding is that every rigorous experiment that tested these implications has confirmed them, in the real physical world ( i should say the physical universe we observe). He came to realize that it was possible, so the maths said, to make a very powerful bomb, and practical useful production of energy. So his equations describe very well our physical universe. everyone of these maths, from Euclid on up, stood on the shoulders of those who came before. and each one of these maths may have been essential to getting from how to add to nuclear power plants.. Silly question prob, but can you get to trig without Euler's equations? or are they essential to a geometry of spinning and moving in circles, and developing calculus, and Einstein's discoveries? prob an ignorant example, but do his equations say anything practical, useful about our real world like, growth of bacteria in a petri dish, spread of a virus in a population, or to make and use an MRI. its a cute equation, but is it practical, or only theoretical. the math works, but where does it work in the real world? or is it just a party trick? Too long, sorry..
You seem to have a very strong understanding of the topic, what if the anchor to x and y were to move in the time axis and falling into a gravity well?
Great breakdown!! For some reason...i see centripetal force in higher dimensional Spacetime spiraling inward. How would you correlate euler's formula with the fine- structure constant???
so if the theta gives us the amount of rotation and i gives us the intrest in the index then why doesnt rotation start from the y axis as value of i is plotted on the y axis
It's easiest to think of e^ix as 1.0 * e^ix. You start at 1.0, and modify that position using e [continuous change], i [rotation], and x [amount of time]. i influences the type change will occur, but not the starting point. If you wanted to start at i and rotate from there, you'd use i * e^ix.
hi--i really liked your book--but if feel like this is very non-intuitive. i think its much easier to set it up by showing how eulers formula appears in converting to the polar form of complex numbers. So e^i is intrinsically rotational. The growth in a circle is interesting--but obscures e as a rotational and oscillatory operator. you should make additional videos from your book--you have great perspective
Kalid, How did you make this video and what software did you use? I like the way you put yourself in the corner, and at the same time explain with the cursor on the screen--just like in a lecture room!
Ah! One thing I just realized--at first I was asking, why don't we accelerate as we go around the circle? The answer is we ARE accelerating--we are CHANGING our velocity. Just like the Moon around the Earth is constantly "accelerating" as it changes direction. It's interesting that that acceleration doesn't accumulate in the velocity the way linear acceleration does. It's very hard to wrap your head around the idea that going 1, 2, 4, 8 is the "same" acceleration as 0, pi/4, pi/2, 3pi/4, pi. That is VERY hard to grasp.
Okay, it finally fully sank in. I was still having trouble with i^i, because I thought we were just "starting" at i, and then rotating from there, rotating at the rate of i. But that's not what we're talking about. The entire entity e^x is the _growth_rate_. It has nothing to with, say, the "principal" we start with. So what "e^x" represents is the direction of the acceleration. If x is real, then e^x is real, so we accelerate along the real numbers. "e" is in a sense to "rate" of change, and the exponent is the "direction" of change. So, for "e^i," the "rate" is still natural, that is, e, but the "direction" of change is now straight up. So again, with i^i, that doesn't mean "start" at 0+i and rotate left. We're still "starting" at 1. When we consider i^i we're still looking at the _vector_ of the change--it's magnitude and direction. If the base component is real, then the imaginary component is straight up. But in i^i, the "rate" of change is i, and then the "direction" is i again, so the vector of the change is straight backward along the real axis. SO, if you had not purely imaginary numbers, but complex numbers, e^(ipi/4)^i, the direction of change would be up and back, 3pi/4 from the x axis. The point, again, is that when looking at y^x in complex numbers, that whole thing is about about the magnitude and direction of change--we haven't even started counting yet, we're just looking at what direction we're going to push the growth. I think that actually, despite the fact we're thinking in polar coordinates in some way here, there are two "components" to the growth vector that you add together. The base and the exponent.
Yes, great insight. It's best to think of e^x as shorthand for "1.0 * e^x" and i^i as "1.0 * i^i". 1.0 is always our implicit starting point: we are going to modify the unit quantity in some way (at some rate, for some amount of time). If we truly wanted to start at (0 + i), we would do "i * e^x" or "i * i^i".
I have a question, how can one define "i" as e^(i*pi/2). I didn't think you could define something with that something. Or, correct me if I'm wrong. Good video.
hi guys. i can easily understand Euler's Identity. However, how to i convince a female to have sexual intercourse with me? can the process be explained mathematically? i have already scanned the math and physics forums as well as the world of warcraft ones.
convincing females? that's like breathing. I am pretty sure he doesn't even need to consciously do that, it just happens when he works as a programming manager for Microsoft, i.e. earning enough in one year for 10 year of your effort. you can make fun of nerds all you want, you can never be one however hard you try.
Compliments young man. I'm a teacher with many years in the job so I know a little of which I speak - you are a loss to the profession.
Please continue to make this kind of video and I'll continue to point my colleagues in the direction of this sort of high-quality teaching resource. At the very least you can take some satisfaction in knowing that many people will benefit from your hard work.
Just coming to this now, thanks so much for the kind words. I really appreciate it.
your point about being 'pulled away' from the real growth by the imaginary growth was such a great explanation! I appreciate the intuitive approach you have taken with this lesson, it makes understanding Euler's identity that much more beautiful!
It's great to see a REAL discussion of this. Most other videos simply show that e^ix = cosx + isinx. Euler's formula is actually a DEFINITION and a convenient convention because the trig version behaves the way one would expect the e^ix version "should" behave (regarding differentiation, exponential behavior, infinite series expansion). While the numbers i and pi may seem superficially unrelated, i represents a rotation in the complex plane and pi represents a rotation (in radians) around the unit circle (with polar coordinates). So i and pi are basically just both units of rotation that are being compared in differents contexts. i and pi / 2 (the argument of i when i is expressed in polar coords) have essentially the same "meaning" (just in different contexts), thus showing that Euler''s formula is not so terribly mysterious, but rather quite intuitive and due to Euler's DEFINITION which simply relates complex plane rotation to trigonometric rotation. Great video! The only video that really gets to the meat of the problem and makes things clear and less "mysterious". That's the talent of a great teacher.
Wait wait wait, i'm profoundly unimpressed by your explanation, I think i'm not getting it. I see how you describe this complex plane rotation with pi and the number i, that's very clear to me, but how does the number e (a number seemingly unrelated to all this) fit so incredibly nicely in this polar coords?
You say Euler's formula is actually a "definition", a "convention" but this makes no sense. The number e was discovered by Bernoulli when he was trying to solve a compound interest financial problem.
When you take the number e and you elevate it with this polar imaginary coordinates then suddenly all the irrational infinite decimals of pi and all the irrational infinite decimals of the number e suddenly vanish. Where is the explanation of this?
lautaa33 Perhaps you're secretly wanting pi and e to be super mysterious. Pi is simply the ratio of the circumference of a circle to its diameter.
en.wikipedia.org/wiki/Circumference
There's nothing super mysterious about a circle, one of the simplest geometric shapes.
e is also not super special. It appears in many contexts in mathematics, both in interest theory and in calculus. This is a long discussion, but there are many simple definitions of e.
en.wikipedia.org/wiki/E_(mathematical_constant)
More importantly, any complex number z = x + iy, can also be defined as z = re^(i * theta), where r is the distance from the origin (in the complex plane), i.e. square_root ( x^2 + y^2) and theta is the angle in radians. As such, an angle for theta of exactly pi (or 2 pi or 3 pi etc.) simply means the number is completely real, i.e. it has no imaginary part. That's it. Nothing more. If theta is 1/2 * pi (or 3/2 * pi or 5/2 * pi etc.), it simply means the number is completely imaginary, i.e. it has no real part.
Is this helpful?
This is not to deny that the universe itself is a great mystery and, as such, every detail of it is mysterious in some way. But, from this perspective, the number 2 is also very mysterious. In fact, e^(i * 2 pi) + 1 = 2. You can make anything mysterious if you are creative enough. ;-)
Douglas Chorpita The number 2 is perfectly senible. The only mysterious thing about e^(i * 2 pi) + 1 = 2 is where did all the irrational decimals go.
I understand where pi and e come from. They come from seemingly unrelated places. I'm not trying to make those numbers mysterious. Is the relantionship of those numbers who are mysterious.
You say "z = re^(i * theta)" but how do you even get to this (I don't distrust that it's true, I'm just saying, why does an irrational number that came from finance relate so perfectly to the polar coords of a complex circle)
lautaa33
e, like the number 0, was always “there”. It was first noticed by humans in the context of interest theory. But that doesn’t mean the number is an "interest theory kind of number". You can define e^x and e^ix in terms of sin x, cos x, sinh x, cosh x and i. And you can define sin x, cos x, sinh x, cosh x in terms of e^x, e^ix and i. So these functions are inherently and deeply related to each other. Where does pi come in? The period of sin x and cos x is 2 pi. Why? Because they represent the rotation about the unit circle, x^2 + y^2 = 1, whose circumference is 2 pi.
e^ix represents the phase (the angle) of a complex number. Here e^i pi = -1 and e^i 2 pi = 1, because e^ix is a periodic function that, just like sin x and cos x, exactly repeats itself after every 2 pi radians.
en.wikipedia.org/wiki/Periodic_function
It might be worth thinking about the unit hyperbola x^2 - y^2 = 1 to understand how sin x, cos x, sinh x, cosh x, e^x and e^ix all fit together nicely.
en.wikipedia.org/wiki/Unit_hyperbola
It might be worth thinking about the Taylor series expansions for all these functions to understand how they are all built of the same kinds of polynomial "components".
en.wikipedia.org/wiki/Taylor_series
It might be worth thinking about differential equations. The derivative of e^x is also e^x. The second, third and fourth derivative of e^x is also e^x. The fourth derivative of sin x is also sin x. The fourth derivative of cos x is also cos x. The fourth derivative e^ix is also e^ix. Such functions that can be expressed in terms of their first (or higher order) derivatives are called Pfaffian functions.
en.wikipedia.org/wiki/Pfaffian_function
sin x, cos x, e^x and e^ix are all such functions.
It is worth mentioning that it's not useful to think in terms of decimal places. Decimal places are just an arbitrary convention that arises from the fact that humans have ten fingers. The number 10 (the number one greater than 9) which is the basis of our most widely accepted number system is just a human convention. If you know computers, you are familiar with base 16 numbers like 3a5c. You could also use base e numbers if you wanted to. In this case 10 would be equal to e and 100 would be e^2. Don't worry about decimals. Numbers don't need to be based on how many fingers you have.
Does that help you now?
Well thank you very much for taking the time to put all that together, but unfortunately I already knew all of it (except Pfaffian functions, but I knew that they both have ciclic derivatives) Although I'm sure I don't have the most profound understanding of all those concepts, I did just pass my Calculus final exam from engineering so I'm fresh in the demonstration that Taylor expansions provides for Euler's identity.
Once you establish the connection between the number e and the sine/cosine functions, then the connection to pi is trivial, I know. In fact, Euler's identity is just an specific case of The Formula of Euler, which relates cosine and sine functions to the number e.
But that's my mistery, why is the number e, when considered in the complex realm of numbers, somehow related to the sine and cosine, which are concepts related to circunferences?
The number e has no business with circunferences, but when you plug the square root of -1 in the equation, then suddenly 2 irrational numbers vanish.
Your dismissing of the decimals profoundly misses the point of the inherent characteristics of an irrational number. Irrational numbers are incommensurable. The reason why this numbers impressed so much the greeks, was not because they had infinite decimals in the decimal base, but because they just can't be expressed, not even as a fraction of integers. Your trick of putting the irrational number as a base is just passing the problem to the base. The fact that pi and e, 2 seemingly unrelated irrational numbers can become a goddamn integer when you plug a square root of -1 is outstanding.
The fact that they are both Pfaffian functions doesn't help me to understand how the complex realm can provide a connection between the number e and the trigonometric functions.
Dude you are a life saver! You know what would be even way cooler? If you could get a list of all high school math topics and create analogies for each topic. You'd be literally bringing joy to the world's math education. You'll change the world.
This guy is a genius!
I've been to a lot of math lectures with many experienced professors. This young man has done an amazing presentation. The best 10 minutes on an academic topic I have ever witnessed.
Wow, what a fantastic explanation. I haven't fully internalized this formula yet, but this was an excellent 10 minutes spent in helping me get there. Well done, and thank you so much!
Best explanation I have ever heard on the subject, thank you so much!
i ve been following your blog for a while and very glad to see a screen cast :)
Its much easier and fun to watch a video before starting on text.
I have come across Benjamins quote on Eulers equation, and took his quote for granted and never made an attempt to understand the equation. Once you have explained all the components of the equation, it doesn't seem to be a mystery anymore !
Excellent work :) subscribing right away .
Got here after noticing that today's Google doodle is celebrating Euler's birthday. I'm not a mathematician, but easily understood your explanation. Very clear and concise.
Great job!
Brilliant explanation.
by far the best explanation fo euler's identity ive ever heard. htanks man!!
You have a great way of conceptualizing and visualizing math! Thank you very much!!
ur actually one of the best explainers on yt and ive watchd a tonnnnnnnnn of vides
I love this channel so much. They should teach this way in schools. I'm hoping to become a lecturer some day and I will try to use your methods as much as possible. These thorough explanations give a much better understanding of the concept and I believe the extra time put into understanding the math intuitively will be saved time on how much students need to revise and will probably be a whole lot enjoyable to learn.
Thank you!
so sin/cos have two properties (may call them x and y or whatever) of which periodically the one looses and the other gains, the one gives the other takes, they hand it over ad infinituum. the same is the case for the definition of imaginary space. you have two properties again, the 'imaginary' dimension of i and the dimension of 'real' numbers. that is just a helpful tool to be able to have something in between of 1 and -1, it is the root of -1 which otherwise couldnt be defined, so if you take i*i you get -1, and if you take i*i*i = -i you are left in the imaginary space between and if you take i*i*i*i you have 1 so you circle around between the two properties again.
thats a beautiful insight. thank you for sharing :)
What an amazing video. Thanks a lot!!!
It is really helpful
I always want to have people like you to be my teachers
Awesome. I love it when math is explained so visually!
Super awesome explanation of the imaginary exponential linked to rotation, thanks a bunch!
Absolutely Excellent! The idea of i being a rotator is very seldom, if ever, discussed. In fact I contend it is a unit of precession and Euler's identity is simply the compact form of Hamilton's quaternions, that is, e to the i pi=i squared=j squared=k squared=ijk=-1.
Helpful. One thing I wonder about is how much of the startling nature of Euler's Identity is due to just the convention of how mathematicians have defined raising something to the "i" power. In other words, does the identity represent some amazing discovery of an inherent property of numbers or is it more of just an artifact of how we define certain mathematical notations (multiplying a number by itself an imaginary number of times) which have no real-world grounding.
Very nice job of explaining -- this combined with the article that it supplements are great.
mind = blown
Thanks for making this. Keep it up!
Thank you bro; I finally grasped the concept. My intuition tells me that Euler's Identity holds the key that will unlock the emergence of consciousness(biological)
@anzatzi Thanks for the note! Yes, there's a few ways to tackle e -- as you say, e^i has rotation built in. I might be confused about your comment -- I think growth following the path of a circle is basically a rotation right? However, there might be another way to state it which gets the point across more clearly too. I'd definitely like to make some more videos from the book :)
@Padenormous Yeah, I don't see it as much as a "definition" and more as showing another way to get to i. It's a bit like saying 4/2 = 2... yes, you used 2 in the explanation of itself, but the idea is to say "You can see 2 on its own, or as half of 4." Similarly, you can see i as a single rotation on its own (just straight "i"), or the result of taking "1" and rotating it upwards 90 degrees (e^i * pi/2).
@consultjohan I used ScreenFlow on Mac -- it makes it easy to both capture and combine your screen and webcam. Yeah, I like having a way to show gestures, etc. vs. just your voice.
@lankyjuggler Thanks, hope you enjoy it! As a chemist you may enjoy the articles on ln and e, as they seem to appear in a ton of chem formulas :). It took a while before I could start to see them intuitively.
This is WONDERFUL. Thank you for expressing in intuitive analogies the functional relationships underlying mathematical symbols.
Please continue this work! You're doing great. (Are you on Patreon? If not, you might wish to look it up!)
Tell me about it, what's great is i've actually recommended 'better explained' to many of my own professors!
fantastic explanation. keep up the awesome work
Well done man. You're a star!
Thank you so much! Very logical explanation!
@dwightivany Thanks for the kind words!
@supergopi Thanks -- yes, that quote really struck a nerve with me because I had resigned myself to not understanding the formula when I first encoutered it also. Appreciate the support :)
Bravo young man. Well done.
From 5:00 to 5:19 , when Kalid talks about 1 unit -- is that same as 1 unit of time? I'm not clear how he is representing 1 unit of time on the circle? At 5:19 , he says "we travel 1 unit" ? What's the unit here ? is it a time unit or distance unit? thanx
@EclecticSceptic Hi, sorry it didn't work for you! Analogies are personal -- for me, I see the equation as relating linear motion (sine + cosine) to rotation (move out, and rotate up).
Great presentation! Props! but i'm confused. Pythagorus turned counting and measuring distances and angles to form geometry, algebra was developed which made it easier to work with equations as units change. Euler led to polar notation (radius of a circle, and angles) which is really useful when things spin or move in a circle. trigonometry develops to make this easier. calculus makes it possible to calculate rates of change. all of this derives from Euclid's 5 axioms.. albert learned all these, but as a physicist, using maths to try to understand some things about the real physical world, realized that Newtons methods didnt work. but working with the maths, discovered equations that energy and mass and time and the speed of light were cleary in a very distinct relationship. he put gravity into the equations, and he resisted believing the implications. but the maths worked!
and I may very well be wrong, but my understanding is that every rigorous experiment that tested these implications has confirmed them, in the real physical world ( i should say the physical universe we observe). He came to realize that it was possible, so the maths said, to make a very powerful bomb, and practical useful production of energy. So his equations describe very well our physical universe. everyone of these maths, from Euclid on up, stood on the shoulders of those who came before. and each one of these maths may have been essential to getting from how to add to nuclear power plants..
Silly question prob, but can you get to trig without Euler's equations? or are they essential to a geometry of spinning and moving in circles, and developing calculus, and Einstein's discoveries? prob an ignorant example, but do his equations say anything practical, useful about our real world like, growth of bacteria in a petri dish, spread of a virus in a population, or to make and use an MRI. its a cute equation, but is it practical, or only theoretical. the math works, but where does it work in the real world? or is it just a party trick?
Too long, sorry..
Concise amazing explanation! 🙏🙏🙏
@MrTheSuperHead Awesome, thanks! I plan on doing this for more articles.
The diagram onscreen starting at 2:22 looks lik the circle is centred on (cosx,0) rather than (0,0) - is that a mistake?
Yep, that was a mistake. The post has been updated, the diagram should be this:
betterexplained.com/wp-content/uploads/euler/circle_traverse.png
@gamesbok This was more of a walkthrough, not really meant to be read. The full article is in the description, hope that helps.
Thanks so much! Very helpful.
@ciminian Thanks, I'd like to make videos for some of the more visual concepts.
I'm glad you think you know what you are talking about.
Exactly what I needed! Thanks
You seem to have a very strong understanding of the topic, what if the anchor to x and y were to move in the time axis and falling into a gravity well?
Thanks for your prompt reply Kalid. It's a pity it doesn't run on Windows and not exactly free, otherwise I might have tried it on VirtualBox.
Great breakdown!! For some reason...i see centripetal force in higher dimensional Spacetime spiraling inward. How would you correlate euler's formula with the fine- structure constant???
so if the theta gives us the amount of rotation and i gives us the intrest in the index then why doesnt rotation start from the y axis as value of i is plotted on the y axis
It's easiest to think of e^ix as 1.0 * e^ix. You start at 1.0, and modify that position using e [continuous change], i [rotation], and x [amount of time]. i influences the type change will occur, but not the starting point.
If you wanted to start at i and rotate from there, you'd use i * e^ix.
i didn't understand immediately. but i do now! thaaanks!!!
hi--i really liked your book--but if feel like this is very non-intuitive. i think its much easier to set it up by showing how eulers formula appears in converting to the polar form of complex numbers. So e^i is intrinsically rotational. The growth in a circle is interesting--but obscures e as a rotational and oscillatory operator. you should make additional videos from your book--you have great perspective
man this is really helpful!
This helped me so much - thank you!!
Thank you young sir.
Kalid, How did you make this video and what software did you use? I like the way you put yourself in the corner, and at the same time explain with the cursor on the screen--just like in a lecture room!
wow. this is great stuff.
Ah! One thing I just realized--at first I was asking, why don't we accelerate as we go around the circle? The answer is we ARE accelerating--we are CHANGING our velocity. Just like the Moon around the Earth is constantly "accelerating" as it changes direction.
It's interesting that that acceleration doesn't accumulate in the velocity the way linear acceleration does. It's very hard to wrap your head around the idea that going 1, 2, 4, 8 is the "same" acceleration as 0, pi/4, pi/2, 3pi/4, pi. That is VERY hard to grasp.
Euler's other exp-trig identity and RH, see "Riemann hypothesis with J.S.Bach".
You need to like Bach, quite. Includes some items from recent travels
Okay, it finally fully sank in. I was still having trouble with i^i, because I thought we were just "starting" at i, and then rotating from there, rotating at the rate of i. But that's not what we're talking about.
The entire entity e^x is the _growth_rate_. It has nothing to with, say, the "principal" we start with. So what "e^x" represents is the direction of the acceleration. If x is real, then e^x is real, so we accelerate along the real numbers. "e" is in a sense to "rate" of change, and the exponent is the "direction" of change.
So, for "e^i," the "rate" is still natural, that is, e, but the "direction" of change is now straight up. So again, with i^i, that doesn't mean "start" at 0+i and rotate left. We're still "starting" at 1. When we consider i^i we're still looking at the _vector_ of the change--it's magnitude and direction. If the base component is real, then the imaginary component is straight up. But in i^i, the "rate" of change is i, and then the "direction" is i again, so the vector of the change is straight backward along the real axis.
SO, if you had not purely imaginary numbers, but complex numbers, e^(ipi/4)^i, the direction of change would be up and back, 3pi/4 from the x axis.
The point, again, is that when looking at y^x in complex numbers, that whole thing is about about the magnitude and direction of change--we haven't even started counting yet, we're just looking at what direction we're going to push the growth.
I think that actually, despite the fact we're thinking in polar coordinates in some way here, there are two "components" to the growth vector that you add together. The base and the exponent.
Yes, great insight. It's best to think of e^x as shorthand for "1.0 * e^x" and i^i as "1.0 * i^i".
1.0 is always our implicit starting point: we are going to modify the unit quantity in some way (at some rate, for some amount of time). If we truly wanted to start at (0 + i), we would do "i * e^x" or "i * i^i".
Buy his book it's awesome
@consultjohan Thanks for the support! :)
Simply awesome.
i love this guy
I have a question, how can one define "i" as e^(i*pi/2). I didn't think you could define something with that something. Or, correct me if I'm wrong. Good video.
@dwightivany Wow, thank you for that message! It's so encouraging to hear that it's helpful -- I hope to make many more like this.
You are brilliant!!! Thank you
OK does this mean the Euler's formula is about finding a position in space?
Extrordinary.. superb..
excellent. Thanks very much.
@pilioff You're welcome :)
Hey Kalid, would be up to caption or transcribe this video? I can sync the captions if you provide a transcript :)
great video!
wow u deserve more subs
Need more volume in your videos, very hard to hear your insights.
Bravo! Thank you!
@PhilosophyAnimation Thanks, I appreciate it!
Can you answer this for me:
6÷2(1+2)
9?
thank you thank you thank you!!!
thanks, very helpful
Great job, thanks. :-)
@TheCancunBaby Thank you!
@MonicaKn17 Thanks!
amazing
thank you for this ... :)
@resal81 Thanks!
ok 10 more times and I think I'll get it
this'll be good for your ratings :p
hi guys. i can easily understand Euler's Identity. However, how to i convince a female to have sexual intercourse with me?
can the process be explained mathematically?
i have already scanned the math and physics forums as well as the world of warcraft ones.
convincing females? that's like breathing. I am pretty sure he doesn't even need to consciously do that, it just happens when he works as a programming manager for Microsoft, i.e. earning enough in one year for 10 year of your effort.
you can make fun of nerds all you want, you can never be one however hard you try.
Havent you herd?? E=MCVagina
I've got a wonderful idea. Why not write this up with excellent graphics, then page through it too fast for anyone to read it.
Cooooooool!!!
PS Thank you for this really great post.
Hurts my eyes to see improper LaTeX. It's \cos, \sin, \ln.
happy math
ROR
@salimk56 Glad it was helpful
He looks like indian Ray William Johnson haha
Is Ray William J. not Indian???
wat .. ill take your word for it lol
A lot of people seem to think this is amazingly explained. I must be honest, it doesn't work for me. I understand maths but I don't like this. Sorry.
Jesus christ on a bike