An interesting motivation for Euler's formula is known as the _moving particle argument_ which is a geometric interpretation of what Euler's formula actually means. If we want to explore what eⁱᵗ (where t is assumed to be real) could actually mean without first defining eˣ for a nonreal x we could start with z(t) = eⁱᵗ which maps the real number line onto the complex plane, at least if we can assign a unique complex value to eⁱᵗ for any real t. If we consider the real variable t to represent _time_ and keep in mind that for any real value t a complex number z(t) = x(t) + i·y(t) is represented by the _point_ with coordinates (x(t), y(t)) in the complex plane, then we can see that z(t) = eⁱᵗ describes the position of a point in the complex plane at any time t, which means that this equation can be interpreted as describing the _trajectory_ of a moving point in the complex plane. Now, the next question obviously is _what_ trajectory this equation could represent. The complex number z(t) gives the position of the point at any time t, but if this point changes with changing values of t then we have a moving point which therefore moves with a certain speed at any time t. The horizontal and vertical positions of the moving point at any time t are given by x(t) and y(t) so z(t) = x(t) + i·y(t) can be seen as the _place vector_ of the moving point at time t. Likewise, the _speed_ at which the point moves at any time t is given by the derivatives x'(t) and y'(t) where the first represents the horizontal and the second the vertical velocity at any time t. Note that any movement of a point in the plane is characterized by both _speed_ and _direction_ at any given time t and can therefore be analyzed as a combination of both horizontal and vertical movement at the same time. This means the the derivative z'(t) = x'(t) + i·y'(t) can be seen as a _velocity vector_ which fully characterizes both the direction of the movement and the speed at which the point moves at any time t. So, to get an idea of the trajectory of our moving point in the complex plane, we should investigate the derivative z'(t) = x'(t) + i·y'(t). But how do we find the derivative of z(t) = eⁱᵗ if we don't yet know what eⁱᵗ represents? Well, if we are to assign any meaningful interpretation to eⁱᵗ we should at least start by assuming that the same rules that apply to differentiating f(t) = eᵃᵗ for any real constant a should also apply to eⁱᵗ because i is also a constant. And since for f(t) = eᵃᵗ we have f'(t) = a·eᵃᵗ = a·f(t) this implies that we may expect to have z'(t) = i·eⁱᵗ which we can also write as z'(t) = i·z(t) So, the derivative z'(t) which represents the velocity vector of the moving point in the complex plane at any time t is equal to the place vector z(t) at that same time multiplied by i. Now, multiplication by i has a very simple but interesting geometric representation in the complex plane. If we take any complex number z = x + iy which is represented by the point (x, y) in the complex plane and multiply that number by i we get iz = i(x + iy) = ix + i²y = ix − y = −y + ix which is a number represented by the point (−y, x) in the complex plane. We can easily see that if we rotate the coordinate axes _counterclockwise_ over a right angle around the origin, then the positive real axis ends up on the original positive imaginary axis and the positive imaginary axis ends up on the original negative real axis. Consequently, if we rotate the point (x, y) representing the complex number z = x + iy counterclockwise over a right angle around the origin, then x will be the new vertical position of the point and −y will be the new horizontal position of the point, that is, the new coordinates of the rotated point are then (−y, x) which is the point representing the complex number iz = i(x + iy) = −y + ix. So, _multiplication by i corresponds to a counterclockwise rotation over a right angle around the origin_ in the complex plane. Returning now to our equation z'(t) = i·z(t) which describes the trajectory of our moving point in the complex plane we can see that this means that at any time the velocity vector z'(t) and, therefore, the direction in which our point moves in the complex plane, is _perpendicular_ to the place vector z(t). Now, the direction in which a point moves along a trajectory is simply the direction of the tangent at the point under consideration on the trajectory. So, we are looking at a trajectory of our moving point where the tangent on any point along the trajectory is perpendicular to the line segment from the origin to that point. But this can only be a _circle_ centered at the origin, because a tangent to a circle is always perpendicular to the radius from the centre of the circle to the point of tangency. So, we have now established that z(t) = eⁱᵗ maps the real number line onto a circle centered at the origin in the complex plane, but what more can we tell about this circle? Well, first of all, if we take t = 0 we have z(0) = e⁰ = 1 so the trajectory of our moving point passes through the point (1, 0) in the complex plane which is the point representing the number 1 + 0i = 1. So, the trajectory of the moving point is the _unit circle_ which is the circle centered at the origin with radius one. Also, since the velocity vector z'(t) is rotated counterclockwise over a right angle relative to the place vector z(t) we can tell that our moving point traverses the unit circle counterclockwise. But what about the speed at which our moving point traverses the unit circle? Well, since z(t) is a complex number represented by a point on the unit circle for any real value of t we have |z(t)| = 1 for any real t. But since z'(t) = i·z(t) this means that we also have |z'(t)| = |i·z(t)| = |i|·|z(t)| = 1·1 = 1. So, the magnitude or absolute value |z'(t)| of the velocity vector is always 1, which means that our moving point travels around the unit circle at a uniform speed of 1 unit per unit of time. Since the moving point passes the point (1, 0) on the unit circle at time t = 0 and since the circumference of the unit circle is 2π and since the point traverses the unit circle at a uniform speed 1 this means that the moving point passes the point (1, 0) at any time t = k·2π = 2kπ for all integer values of k. By now, this should all start to sound very familiar if you remember the unit circle definitions of the sine and cosine. In fact, if we have a point that traverses the unit circle counterclockwise at unit speed and which passes the point (1, 0) on the unit circle at time t = 0 then the unit circle definitions of the sine and cosine imply that we have x(t) = cos t and y(t) = sin t. Since the point (x(t), y(t)) represents the complex number z(t) = x(t) + i·y(t) this means that we have z(t) = cos t + i·sin t and since we started with z(t) = eⁱᵗ we can now conclude that for any real value of t we have eⁱᵗ = cos t + i·sin t which is of course Euler's formula.
This here is absolutely beautiful. I have never heard of the moving particle argument, but you have explained it incredibly well and it makes a lot of sense. I have been inspired to perhaps even make a video that shows this identity using this argument, but no promises yet. However, thank you so much for sharing this and I’ll pin this comment so that others can learn about this method as well.
Few corrections: At 1:25, there is a negative sign which should be a plus sign. The term x^6 / 6! should be -x^6/6! at 1:27 and 1:29. At 2:01, the labels should be 2.5cos(alpha) and 2.5isine(alpha), this is because we are scaling by the magnitude 2.5. At 2:16, I say all the possible values of e^ (i pi), but i meant all the possible values of e ^ (ix)- e^ (i pi) lies on the point -1.
Good one. I always like to see the equation with 1 added to both sides of the representation here. That way, zero appears on the right side. There's something nice about having both the additive and multiplicative identity elements for the real numbers present along side the important numbers i, e, and pi. Thanks for sharing the video!
You should see the version with τ (tau). τ is the ratio of a circle's circumference to its radius, equal to the circumference of the unit circle and the number of radians in a full revolution, or one turn. It is about 6.28 and equal to 2π. τ is very nice for working with radians: 1/2 turn is τ/2 radians, 1/12 turn is τ/12 radians, and so on. In turn, this makes it nice for working with trigonometric functions. Taking Euler's formula, e^(iθ) = cos θ + i sin θ, and substituting θ = τ gives: e^(iτ) = cos τ + i sin τ Going τ units around the unit circle is a full revolution, meaning you land back at the point (1, 0). This means that cos τ = 1, and sin τ = 0, so: e^(iτ) = 1 + 0i Note that 1 + 0i is a complex number written in a + bi form. So, this version relates the important numbers WITHOUT rearrangement.
You raise a good point. This equation is definitely also written in the format that you have just mentioned. In fact, I think that is the more common way of writing this identity. That way we have 5 of the arguably most important constants in the same equation. And of course, my pleasure for sharing the video!
@@isavenewspapers8890although I did not mention it directly, if you look at the penultimate animation, the spiral that is formed is very much a hint at what you just said. Tau is nowhere near as common as pi, therefore I didn’t make too much of a fuss about it-but you are definitely right, and e^ ( I times tau) = 1.
@@MathsWithMuza It's precisely because τ is relatively obscure that I would've liked to see it mentioned in this video. I genuinely believe in the good that τ can do for mathematical education, so it's nice when people help get the word out.
Corrections: 0:43 Euler did not discover the number e. The first known explicit reference to the number e was by Jacob Bernoulli in 1683, before Euler was even born. 1:22 The way this is written implies that x^n / n! is the last term, implying that there *is* a last term, which is false for a series. It probably would have been better to add on another plus sign and ellipsis at the end. 1:25 The minus sign should be a plus sign. 1:27 The term x^6 / 6! should actually be -x^6 / 6!. The same issue reoccurs in the next equation. 1:29 Why does the cosine series just cut off? I guess the remaining terms could be contained in the ellipsis at the end, but that seems like a confusing way of writing it. I would suggest adding an ellipsis after the cosine terms to indicate that it's a series, then enclosing the whole series in parentheses, like so: e^(ix) = (1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...) + (i * x - i * x^3 / 3! + i * x^5 / 5! - i * x^7 / 7! + ...) By the way, this isn't an error, but why did you write out 4 terms of cosine and only 3 of sine? I guess 4 each would've taken up a bit too much space, but you can just do 3 each. 2:01 You have to scale these by the magnitude of the complex number, so the horizontal is actually 2.5 cos(α), and the vertical is 2.5i sin(α). 2:12 A whole new algebraic structure? Are you referring to the complex numbers and the operations on them? I thought this video already assumed prior knowledge of complex numbers, so I don't know why you'd call them new. 2:16 You mean all the possible values of e^(ix), for real-valued x. There is only one possible value of e^(iπ), and it lies on the point -1. 2:34 ix, not i. 2:39 "In other words" seems to imply that this is a restatement of the previous sentence, even though it's actually a different statement. 3:00 Surprise is subjective, but I don't see what's surprising about the fact that we got the exact same result as before. 3:04 I'd argue that this is just as much of a proof as the Taylor series argument. Both can be naturally adapted into a rigorous mathematical proof.
Thanks for your comment! Let me address your concerns one at a time: Never said Euler discovered the constant e- it’s just a gift from him in the sense that it is named after him. Even if he didn’t discover it, he’s the one who made it famous. You raise a good point and I probably should’ve added ellipses to show that this series is never ending. I’ll keep that in mind for next time. Ahh yes you are right it should be a plus sign. That’s my mistake. Good catch again. It should be negative x^6/6!. I could’ve done 3 each yes. That would’ve worked too. But I wanted to do 4. The cosine series cuts off but then I bring it back in summation notation to show that it’s a series. You are right. It’s 2.5 cos alpha and 2.5 I sine alpha. Good catch. It doesn’t assume a lot of knowledge about complex numbers. For some people this might be an introduction to them and they might just watch it for fun, while others watch it to learn more about this identity. So it’s a new algebraic structure for some people. Ohh yeah a mistake a in the scripting. I meant e^(ix) but e^ (i pi) lies on -1. It does say ix. The i is highlighted in blue. Surprising is indeed subjective. That’s why for some people it’s surprising if they see it for the first time. Of course at the end of the day they are both beautiful proofs of this identity. One is more mathematically rigorous in the sense that it is algebra based, while the other takes on a more visual approach. Beauty is in the eye of the beholder. Thanks for the corrections and your input. Very much appreciated. I’ll add the important corrections as a comment at the top of all comments!
@@MathsWithMuza Thank you. I have further thoughts to share regarding your response. 1) Your stance on the number e feels super weird to me. Like, so what if Euler is the one who made the constant popularly known? If people put up advertising for a TV show, do we have them to thank for gifting us the TV show? I dunno, it's just a strange line to me. It genuinely feels like that was not your intended meaning when you first put the line in the video. 2) My point is that you didn't do 4 terms each; you only did the first three terms for the cosine series. Also, just because you have a good representation of the series later in the video, that doesn't make your earlier confusing representation okay. 3) Obviously, the video can be very educational in terms of teaching the viewer about complex numbers. I get that. What I'm saying is that, while this is a lot of new information about this particular algebraic structure, it's not a new algebraic structure altogether. Even if the viewer only possessed elementary knowledge about it beforehand, they still did know about it. 4) The visual is correct, but your speech is wrong. You said, "And the question is, what happens when we plug i in place of x?" But you're not changing e^x to e^i; you're changing it to e^(ix). 5) I pointed out how strange it is to call it surprising specifically because this ISN'T the first time we've seen it. It's the exact same identity we saw earlier in the video. 6) Regarding the second proof, I'm not talking about the approach of, "Look at the visuals and see what feels right," because you are correct that that is not a proof. But the limit you used is a well-defined mathematical object, and we can use valid mathematical logic on it in order to deduce its value.
@@isavenewspapers8890 1) "His GIFT to us is the constant e, regarded as the second most beautiful constant...". I did not say he discovered it. If that is the impression that you got then that is a misunderstanding, and I could have been more clear. Also, euler has given us so much that we name things after the second person who made them famous after him, which is slight injustice to him for our own good. 2) That's all good. I have mentioned that Taylor series gives us INFINITE sums of simpler terms before anything else, so maybe ellipses would've helped out but you should not be too worried. 3) Some people might have never heard of the term complex numbers before, and only clicked because the thumbnail looked interesting. You never really know. Yes, it isn't a 'new' algebraic structure in the sense that this has been discovered. 4)Ahh okay I see. Yes, it should be ix in place of x. I see what you mean thank you for correcting it! 5)Debatable. 6 )I see your point. Fair enough. Thanks for your comments and feedback. This is what makes the channel grow!
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An interesting motivation for Euler's formula is known as the _moving particle argument_ which is a geometric interpretation of what Euler's formula actually means. If we want to explore what eⁱᵗ (where t is assumed to be real) could actually mean without first defining eˣ for a nonreal x we could start with
z(t) = eⁱᵗ
which maps the real number line onto the complex plane, at least if we can assign a unique complex value to eⁱᵗ for any real t. If we consider the real variable t to represent _time_ and keep in mind that for any real value t a complex number
z(t) = x(t) + i·y(t)
is represented by the _point_ with coordinates (x(t), y(t)) in the complex plane, then we can see that z(t) = eⁱᵗ describes the position of a point in the complex plane at any time t, which means that this equation can be interpreted as describing the _trajectory_ of a moving point in the complex plane.
Now, the next question obviously is _what_ trajectory this equation could represent. The complex number z(t) gives the position of the point at any time t, but if this point changes with changing values of t then we have a moving point which therefore moves with a certain speed at any time t.
The horizontal and vertical positions of the moving point at any time t are given by x(t) and y(t) so z(t) = x(t) + i·y(t) can be seen as the _place vector_ of the moving point at time t.
Likewise, the _speed_ at which the point moves at any time t is given by the derivatives x'(t) and y'(t) where the first represents the horizontal and the second the vertical velocity at any time t. Note that any movement of a point in the plane is characterized by both _speed_ and _direction_ at any given time t and can therefore be analyzed as a combination of both horizontal and vertical movement at the same time. This means the the derivative z'(t) = x'(t) + i·y'(t) can be seen as a _velocity vector_ which fully characterizes both the direction of the movement and the speed at which the point moves at any time t.
So, to get an idea of the trajectory of our moving point in the complex plane, we should investigate the derivative z'(t) = x'(t) + i·y'(t). But how do we find the derivative of z(t) = eⁱᵗ if we don't yet know what eⁱᵗ represents? Well, if we are to assign any meaningful interpretation to eⁱᵗ we should at least start by assuming that the same rules that apply to differentiating f(t) = eᵃᵗ for any real constant a should also apply to eⁱᵗ because i is also a constant. And since for f(t) = eᵃᵗ we have f'(t) = a·eᵃᵗ = a·f(t) this implies that we may expect to have
z'(t) = i·eⁱᵗ
which we can also write as
z'(t) = i·z(t)
So, the derivative z'(t) which represents the velocity vector of the moving point in the complex plane at any time t is equal to the place vector z(t) at that same time multiplied by i.
Now, multiplication by i has a very simple but interesting geometric representation in the complex plane. If we take any complex number z = x + iy which is represented by the point (x, y) in the complex plane and multiply that number by i we get iz = i(x + iy) = ix + i²y = ix − y = −y + ix which is a number represented by the point (−y, x) in the complex plane.
We can easily see that if we rotate the coordinate axes _counterclockwise_ over a right angle around the origin, then the positive real axis ends up on the original positive imaginary axis and the positive imaginary axis ends up on the original negative real axis. Consequently, if we rotate the point (x, y) representing the complex number z = x + iy counterclockwise over a right angle around the origin, then x will be the new vertical position of the point and −y will be the new horizontal position of the point, that is, the new coordinates of the rotated point are then (−y, x) which is the point representing the complex number iz = i(x + iy) = −y + ix. So, _multiplication by i corresponds to a counterclockwise rotation over a right angle around the origin_ in the complex plane.
Returning now to our equation
z'(t) = i·z(t)
which describes the trajectory of our moving point in the complex plane we can see that this means that at any time the velocity vector z'(t) and, therefore, the direction in which our point moves in the complex plane, is _perpendicular_ to the place vector z(t). Now, the direction in which a point moves along a trajectory is simply the direction of the tangent at the point under consideration on the trajectory.
So, we are looking at a trajectory of our moving point where the tangent on any point along the trajectory is perpendicular to the line segment from the origin to that point. But this can only be a _circle_ centered at the origin, because a tangent to a circle is always perpendicular to the radius from the centre of the circle to the point of tangency.
So, we have now established that z(t) = eⁱᵗ maps the real number line onto a circle centered at the origin in the complex plane, but what more can we tell about this circle? Well, first of all, if we take t = 0 we have z(0) = e⁰ = 1 so the trajectory of our moving point passes through the point (1, 0) in the complex plane which is the point representing the number 1 + 0i = 1. So, the trajectory of the moving point is the _unit circle_ which is the circle centered at the origin with radius one. Also, since the velocity vector z'(t) is rotated counterclockwise over a right angle relative to the place vector z(t) we can tell that our moving point traverses the unit circle counterclockwise.
But what about the speed at which our moving point traverses the unit circle? Well, since z(t) is a complex number represented by a point on the unit circle for any real value of t we have |z(t)| = 1 for any real t. But since z'(t) = i·z(t) this means that we also have |z'(t)| = |i·z(t)| = |i|·|z(t)| = 1·1 = 1. So, the magnitude or absolute value |z'(t)| of the velocity vector is always 1, which means that our moving point travels around the unit circle at a uniform speed of 1 unit per unit of time. Since the moving point passes the point (1, 0) on the unit circle at time t = 0 and since the circumference of the unit circle is 2π and since the point traverses the unit circle at a uniform speed 1 this means that the moving point passes the point (1, 0) at any time t = k·2π = 2kπ for all integer values of k.
By now, this should all start to sound very familiar if you remember the unit circle definitions of the sine and cosine. In fact, if we have a point that traverses the unit circle counterclockwise at unit speed and which passes the point (1, 0) on the unit circle at time t = 0 then the unit circle definitions of the sine and cosine imply that we have x(t) = cos t and y(t) = sin t. Since the point (x(t), y(t)) represents the complex number z(t) = x(t) + i·y(t) this means that we have
z(t) = cos t + i·sin t
and since we started with
z(t) = eⁱᵗ
we can now conclude that for any real value of t we have
eⁱᵗ = cos t + i·sin t
which is of course Euler's formula.
This here is absolutely beautiful. I have never heard of the moving particle argument, but you have explained it incredibly well and it makes a lot of sense. I have been inspired to perhaps even make a video that shows this identity using this argument, but no promises yet. However, thank you so much for sharing this and I’ll pin this comment so that others can learn about this method as well.
Yep, 3Blue1Brown has a great video on the topic.
Few corrections:
At 1:25, there is a negative sign which should be a plus sign.
The term x^6 / 6! should be -x^6/6! at 1:27 and 1:29.
At 2:01, the labels should be 2.5cos(alpha) and 2.5isine(alpha), this is because we are scaling by the magnitude 2.5.
At 2:16, I say all the possible values of e^ (i pi), but i meant all the possible values of e ^ (ix)- e^ (i pi) lies on the point -1.
@@MathsWithMuza Nice.
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Good one. I always like to see the equation with 1 added to both sides of the representation here. That way, zero appears on the right side. There's something nice about having both the additive and multiplicative identity elements for the real numbers present along side the important numbers i, e, and pi. Thanks for sharing the video!
You should see the version with τ (tau). τ is the ratio of a circle's circumference to its radius, equal to the circumference of the unit circle and the number of radians in a full revolution, or one turn. It is about 6.28 and equal to 2π.
τ is very nice for working with radians: 1/2 turn is τ/2 radians, 1/12 turn is τ/12 radians, and so on. In turn, this makes it nice for working with trigonometric functions. Taking Euler's formula, e^(iθ) = cos θ + i sin θ, and substituting θ = τ gives:
e^(iτ) = cos τ + i sin τ
Going τ units around the unit circle is a full revolution, meaning you land back at the point (1, 0). This means that cos τ = 1, and sin τ = 0, so:
e^(iτ) = 1 + 0i
Note that 1 + 0i is a complex number written in a + bi form. So, this version relates the important numbers WITHOUT rearrangement.
You raise a good point. This equation is definitely also written in the format that you have just mentioned. In fact, I think that is the more common way of writing this identity. That way we have 5 of the arguably most important constants in the same equation. And of course, my pleasure for sharing the video!
@@isavenewspapers8890although I did not mention it directly, if you look at the penultimate animation, the spiral that is formed is very much a hint at what you just said. Tau is nowhere near as common as pi, therefore I didn’t make too much of a fuss about it-but you are definitely right, and e^ ( I times tau) = 1.
@@MathsWithMuza It's precisely because τ is relatively obscure that I would've liked to see it mentioned in this video. I genuinely believe in the good that τ can do for mathematical education, so it's nice when people help get the word out.
@@isavenewspapers8890ahhh i see. Yes, we should get the word out. Maybe a short video on pi vs tau will come out soon
Corrections:
0:43 Euler did not discover the number e. The first known explicit reference to the number e was by Jacob Bernoulli in 1683, before Euler was even born.
1:22 The way this is written implies that x^n / n! is the last term, implying that there *is* a last term, which is false for a series. It probably would have been better to add on another plus sign and ellipsis at the end.
1:25 The minus sign should be a plus sign.
1:27 The term x^6 / 6! should actually be -x^6 / 6!. The same issue reoccurs in the next equation.
1:29 Why does the cosine series just cut off? I guess the remaining terms could be contained in the ellipsis at the end, but that seems like a confusing way of writing it. I would suggest adding an ellipsis after the cosine terms to indicate that it's a series, then enclosing the whole series in parentheses, like so:
e^(ix) = (1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...) + (i * x - i * x^3 / 3! + i * x^5 / 5! - i * x^7 / 7! + ...)
By the way, this isn't an error, but why did you write out 4 terms of cosine and only 3 of sine? I guess 4 each would've taken up a bit too much space, but you can just do 3 each.
2:01 You have to scale these by the magnitude of the complex number, so the horizontal is actually 2.5 cos(α), and the vertical is 2.5i sin(α).
2:12 A whole new algebraic structure? Are you referring to the complex numbers and the operations on them? I thought this video already assumed prior knowledge of complex numbers, so I don't know why you'd call them new.
2:16 You mean all the possible values of e^(ix), for real-valued x. There is only one possible value of e^(iπ), and it lies on the point -1.
2:34 ix, not i.
2:39 "In other words" seems to imply that this is a restatement of the previous sentence, even though it's actually a different statement.
3:00 Surprise is subjective, but I don't see what's surprising about the fact that we got the exact same result as before.
3:04 I'd argue that this is just as much of a proof as the Taylor series argument. Both can be naturally adapted into a rigorous mathematical proof.
Thanks for your comment! Let me address your concerns one at a time:
Never said Euler discovered the constant e- it’s just a gift from him in the sense that it is named after him. Even if he didn’t discover it, he’s the one who made it famous.
You raise a good point and I probably should’ve added ellipses to show that this series is never ending. I’ll keep that in mind for next time.
Ahh yes you are right it should be a plus sign. That’s my mistake. Good catch again. It should be negative x^6/6!.
I could’ve done 3 each yes. That would’ve worked too. But I wanted to do 4. The cosine series cuts off but then I bring it back in summation notation to show that it’s a series.
You are right. It’s 2.5 cos alpha and 2.5 I sine alpha. Good catch.
It doesn’t assume a lot of knowledge about complex numbers. For some people this might be an introduction to them and they might just watch it for fun, while others watch it to learn more about this identity. So it’s a new algebraic structure for some people.
Ohh yeah a mistake a in the scripting. I meant e^(ix) but e^ (i pi) lies on -1.
It does say ix. The i is highlighted in blue.
Surprising is indeed subjective. That’s why for some people it’s surprising if they see it for the first time.
Of course at the end of the day they are both beautiful proofs of this identity. One is more mathematically rigorous in the sense that it is algebra based, while the other takes on a more visual approach. Beauty is in the eye of the beholder.
Thanks for the corrections and your input. Very much appreciated. I’ll add the important corrections as a comment at the top of all comments!
@@MathsWithMuza Thank you. I have further thoughts to share regarding your response.
1) Your stance on the number e feels super weird to me. Like, so what if Euler is the one who made the constant popularly known? If people put up advertising for a TV show, do we have them to thank for gifting us the TV show? I dunno, it's just a strange line to me. It genuinely feels like that was not your intended meaning when you first put the line in the video.
2) My point is that you didn't do 4 terms each; you only did the first three terms for the cosine series. Also, just because you have a good representation of the series later in the video, that doesn't make your earlier confusing representation okay.
3) Obviously, the video can be very educational in terms of teaching the viewer about complex numbers. I get that. What I'm saying is that, while this is a lot of new information about this particular algebraic structure, it's not a new algebraic structure altogether. Even if the viewer only possessed elementary knowledge about it beforehand, they still did know about it.
4) The visual is correct, but your speech is wrong. You said, "And the question is, what happens when we plug i in place of x?" But you're not changing e^x to e^i; you're changing it to e^(ix).
5) I pointed out how strange it is to call it surprising specifically because this ISN'T the first time we've seen it. It's the exact same identity we saw earlier in the video.
6) Regarding the second proof, I'm not talking about the approach of, "Look at the visuals and see what feels right," because you are correct that that is not a proof. But the limit you used is a well-defined mathematical object, and we can use valid mathematical logic on it in order to deduce its value.
@@isavenewspapers8890 1) "His GIFT to us is the constant e, regarded as the second most beautiful constant...". I did not say he discovered it. If that is the impression that you got then that is a misunderstanding, and I could have been more clear. Also, euler has given us so much that we name things after the second person who made them famous after him, which is slight injustice to him for our own good.
2) That's all good. I have mentioned that Taylor series gives us INFINITE sums of simpler terms before anything else, so maybe ellipses would've helped out but you should not be too worried.
3) Some people might have never heard of the term complex numbers before, and only clicked because the thumbnail looked interesting. You never really know. Yes, it isn't a 'new' algebraic structure in the sense that this has been discovered.
4)Ahh okay I see. Yes, it should be ix in place of x. I see what you mean thank you for correcting it!
5)Debatable.
6 )I see your point. Fair enough.
Thanks for your comments and feedback. This is what makes the channel grow!
@@MathsWithMuza "Debatable."
Then debate it.
Nothing much. Surprising for some people. And they have told me so. Maybe not surprising for others.
3:43 This would've been a great time to mention τ.
Good idea! Maybe a small reference would’ve been good for sure.