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Why does e to pi i equal -1? (2015) [video]

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Re: Why does e to pi i equal -1? (2015) [video]

#91

Post by jacobolus »

mtourne wrote: post #63
gizmo686 wrote: post #60

Earlier quoted context omitted.

>To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. No you don't. You might notice this by simply working with imaginary numbers. You might invent imaginary numbers for rotation [0]. Alternatively, consider the multiplication (x + yi)(a + bi), as the value (a + bi) performing a transformation on (x + yi). We want (x + yi)(a + bi) = xy - by + ayi + bxi. If we consider (x +…

Yay skew symmetric matrixes. It's also fun to introduce e as a matrix operator for 3d rotations. It's useful for kinematics and having compact representations for axis angle notations. It's a little far in my head but at some point it felt like a ha-ha moment with the Euler identity, in the "2d version".

In the three-dimensional case, for any practical purpose, use quaternions. https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

(At least if you’re ever needing to compose rotations; to apply a quaternion to a big array of 3-vectors, go ahead and convert it to a 3x3 matrix first, which should end up slightly more efficient.)

3x3 rotation matrices are really hard to keep normalized properly, whereas quaternions are trivial to normalize (just divide by the norm).

If you need to compress a unit quaternion down to 3 numbers, instead of taking the logarithm (which is computationally expensive and a pain to deal with) use the stereographic projection.

Re: Why does e to pi i equal -1? (2015) [video]

#93

Post by [deleted] »

amelius wrote: post #84
cousin_it wrote: post #3

1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.

I think you should explain step 2 in more detail.

[deleted]

Re: Why does e to pi i equal -1? (2015) [video]

#94

Post by [deleted] »

amelius wrote: post #84
cousin_it wrote: post #3

1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.

I think you should explain step 2 in more detail.

[deleted]

Re: Why does e to pi i equal -1? (2015) [video]

#95

Post by AndrewOMartin »

hueving wrote: post #89
AndrewOMartin wrote: post #80

Earlier quoted context omitted.

And who would you prefer in the pilot's seat? To say suggest that you don't understand something without a formal underlying theory is one view, but not mine. I feel like I understand English (as in a deep understanding, not just the ability to interpret sentences) without anything like a set of rules, and everything like a set of experiences akin to a pilot's experience with flight.

What does that have to do with anything I said? I'm pointing out the difference between "understanding" and "getting used to".

I'm refuting that difference.

Re: Why does e to pi i equal -1? (2015) [video]

#96

Post by sandyjames »

cousin_it wrote: post #3

1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.

I'm making over $7k a month working part time. I kept hearing other people tell me how much money they can make online so I decided to look into it. Well, it was all true and has totally changed my life. This is what I do... http://bit.ly/2jdTzrM

Re: Why does e to pi i equal -1? (2015) [video]

#97

Post by k__ »

godmodus wrote: post #48
kmill wrote: post #5

Earlier quoted context omitted.

In a linear algebra course I helped teach last semester, I had the students go through the exercise of writing down the matrix for "multiplication by i," where the complex numbers are thought of as a two-dimensional vector space with {1,i} as a basis. Then I asked them to recognize the matrix (it's a 90-degree rotation of the plane), and then I asked them for an interpretation of that matrix squared (a 180-degree rot…

I love this answer. For some reason, I've been dreaming about negative numbers lately. I think they deserve their own number set notation.

Speaking of number sets.

I never understood why complex numbers are considered one. I mean, yes they "are" a set, but besides that they are completely different.

All number sets I learned about did fill some gaps in one dimension, but complex numbers somehow added a new dimension.

Like real numbers stood in an entirely different context to rational numbers than complex numbers stood to real numbers.

Re: Why does e to pi i equal -1? (2015) [video]

#98

Post by taw1 »

dnautics wrote: post #73
espeed wrote: post #67

Earlier quoted context omitted.

Have you seen "Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic"? https://www.youtube.com/watch?v=aP0Y1uAA-2Y

Well, I'm the guy in the black shirt who did the demo. If you liked that lecture, I'm already starting on some verilog implementations. This is an example multiplication 8 bit * 8 bit -> 16 bit unpacked (20 bits). It differs from standard floating point in that the fractions are stored as two's complement. It takes a little bit of wrapping your head around, but the hidden bit for negative numbers is actually -2 ! Mom…

Posits look too good to be true! Is there a reason why regime bits do not include the sign bit? Then both "0 0001" and "1 1110" could be interpreted as 4 regime bits. Even better we could include the last flipped bit as well and we would have 5 bits.

Edit: Well. I see it would result in losing the values 0 and 1. Another question: Since it is fixed length of 4 bits (for N=32) why don't we just extract the 4 bit value, then we could represent 2^4 regimes this time without losing 0 and 1.

Re: Why does e to pi i equal -1? (2015) [video]

#99

Post by jameshart »

E6300 wrote: post #34
johnfn wrote: post #31

Earlier quoted context omitted.

I don't really like this, because it's a self limiting quote that leads to complacency over real understanding. Especially in the case of imaginary numbers and e where intuitive understanding only doesn't exist because mathematics has a history of being poorly taught.

Honestly, after spending months studying the subject, I don't think it's really possible to "get" complex numbers. I just view them as affine transformations written in an unusual notation. I don't think they make sense as anything but a recontextualization of R^2.

The problem with that is that complex numbers initially emerge as roots of polynomials with real coefficients. Getting to affine transformations from there seems a much bigger leap than asserting there is a square root of -1.
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