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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]

#81

Post by espeed »

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…

Way cool -- I realized that when I checked out your profile just after posting the comment -- what timing -- I discovered the video a few days ago when looking for precise/compact interval representations. Interesting work indeed.

Has there been much traction for getting major chip manufacturers to implement this? I know they're all looking for the next big thing and Intel is working on specialized neuromorphic chips. A general "drop-in" replacement for floating point seems like an opportunity for a general-purpose win from low-hanging fruit:

Intel Gets Serious About Neuromorphic, Cognitive Computing Future https://news.ycombinator.com/item?id=13623846

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

#82

Post by johnfn »

pyromine wrote: post #36
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.

I think there is both a good and bad aspect of this quote, you've done a great job highlighting the bad. The good, is in essence that you don't always need to be fully comfortable and in fact will not always be comfortable. I find in mathematics if you try to always strive for total comfort you will never progress, as a lot of the comfort comes from advancing past a topic and upon revisiting it you realize you unders…

Agreed, but isn't that irrelevant here? e^i*pi has an intuitive meaning, so we shouldn't be content with our lack of understanding.

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

#83

Post by mitchtbaum »

e^τi = 1.

In words, this equation makes the following fundamental observation:

The complex exponential of the circle constant is unity.

Geometrically, multiplying by e^iθ corresponds to rotating a complex number by an angle θ in the complex plane, which suggests a second interpretation of Euler’s identity:

A rotation by one turn is 1.

The Tau Manifesto http://tauday.com/tau-manifesto#sec-euler_s_identity

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

#84

Post by amelius »

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.

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

#86

Post by wodenokoto »

mitchtbaum wrote: post #83

e^τi = 1. In words, this equation makes the following fundamental observation: The complex exponential of the circle constant is unity. Geometrically, multiplying by e^iθ corresponds to rotating a complex number by an angle θ in the complex plane, which suggests a second interpretation of Euler’s identity: A rotation by one turn is 1. The Tau Manifesto http://tauday.com/tau-manifesto#sec-euler_s_identity

Thanks for posting this. I recently saw another video about this equation than the one posted here and thought "maybe this is one of those cases where tau doesn't work out nicely", but didn't have the intuition to see if it was or not.

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

#87

Post by mstade »

timlod wrote: post #69

I love the Khan Academy video on this: https://www.youtube.com/watch?v=mgNtPOgFje0 Sal's enthusiasm at the end is contagious!

I've learned so much from Khan Academy – not just whatever is explained by the video I'm watching, but different ways of educating people as well. In fact, I helped out with tutoring basic math a while back and I struggled at first because I didn't realize how much I wasn't teaching simply because it was second nature to me. After watching a bunch of Khan Academy videos on the same topics, I found different ways of explaining things, and was much more successful. Khan Academy has really challenged my own way of thinking about many things, math in particular.

It's a great resource, and while I don't use Khan Academy much these days I still donate $100 every year.

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

#88

Post by mstade »

gydfi wrote: post #9
hprotagonist wrote: post #4

This seemed like voodoo until i took signals and systems. Then it's just a normal observation about vector sums on the complex unit circle.

It seems like voodoo when you have high-school mathematics, because you've been taught about "to the power of" in terms of multiplying something by itself a certain number of times, and multiplying something by itself an imaginary number of times is nonsensical. Once you've done enough maths you think of it in a different way and e^ipi seems obvious. It's just a different conception of what the symbols mean, I suppos…

There are so many assumption that were hammered into me in high school (and below) that I find hold back my thinking, rather than help me. I kind of feel like the education I received was full of hacks to get me past whatever test was coming up, rather than truly teaching me. I've spent years playing what feels like catch up to actually truly learn the things I thought I knew – a lot of high school mathematics included.

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

#89

Post by hueving »

AndrewOMartin wrote: post #80
hueving wrote: post #46

Earlier quoted context omitted.

Not quite, you can get used to something without understanding it at all. Many people that fly are used to it but they don't understand the basic physics of flight at all.

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".

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

#90

Post by akvadrako »

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.

So what don't you get? Are you implying there is something more to complex numbers?
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