# 3Blue1Brown: Reminding the world that math makes sense

June 6, 2015 11:42 AM Subscribe

Understanding e to the pi i - "An intuitive explanation as to why e to the pi i equals -1 without a hint of calculus. This is not your usual Taylor series nonsense." (via via; reddit; previously)

More videos from 3Blue1Brown: "3Blue1Brown is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations, for difficult problems to be made simple with changes in perspective, and for philosophizing to be limited to the brevity and semantic constraints of silly poetry. Basically, math sits in an ivory tower it built itself out of jargon and impossibly long sequences of (seemingly) logical steps, and I would like to take it out for a walk to meet everyone."

More videos from 3Blue1Brown: "3Blue1Brown is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations, for difficult problems to be made simple with changes in perspective, and for philosophizing to be limited to the brevity and semantic constraints of silly poetry. Basically, math sits in an ivory tower it built itself out of jargon and impossibly long sequences of (seemingly) logical steps, and I would like to take it out for a walk to meet everyone."

The animations are quite pretty, but I'm not convinced that this video will be helpful for people that don't already know that Euler's equation is a rotation in the complex plane.

As for why multiplying by

posted by tickingclock at 1:27 PM on June 6, 2015 [11 favorites]

As for why multiplying by

*i*is equivalent to rotation in the complex plane, I like this link. tl;dr: Imagine a real number line stretching from -Inf to Inf. You have a point at +1. Define "multiplication by*i*, twice in a row" as an operation that takes +1 and puts it at -1. What operation does this? This a rotation by 180 degrees. So, "multiplication by*i*, once" is a rotation by 90 degrees.posted by tickingclock at 1:27 PM on June 6, 2015 [11 favorites]

A lovely video, and an interesting way to think of things. I particularly like the idea of "turning adders into multipliers." That's pretty neat.

But I'm not really convinced that it can be called more "intuitive" than the standard way of explaining things--there's an awful lot of stuff you have to absorb to get to the end, and it's not at all clear (to me anyway) that that's any easier.

The real breakdown of the "intuitive" claim is the point in the video (about 3:20) when he says that when one tries to define functions with the property

posted by mondo dentro at 1:28 PM on June 6, 2015 [2 favorites]

But I'm not really convinced that it can be called more "intuitive" than the standard way of explaining things--there's an awful lot of stuff you have to absorb to get to the end, and it's not at all clear (to me anyway) that that's any easier.

The real breakdown of the "intuitive" claim is the point in the video (about 3:20) when he says that when one tries to define functions with the property

*f*(*x*+*y*)=*f*(*x*)*f*(*y*), "one stands stands out as being the most natural, and we express it with this infinite sum." Um. OK. Will non-calculus people feel comfortable with infinite sums? He does show how to do it without calculus, but it's not likely to be something non-math-centric people will find "intuitive."posted by mondo dentro at 1:28 PM on June 6, 2015 [2 favorites]

"As for why multiplying by i is equivalent to rotation in the complex plane, I like this link. tl;dr: Imagine a real number line stretching from -Inf to Inf. You have a point at +1. Define "multiplication by i, twice in a row" as an operation that takes +1 and puts it at -1. What operation does this? This a rotation by 180 degrees. So, "multiplication by i, once" is a rotation by 90 degrees."

the video I just watched showed exactly that

posted by idiopath at 1:32 PM on June 6, 2015

the video I just watched showed exactly that

posted by idiopath at 1:32 PM on June 6, 2015

I guess you can count me among those who think that, yes, this seems an intuitive explanation but only because I already have a good grasp of calculus.

posted by The Great Big Mulp at 1:54 PM on June 6, 2015 [3 favorites]

posted by The Great Big Mulp at 1:54 PM on June 6, 2015 [3 favorites]

*the video I just watched showed exactly that*

I know, but I feel like it wasn't as explicit about it. Also, there was no reason for it to be a video given that the same information could be easily conveyed using static images. I found the animation to be distracting.

posted by tickingclock at 2:00 PM on June 6, 2015 [2 favorites]

But I sent the video link to my friend/former roommate who probably hasn't done this stuff since high school, and she really liked it, so I guess it works for some people. (shrug)

posted by tickingclock at 2:02 PM on June 6, 2015 [2 favorites]

posted by tickingclock at 2:02 PM on June 6, 2015 [2 favorites]

Different perspectives are always a good thing, because different explanations are useful to different people, but sometimes with these videos it seems like you typically substitute the traditional set of jargon with an alternate one and choose to obscure a different part of the issue.

Here, he seemed to gloss over why exp(i*pi) = -1, but not 5^(i*pi)

posted by rocketbadger at 2:02 PM on June 6, 2015 [1 favorite]

Here, he seemed to gloss over why exp(i*pi) = -1, but not 5^(i*pi)

posted by rocketbadger at 2:02 PM on June 6, 2015 [1 favorite]

You know what really made this click for me? I had a professor who called "e^(ix)" the "rotini function."

He drew a nice helical graph on the board the first week of class: Re[e^(ix)] on the horizontal axis, Im[e^(ix)] on the vertical axis, and "x" on the "tilted into the board" direction. Like a piece of rotini.

When I finally got how circles were connected to waves, and to phase (a nice illustration of the shadow cast by a clock hand against a perpendicular wall) it was another big breakthrough. These days I tend to think that the pretty much the whole point of complex numbers is to allow us to compare waves which have different phases using regular multiplication rules instead of complex trig identities. (At least, that's what I pretty much always use them for.) It's not that profound. It's mostly just a shorthand, a convenience.

posted by OnceUponATime at 2:23 PM on June 6, 2015 [4 favorites]

He drew a nice helical graph on the board the first week of class: Re[e^(ix)] on the horizontal axis, Im[e^(ix)] on the vertical axis, and "x" on the "tilted into the board" direction. Like a piece of rotini.

*This is the whole point of e^(ix)*. It's got this relationship to circles because that's what it was invented to do. Of course it was. It's the "rotini function."When I finally got how circles were connected to waves, and to phase (a nice illustration of the shadow cast by a clock hand against a perpendicular wall) it was another big breakthrough. These days I tend to think that the pretty much the whole point of complex numbers is to allow us to compare waves which have different phases using regular multiplication rules instead of complex trig identities. (At least, that's what I pretty much always use them for.) It's not that profound. It's mostly just a shorthand, a convenience.

posted by OnceUponATime at 2:23 PM on June 6, 2015 [4 favorites]

*there was no reason for it to be a video*

I've always found myself grasping math concepts more quickly when looking at an animation that showed the change from one step to the next as a meaningful movement. It fills in the 'how do you get there from here?' problem that I always had trying to understand the concepts in my math texts. I agree the end of this video kind of rushed to the conclusion, and I'd also have like a little more playing around with moving and stretching the complex grid just for funnies.

posted by Space Coyote at 2:39 PM on June 6, 2015 [1 favorite]

As soon as he rotated the axis, I lost my shit and exclaimed at the top of my lungs, "Holy shit!" That in and of its self isn't really anything new, but when you say that accidentally in front of a six year old and loud enough so that your wife pops her head out of your daughter's story time...

Eh, let's just say I've spent the past half hour trying to teach my son about the circumference of a circle, talking about number planes instead of just number lines, making him watch the multiplication portion of that and otherwise probably interrupting his coloring and writing for just some silly math stuff.

While common core is something I know a lot of people hate, I think that conversation went better than what will happen when we talk about puberty.

posted by Nanukthedog at 3:04 PM on June 6, 2015 [2 favorites]

Eh, let's just say I've spent the past half hour trying to teach my son about the circumference of a circle, talking about number planes instead of just number lines, making him watch the multiplication portion of that and otherwise probably interrupting his coloring and writing for just some silly math stuff.

While common core is something I know a lot of people hate, I think that conversation went better than what will happen when we talk about puberty.

posted by Nanukthedog at 3:04 PM on June 6, 2015 [2 favorites]

as a "non-math person" (I know that that's a flawed/fraught way of looking at things...), I will say that that was pretty, but still way over my head. I'd probably have to watch each part of the video on its own, and try to grasp each part, but it would probably take time for each part.

posted by subversiveasset at 4:38 PM on June 6, 2015

posted by subversiveasset at 4:38 PM on June 6, 2015

STOP WITH THE DAMN IRISH FIDDLE ALREADY.

I didn't follow the jump that he called "turning adders into multipliers" and the fecking drone of that damn fiddle means I'd rather pull my fingernails out than watch it again.

posted by Joe in Australia at 5:45 PM on June 6, 2015 [1 favorite]

I didn't follow the jump that he called "turning adders into multipliers" and the fecking drone of that damn fiddle means I'd rather pull my fingernails out than watch it again.

posted by Joe in Australia at 5:45 PM on June 6, 2015 [1 favorite]

geeze, I found the calculus made a LOT more sense than this adders and stretcher nonsense.. The symbology of the derivation is just freaking beautiful.

posted by k5.user at 6:29 PM on June 6, 2015 [2 favorites]

posted by k5.user at 6:29 PM on June 6, 2015 [2 favorites]

I used to think math was no funposted by maryr at 7:53 PM on June 6, 2015 [5 favorites]

'Cause I couldn't see how it was done

Now Euler's my hero

And I see why zero

Equals e^{iπ+1}

This reminds me of when I was at a party with a bunch of math majors and when I asked them a question about math they all said, "oh that's not my field," and then when I asked what their field was they said some things that were incomprehensible.

posted by Panjandrum at 8:17 PM on June 6, 2015 [4 favorites]

posted by Panjandrum at 8:17 PM on June 6, 2015 [4 favorites]

I used to love math, including calculus, but this lost me fast. Maybe my brain's too old.

posted by anadem at 8:42 PM on June 6, 2015

posted by anadem at 8:42 PM on June 6, 2015

That went from "oh, this is neat" to "ok now I'm being assaulted with scary equations and symbols" way too fast.

posted by treepour at 8:51 PM on June 6, 2015 [1 favorite]

posted by treepour at 8:51 PM on June 6, 2015 [1 favorite]

"An intuitive explanation" no no no just back it up.

By 'intuition' 3Blue 1Brown writes in the assumption that the viewer has intuition for a great many things, be it the number 'e', the hierarchy of mathematical operations, that a trusted knowledgeable voice works with an NPR-style cadence...

There is extremely solid stuff here -- I love the attempts to describe "numbers as actions" and distinguishing "adders" from "multipliers". These are fantastic qualitative points to make: "1" can represent "next in line" or it can mean "keep the status quo" depending on whether you're operating with addition or multiplication in mind.

But only in thinking about mathematical structures (is that intuitive for 99% of folks?) can we extend this notion, thinking of numbers as translations and rotations, bringing the complex plane into the mix, sending us down the road to $e^{\pi i} = -1$.

Structures =/= intuition: go ahead an call this a calculus-less explanation of Euler's formula. But, intuitive... I'm dubious.

posted by Theophrastus Johnson at 11:16 PM on June 6, 2015 [4 favorites]

By 'intuition' 3Blue 1Brown writes in the assumption that the viewer has intuition for a great many things, be it the number 'e', the hierarchy of mathematical operations, that a trusted knowledgeable voice works with an NPR-style cadence...

There is extremely solid stuff here -- I love the attempts to describe "numbers as actions" and distinguishing "adders" from "multipliers". These are fantastic qualitative points to make: "1" can represent "next in line" or it can mean "keep the status quo" depending on whether you're operating with addition or multiplication in mind.

But only in thinking about mathematical structures (is that intuitive for 99% of folks?) can we extend this notion, thinking of numbers as translations and rotations, bringing the complex plane into the mix, sending us down the road to $e^{\pi i} = -1$.

Structures =/= intuition: go ahead an call this a calculus-less explanation of Euler's formula. But, intuitive... I'm dubious.

posted by Theophrastus Johnson at 11:16 PM on June 6, 2015 [4 favorites]

While viewing this on the Tube, I noticed that someone had put pi to music in base 12, beautiful Gershwin sounding music

posted by Narrative_Historian at 2:55 AM on June 7, 2015 [1 favorite]

posted by Narrative_Historian at 2:55 AM on June 7, 2015 [1 favorite]

What teachers ought to do, I think, is first make the point that complex numbers have a "direction" as well as a magnitude.

Then introduce the function cos (x) + i sin (x) and show how the direction along which it lies in the complex plane depends on x. (This is a good time to draw the helix I mentioned above, and also its "shadow" or projection in one plane looking like a circle, in another like a sine wave, and in another like a cos wave... Of course this will be much easier to explain if the student already understands what sin and cos do, and especially if they've seen vectors before, which work just the same way...)

Then and only then, after the student has grasped how that function behaves and what it does, you can try to relate it to e^(ix), which is just a more convenient way of writing it.

If the student knows calculus, a good way to make this connection is to point out that while e^x is defined as the function which is its own derivative, cos (x) is

Show that taking derivatives of cos x + i sin x gives you this pattern of getting the same function back but multiplied by i, and then it becomes pretty believable that e^ix is exactly the same function, since it works exactly the same way.

Then do some simple examples showing how easy this e^ix notation makes it to express the relationship between two different sine waves that have different phases, to motivate why you would want to write it this way. Handy, huh?

I don't really see the point of trying to explain anything about e^x or e^ix without first explaining derivatives, because that "it's the derivative of itself" thing is the only thing that's special about "e," and is the whole point of using it. But fortunately, derivatives are pretty easy to explain. Still and all, there are reasons math classes have pre-requisites, and I think the video is kind of doomed to come off as incomprehensible if you don't have that background info.

Focusing on x=pi like the video does seems counterproductive

posted by OnceUponATime at 4:15 AM on June 7, 2015 [1 favorite]

Then introduce the function cos (x) + i sin (x) and show how the direction along which it lies in the complex plane depends on x. (This is a good time to draw the helix I mentioned above, and also its "shadow" or projection in one plane looking like a circle, in another like a sine wave, and in another like a cos wave... Of course this will be much easier to explain if the student already understands what sin and cos do, and especially if they've seen vectors before, which work just the same way...)

Then and only then, after the student has grasped how that function behaves and what it does, you can try to relate it to e^(ix), which is just a more convenient way of writing it.

If the student knows calculus, a good way to make this connection is to point out that while e^x is defined as the function which is its own derivative, cos (x) is

*almost*its own derivative... in that taking a derivative turns it into sin (x), but taking another one gets you to -cos (x)...Show that taking derivatives of cos x + i sin x gives you this pattern of getting the same function back but multiplied by i, and then it becomes pretty believable that e^ix is exactly the same function, since it works exactly the same way.

Then do some simple examples showing how easy this e^ix notation makes it to express the relationship between two different sine waves that have different phases, to motivate why you would want to write it this way. Handy, huh?

I don't really see the point of trying to explain anything about e^x or e^ix without first explaining derivatives, because that "it's the derivative of itself" thing is the only thing that's special about "e," and is the whole point of using it. But fortunately, derivatives are pretty easy to explain. Still and all, there are reasons math classes have pre-requisites, and I think the video is kind of doomed to come off as incomprehensible if you don't have that background info.

Focusing on x=pi like the video does seems counterproductive

posted by OnceUponATime at 4:15 AM on June 7, 2015 [1 favorite]

All that stuff about adders and multipliers does seem like a nice way to get across why we use exponential notation to mean more than just "multiply by itself this many times," though. I kinda think they should have just focsed on making that one point as clearly as possible.

For extra credit, they could go on ti use that approach to explain logarithms, especially if they go one one step farther and turn the sliding number line into a slide rule...

That would be really cool, and requires a lot less background knowledge to understand.

posted by OnceUponATime at 5:21 AM on June 7, 2015

For extra credit, they could go on ti use that approach to explain logarithms, especially if they go one one step farther and turn the sliding number line into a slide rule...

That would be really cool, and requires a lot less background knowledge to understand.

posted by OnceUponATime at 5:21 AM on June 7, 2015

*The animations are quite pretty, but I'm not convinced that this video will be helpful for people that don't already know that Euler's equation is a rotation in the complex plane.*

There was an interesting visualization of the complex plane here (I don't recall if that link discussed the Euler identity, though).

posted by thelonius at 7:15 AM on June 7, 2015

It took me many viewings to understand 3Blue1Brown's videos (a few months ago when I first ran across them) and I felt stupid to have this much trouble when everyone was saying how intuitive they were. I say this as a math person (I was on my college's Putnam team) but I'm posting about it here, because I'm a psychotherapist and know that what's intuitive to one person might be opaque to someone else.

So don't feel bad if the adders-into-multipliers point of view doesn't work for you. It didn't work for me either.

posted by Obscure Reference at 10:06 AM on June 7, 2015 [1 favorite]

So don't feel bad if the adders-into-multipliers point of view doesn't work for you. It didn't work for me either.

posted by Obscure Reference at 10:06 AM on June 7, 2015 [1 favorite]

Now I know how 4th grade students feel when I'm telling them something that I think is perfectly obvious.

posted by argybarg at 2:08 PM on June 7, 2015 [1 favorite]

posted by argybarg at 2:08 PM on June 7, 2015 [1 favorite]

When the music kicked in I thought I was going to see lots of panning around old sepia-toned pictures.

(Put me down as someone who prefers the calculus method, but whatever floats your boat)

posted by dirigibleman at 2:20 PM on June 7, 2015

(Put me down as someone who prefers the calculus method, but whatever floats your boat)

posted by dirigibleman at 2:20 PM on June 7, 2015

The pi connection hit me as soon as they mentioned rotation the plan. Rotation = circles oh so tight with triangles that feel just right.

posted by BiggerJ at 6:59 PM on June 7, 2015

posted by BiggerJ at 6:59 PM on June 7, 2015

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I do wish it had been about one minute longer though, because the final "And there you have it" seemed like a larger intuitive jump than everything else leading up to that point.

posted by 256 at 1:10 PM on June 6, 2015 [1 favorite]