| by Kenneth Chase | 100 comments

e to the pi i for dummies


Recently we did a video on the most
mysterious and beautiful identity in mathematics e to the pi i is equal to minus one.
Comes up three times in the Simpsons which, of course, makes it even more
important. Now, afterwards a few people challenged me to
come up with an explanation that even Homer can understand and I’ve actually been agonizing over this ever since. And today I want to do just that, I want
to explain e to the pi is equal to minus 1 to
someone like Homer. Ok, someone like Homer who can only do addition, subtraction,
multiplication, division. So, we have to remind him or tell him
two things. The first one is that i is this strange complex number square root of -1, i squared is -1. Second thing is just
kind of a reminder. If you’ve got a semicircle of radius 1 then the length of
the semicircle is pi. Ok, so keep those two things in mind, we
have to use them later on. The first thing I have to explain to
Homer is what is e. So to do that I give him a dollar and tell him “Go to the
bank”. Now I’ve arranged with the manager here to give him 100% interest
over a year, ok. So what happens to this one dollar
when Homer puts it in? Well after one year he has 1+1=2 dollars. Now this is actually not the best you can do with a 100% interest, you can do better if
you find a better bank. And we found a better bank, the Second Bank of
Springfield. At the Second Bank of Springfield they calculate and credit
interest twice a year. So after six months what happens? You get fifty percent on what you’ve got
there. So that’s 50 cents which gives 1.5 dollars. Now another six months pass. Half of 1.5 is 0.75, so you have to add that to 1.5 and that gives you 2.25. That’s what you’ve got at the end of the year if you calculate
and credit twice. Now, at the Third Bank of Springfield they do it three times. So what do you get? After four months you
get that, after eight months you get that and at the end
of the year you’ve got that, even more. And it’s actually quite easy to figure
out the general formula for this. Well, maybe Homer cannot do it, … , but I can do it. So
it’s this one here and you can probably do it, too, if you’re watching this video. So, it’s (1+1/n^n. So, if you credit n times throughout the year that’s how much money you have at the
end of the year. Let’s just check this for the simplest cases 2 and 2.25. So for n=1, we’ve got 1+1=2 hmm is 2 – okay. 1+1/2=1.5 squared is 2.25, ok,
works. Ok, works in general. Now, this is really
good news but maybe what you think now, Homer definitely thinks this is:
Well I divide more and I get more money. So, if I just divide enough maybe I get a
trillion dollars at the end of the year. Sadly that doesn’t work. So, for example, if you divide in 125
parts you get that much money at the end of the year, or have that much money
at the end of the year. Now, if you crank up the n what happens
is, well, that number goes up but it goes up very slowly and actually settles down to
a number. So, if we push the whole thing to infinity, take the limit of this, we get this
number here: 2.718… dollars and that’s the absolute maximum,
that’s “continuously compounding interest”, that’s what it is, so we can’t do any
better than this, that’s e. And that’s also where e comes up for the first time historically, exactly this sort of
consideration, ok. Cool, so now we’ve got e. We’re ready to move on.
e^(pi i). Well, not so fast. Let’s just go to e^pi first which is
actually almost as mysterious as e to the pi i. Why is that? Well it’s got a
special name it’s called Gelfond’s constant and eventually
I’ll definitely make a video about this one, but just for today just ponder it a little bit. What
does this actually say. Well it says weird number to the power of another weird
number and you supposed to calculate this. How
do you calculate something like this? I give it to you on a piece of paper and
you don’t have a calculator. That’s … strange. I think nobody will be able to do this. Well, it would be doable if pi was equal to 3 because then we know we just
have to multiply, you know, maybe chopped off bits here (at the dots),
three times together and we get a rough approximation to what we are
looking for. But, no, we have this one here. So we really want to calculate this, we want to really know for some strange
reason how much money Homer has after pi
years if you’re compounding interest continuously. That’s what I want to know, I can’t go
to sleep tonight if I don’t know. Ok, now the trick here is, we have got this
bit here, which gets us closer close to e the more we crank up the n. Ok, and so if I put that one up here and
put a large number in here we get the right thing, or approximately the right
thing, or as close to the right as we want. Alright now that looks still pretty
awful, okay, and, well, let’s muck around a little bit with it. So, the first thing we do is we multiply by pi here and there, and
see when you do this on the top and the bottom obviously nothing changes and it actually looks a bit uglier than before. But what’s nice is that these two bits are the same and,
you know, what you have to do now to get this is to just crank up the bit in
the box. So n=1 we have this, n=2 we
have that, and then that, and …. that’s still pretty awful ! Except what’s really important here, and
that’s a really really nice trick, is, what’s really essential here is, that
we’re going up. It doesn’t matter how we’re going up, as
long as we’re going up towards infinity, we can go up via nice numbers: 1, 2, 3, … That will also get us there, and that’s
actually what we do and this here is exactly what we’re looking for. So here
it’s like really awful to the power of awful, but now we’ve just got addition,
division, multiplication that’s all we have to do … just a lot.
But that’s basically all we have to do, so we are getting there. And, of course, the pi here stands for
really any number whatsoever. So what we’ve done is actually
we’ve figured out how to calculate the exponential function with basically
nothing, with just this. That’s what we’ve just figure out, that’s a pretty pretty good effort. Alright, so let’s graph this (e^x) and a
couple of those guys (the functions on the right) and see what happens. So I’ve graphed the exponential function
and I graphed the first one of these guys, well the second one really where we take m=2. Not a terribly good fit but if you crank
up the m you can really see how good this gets.
And, actually, when you press, you know, the button on the calculator that’s what
your calculator does at some level. It just adds and multiplies and divides
and these sorts of things. That’s all you can do. Anything, anything complicated in
mathematics, you know, when you do it numerically has to be reduced to just basic
arithmetic otherwise it doesn’t work. Okay here we
go. Almost there now. Just chuck in your pi i, that’s what we’re interested in, and go for it. And actually we could go for it at
this stage. It’s actually not very hard to multiply things like this. Well, this is basically a complex number in
here, so we’ve got a nice number plus a nice number times i. It’s actually not that hard to multiply a couple of those things together I could
teach you in a second actually i’m going to teach you in a second. Let’s just do it on Mathematica and see
what Mathematica spits out. So for m=1 we get this number,
it’s also a complex number. Doesn’t matter what you put up there, doesn’t matter what m is, the result is
always going to be a complex number. So, let’s crank it up now. Crank it up, crank it up, crank it up, all
the way to … what did I do, a hundred. And you can see that this first bit here
gets closer and closer to -1 and the second bit here, that nice
number in front of the i goes to 0. So basically the ugly part goes away and
were left with the -1 if you kind of go to infinity. We could stop here,
but actually I’ve got this really, really nice way of multiplying complex numbers
which we can apply to this, multiplying complex numbers with triangles. Let me just show you. Ok so here we go. Now complex numbers you can draw. Real numbers you can draw on the number line, complex numbers you can draw in the xy-plane. Actually Homer stands right on
top of the xy-plane so we might as well use it and he can really relate to it at
this point in time. So here we’ve got the real number line
there is 0, there is 1, there is 2, and so on. And, well, we’ve extended this real line by the complex
plane. It’s just this whole thing. Every complex number corresponds to a
point in here. For example, 1.5+i is just the point where you go 1.5 over here along
the x-axis and then one up in the direction of the y-axis. And then this guy here,
for example, 1+2i. Well, 1 over here and then 2 units up. Ok, now multiply those two things
together. So what do we do? Well, we do 1 x 1.5 is
1.5 then 1 x i is i. 2i x 1.5 is 3i and then the last one that’s where we have to remember
that i squared is equal to -1, so this is -1 x 2 is -2. Now we just combine things together in the
obvious way, so there and there and that’s the product. And, of course, that
corresponds to a point, that guy out there. Well how do you get from here to there? Not
obvious, right, we can do this, but you can actually see at a glance, you can see at
a glance that these two guys get you up there. How? With triangles! Okay, so to every point, to every complex
number we associated a triangle and the corners are 0, 1 and that point here. So that’s the first triangle. Let’s just
save it. Second triangle 0, 1, point. Now we align them like that, stretch this one, the red one,
so that these two sides are the same and there is your product. Brilliant isn’t it. So you just kind of
aline and stretch these triangles and you know what happens. And actually if
you know the triangles it’s pretty easy to predict where the
product is going to be. Let’s do another example. Let’s do this one here
squared. So what you do for squared is this triangle twice. Ok stretch it, that’s the square. Now, cubing and we’re going to have higher
powers so we need to see what happens here, so just make another triangle,
stretch it, that’s the cube of this number here. Alright now higher powers. That’s the
complex plane. For the higher powers this circle here, the unit circle, the circle
of radius 1, around 0 plays a very very special role. Why is that? Well here is a complex number
on that unit circle. The triangle that corresponds to it has two equal sides, there and there. So, when you align two such triangles what happens? Well you don’t have to stretch, right. Let’s see what happens when I kind of
raise this to the power of 8 … 2nd power, 3rd power, 4th, 5th, 6th, 7th, 8th. That’s the 8th
power of that guy here. So this power spiral, or whatever you want to call it, is just kind of wrapping around the unit circle. So it doesn’t matter how high a power
you choose, it’s just going to end up somewhere here on that circle. And what happens when you move
that guy here off the unit circle? Let’s just move it inside. So we move it
inside, what do we get? Well, we get this nice spiral here, kind of spiraling inside. And,
actually, to go higher and higher it goes closer and closer to 0. If we move this guy outside, well it’s
always going to be a spiral, but the spiral kind of spirals outside. Main
lesson to take away from this is that the closer you start at the unit circle
the closer the spiral, this power spiral will wrap around the unit circle. So now
let’s go for the real thing, the one we’re really interested in. Ok, this guy here. So there’s the complex
number, here in the middle. What is that? Well it’s 1 over here and
then you have to go up pi /m so that’s kind of going
up there. Let’s go for m=3. Ok, let’s just draw this. There we go 1 over here, pi/3 up
there, and then we have to do cubes, right. So 3 times same triangle, scaling, and so
on, what we’ve just done. And it gets us over there. Ok, right. Now what’s going to happen when
I make this m bigger? Well, the 1 stays the same. I make the m bigger, that means that this number here gets ….
smaller, right? That means that it’s going to wander down
here, that it’s going to get closer and closer to this point, and actually I can make it
as close to this point as I want, as close to 1 as I want by making m bigger and bigger. It’s just going to wander down here, down here, down here. This means that the spiral is going to
wrap close to the unit circle. Well let’s do it. So, crank up to
4, four triangles. Now crank up to 5, five triangles now. It’s wrapping closer, right. 5, 6 now let’s just let it go and see how
that guy here gets closer and closer to -1.
It’s real magic about to happen. Ready to go for the magic? There we go, cranking it up all the
way. Well not all the way, up to a hundred 🙂 We can see it’s really getting closer
and closer to -1 and it’s pretty obvious why, right? I mean the bit that’s obvious so far is that
because that guy here wanders down and down, it gets closer and closer to the
unit circle we should get a closer and closer wrap
around the unit circle but what’s not clear at the moment, maybe, is why we
don’t wrap further or closer. Why do we only go halfways around. And for that you have
to remember what I said at the very beginning this reminder about the length
of this semicircle. What’s the length of this semicircle again? It’s pi, okay it’s pi. And what is this? This is
the mth part of pi. So, basically we’re starting out with the mth part pi here and then we’re doing this m times so pi/m times m
is pi. So we’re going to eventually wrap around
halfways, smack on, and we’re going to get e^(pi i)=-1 and I think this is the way to explain it. Hopefully, well I don’t know about Homer but
you know hopefully you who are masters of plus, minus, multipl,y and so on got something out of it.

100 Comments

ALiNa DoNG

Jul 7, 2019, 11:29 am Reply

See ya numberphile

pepe6666

Jul 7, 2019, 5:33 am Reply

BAHAHA holy shit where he shows you how to multiply complex numbers by drawing the triangles – what a fucking hero. you sir are a champion of the human race.

Witold Domeyko

Jul 7, 2019, 9:11 am Reply

Is it correct to deduce that e to any complex number = -1 ?

Watching The Earth

Jul 7, 2019, 5:11 pm Reply

I guess this video named as : " pi over ei for dummies " but youtube auto-translate name to different languages as " pi over ei for stupids " . So maybe manual translation required.

FuckYouWhosNext

Jul 7, 2019, 3:50 pm Reply

you should wear tight black turtle necks and dance around like Gunter on Sprockets!

Naz Doga

Jul 7, 2019, 9:18 pm Reply

8.32 / How ( 1 + … ) suddenly becomes ( -1 + … ) ?? That's the key point I needed but missed and so couldn't follow the rest.

tsiggy

Jul 7, 2019, 11:00 pm Reply

Mathologer, you are a teacher.

Александр Захаров

Jul 7, 2019, 5:03 pm Reply

Хуйня какая-то

Tony Detroit

Jul 7, 2019, 2:59 am Reply

I'm confused by the proof strategy. Can someone explain to me why e^pi was explored as part of the proof. The pi seemed to play no role. He just as easily could have considered e^5. Or could have begun with e^x, no?

name2

Jul 7, 2019, 3:36 pm Reply

e^pi*i is equal to -1. Which, coincidentally, is i squared. So that means e^pi*i = i^2. Then with this, you can figure out that:
e^pi = i^(2/i)
e^i = i^(2/pi)
And finally…
e = i^(2/pi*i).
And when I checked to see if this is legit, it is! (e = 2.72…)
So not only e is transcental, it's also made from an imaginary number!

Bryan Carter

Jul 7, 2019, 12:06 am Reply

Great video, except he completely lost me in the last 30 seconds, he just re-stating the formula. There's so many cool concepts here that I do understand, but none of it helped me understand i as an exponent. I learned a lot from this video, but I feel like I'm no closer to understanding e to the pi i.

Rajesh Sharma

Jul 7, 2019, 1:51 am Reply

You are a genius sir

Mark Durham

Jul 7, 2019, 10:20 pm Reply

Seems relevant somehow that "pie-eyed" means "very drunk." 🙂

TheJ4RyD

Jul 7, 2019, 11:27 pm Reply

"i squared is 1". He means negative 1, right?

HBH ZTH

Jul 7, 2019, 5:01 am Reply

At first I wondered where I had heard the intro tune first. Then I remembered Kate Bush.

priyank2626

Jul 7, 2019, 2:16 pm Reply

In the final case we substituted pi i with m . So a imaginary number with real number.

That Guy

Jul 7, 2019, 7:08 pm Reply

I'm shocked how dumb I am

Falling Oranges

Jul 7, 2019, 12:11 am Reply

The intro chords remind me of the intro to Babushka by Kate Bush

도꼬마리

Jul 7, 2019, 12:06 pm Reply

well now i know!

Chuck Desylva

Jul 7, 2019, 7:32 am Reply

Ok, please do in 3 dimensions.

Angel IA

Jul 7, 2019, 2:40 pm Reply

3.17157

Rajpal singh Dawar

Jul 7, 2019, 1:46 pm Reply

I really want to know what type of calculator you used for the calculation of infinite sums of the pattern and plus one by N to the power n

Rajpal singh Dawar

Jul 7, 2019, 1:47 pm Reply

At the time 3.43 seconds what type of software or calculator you used please let me know this is very important

JeevS GOne

Jul 7, 2019, 4:09 pm Reply

e^{i x} = cos x + i sin x

Khang Nguyễn Trường

Jul 7, 2019, 3:21 am Reply

stonk

WireDog

Jul 7, 2019, 4:10 am Reply

My IQ is 8 points short of genius level. Literally. I can’t understand this.

Nilo Alvarado

Jul 7, 2019, 5:01 am Reply

Se agradecen los subtítulos, renace nuevamente el deseo de perderme en el universo matemático

Cocoabine

Jul 7, 2019, 1:19 pm Reply

Sooooo.. because triangles?

Andreas Andreotti

Jul 7, 2019, 6:08 am Reply

Thank you sir, it could not be more pedagogical presented. I like mathematics because it has always given me peace of mind. I often try to I solve problems on algebra, geometry and trigonometry in order to get away of everyday concerns. The fascination of it is to reduce a complex problem into the four main calculation forms as you wisely pointed out. You call them tricks, I call it maneuvering, or favorable manipulation. Being a Greek I always have the Aristotelian logical categories in mind that help a lot in mathematical thinking.

vLinh dh

Jul 7, 2019, 10:12 am Reply

ho ho ho

Ryan Fuxa

Jul 7, 2019, 7:59 am Reply

That awkward moment when you realise Homer Simpson is smarter than you….He works at a nuclear facility….?

Antivlog

Jul 7, 2019, 8:15 pm Reply

TEH UNIT CIRCLLLE!,!!!

Wang Wei

Aug 8, 2019, 3:34 pm Reply

What is the curve if you connect those points from m=1 to m=∞?

rajesh sarkar

Aug 8, 2019, 4:08 am Reply

OUTSTANDING ……

Leon Ravenclaw

Aug 8, 2019, 3:39 pm Reply

4:44 Engineers entered the chat

jstarks21

Aug 8, 2019, 2:45 am Reply

Oh, that's what Elon Musk put on his Twitter

Red Rock

Aug 8, 2019, 9:15 am Reply

Instructions unclear.
Divided everything by 0.

Brett Ess

Aug 8, 2019, 5:54 am Reply

Brilliant! I wish my mathematics lecturer was this good. 😎

gore buster

Aug 8, 2019, 6:41 am Reply

Soo if I deposit one dollar with a banker at an imaginary rate of interest compounded continuously, at the end of 3 years, 1 month, 21 days, 16 hours, 21 minutes, 5 seconds, 923.60771908 milliseconds I owe him a dollar?

(Bcz the money the banker owes me is equal to my principal amount * e^(rate of interest * time) so e^(i*pi) = -1 so the banker owes me $-1 which means I owe him $1.

Jimin Bang

Aug 8, 2019, 6:36 am Reply

"Where's the pie?"

PRITIPRIYA DASBEHERA

Aug 8, 2019, 4:23 pm Reply

it is so neatly "pi"ed

Avinash SOOKUR

Aug 8, 2019, 7:22 pm Reply

You are the GOD of Mathematics …….

Floofy shibe

Aug 8, 2019, 9:28 pm Reply

3:14

Muttley

Aug 8, 2019, 3:52 am Reply

that t-shirt is both transcendental and imaginary

Muttley

Aug 8, 2019, 4:13 am Reply

Mmmmmmm pie

Payge

Aug 8, 2019, 9:21 am Reply

What’s the equation for being dumber than homer ?

Joan Ferran

Aug 8, 2019, 10:52 pm Reply

Excellent explanation tks

Mike Gordon

Aug 8, 2019, 11:09 pm Reply

Fascinating, thank you. But one question: By the end, you seemed to have eliminated e from the function. So does it work for any other number in place of e? What is the connection with e as the ‘number of natural growth’?

John Byrd

Aug 8, 2019, 10:46 pm Reply

"for dummies" and then blazes through multiplication of two algebraic functions. Bad teacher. No Twinkie for you.

Alive Afterall

Aug 8, 2019, 6:55 pm Reply

e^i*pi = -1;
e^2*i*pi =1;
ln(e^2*i*pi) =ln(1);
2*i*pi =0;
WTF??))

DIY home tricks automation

Aug 8, 2019, 7:18 pm Reply

he is from NZ or Aus??

The Sahil

Aug 8, 2019, 8:40 pm Reply

What software do you use?

13IG 0NE THE

Aug 8, 2019, 1:03 pm Reply

Wait… if we assumed m=n×pi×i, doesn't that make m itself an imaginary number?

Hjtunfgb

Sep 9, 2019, 12:17 am Reply

If you went a little bit beyond you could easily prove the more general form e^i*theta… genious

David Ramadeen

Sep 9, 2019, 4:08 pm Reply

Why you say you do not know about Homer? YOU ARE HOMER. 15:34. Just suck the bottom lips in. Thanks for nice job. BUT WHAT HAPPEN WHEN YOUR M=Pi?

Technical YADAV

Sep 9, 2019, 4:32 am Reply

Puzzle lover
Nice! Content
#HUMANTHINGS

Clive Goodman

Sep 9, 2019, 9:10 am Reply

This identity is actually a special case of exp(ix)=cos(x)+sin(x)i

Colin Holloway

Sep 9, 2019, 1:16 pm Reply

I got some of it. That will do for now. Love this explanation and REALLY like the graphics. So good to add some concrete explanation to the abstractions.

italianpro 2005

Sep 9, 2019, 12:18 pm Reply

so, can we say that (1+e) × (1)^i pi is still equal to 0??

Aqua rius

Sep 9, 2019, 1:16 pm Reply

Thank you very much! Now I understand Euler's formula!!!

Imagine Existance

Sep 9, 2019, 1:40 am Reply

An easier way to understand this is
x^yi=cos(y*log(x))+i*sin(y*log(x))
Plug x to be e and y to be pi
so -1+0
Or cos(pi)+i*sin(pi)

You can cancel out log and e because log(e)=1

Darki i

Sep 9, 2019, 8:20 pm Reply

1:09

OliBint

Sep 9, 2019, 7:55 am Reply

I'm pretty sure Homer would have lost you with the funky shifting triangles on a graph where the y axis represents complex numbers…. at around 11 mins.

NotValik

Sep 9, 2019, 7:54 pm Reply

“This guide can be understood by anyone that knows how to do addition, subtraction, multiplication and division!”

Uses exponents, graphs, imaginary numbers, funcions…

Laureano Luna

Sep 9, 2019, 8:39 am Reply

Far from being a proof.

Jose Lozano

Sep 9, 2019, 1:56 pm Reply

I had a hard time decoding the title of this video!

Marie Meyer

Sep 9, 2019, 4:46 pm Reply

I freaking love that video

omar elbendary

Sep 9, 2019, 7:44 pm Reply

Thanks junji ito

дан нестеров

Sep 9, 2019, 8:17 pm Reply

Thx, you helped a lot

Sherine Victoria

Sep 9, 2019, 2:11 pm Reply

Can believe they made a video just for me lol

Mark Odern

Sep 9, 2019, 5:24 pm Reply

Still not a single word about 2π (tau). Do you want Martians to laugh at us cause we still use a half of the number?
e^2πi = 1
e^ti = 1
And graphically it would make even more sense, since we make 1 revolution, not a half (why we would even do that?)
And I meet this misconception in every YouTube video on the topic.

Mustafa Malik

Sep 9, 2019, 7:26 am Reply

0:51 HOMER NO UNDERSTAND

VronZ

Sep 9, 2019, 12:29 pm Reply

I knew it was beautiful but I didn't know it was this beautiful

Arttu Kettunen

Sep 9, 2019, 2:31 pm Reply

Different school subjects' levels taught by youtubers:
Anything else: middle school
Math: University

Floofy shibe

Sep 9, 2019, 10:37 am Reply

at 5:07

mathologer: e^pi
Me: you mean 27??

Jean-Yves BOULAY

Sep 9, 2019, 5:29 pm Reply

The number Pi and the Golden Number (φ) and the inverse of these numbers are made up of a seemingly random digits. This article is about order of the first appearance of the ten figures of decimal system in these fundamental numbers of mathematics. There turns out that the ten digits decimal system (combined here with their respective numbers: figure 1 = number 1, figure 2 = number 2, etc..) do not appear randomly in the digits sequence of Pi and the digits sequence of Golden Number (φ). The same phenomenon is also observed for the inverse of these two numbers (1/Pi et 1/φ).

https://youtu.be/NDj0rJhnO_g

PDF here: http://jyboulaypublications.e-monsite.com/

Kenneth Payne

Sep 9, 2019, 4:19 pm Reply

I like mathologer a lot, but on this topic, I’m afraid that he gets blown away by the 3 blue 1 brown guys, whose 3 minute video is way more understandable.

Thibo Van België

Oct 10, 2019, 2:19 pm Reply

You lost me at point: 00.00

wsquix58

Oct 10, 2019, 11:14 am Reply

Brilliant in its simplicity in explaining the complex…No pun intended.

Rednaw

Oct 10, 2019, 3:58 pm Reply

3:28 i broke it, if n = 999999999999999 then the value will be 3.03503…

S T

Oct 10, 2019, 8:48 pm Reply

Yeaaa … no.

asmr lover

Oct 10, 2019, 2:19 pm Reply

What if you listen to something you cant understand but you think that you understands it

Nimesh Poudel

Oct 10, 2019, 4:45 pm Reply

How (1/2)! = √π/2

Slawek Zawislak

Oct 10, 2019, 9:59 pm Reply

i think we should ask Homer te explane it to us

Girls Girls Girls

Oct 10, 2019, 8:59 pm Reply

awsome, it really is for dummies, but i finally understood it. Thanks

NUKE

Oct 10, 2019, 6:16 am Reply

e^iπ=-1 is so poetic.
All the famous weird numbers in mathematics, 2 transcendental, one imaginary and one negative: e, π, i and -1 all related to each other directly!

Karl Jo

Oct 10, 2019, 1:50 pm Reply

your explanations are so clear and understandable!

Sidhant Srivastava

Oct 10, 2019, 4:55 am Reply

0:53 i^2 is not 1 lol

JustAnotherYoutuber

Oct 10, 2019, 9:48 pm Reply

So my question is, why pi? Seems like you are just equating it as a constant * pi = some arbitrary constant. Couldnt this work with any number other than pi?

Sergio Andres Solis

Oct 10, 2019, 8:10 pm Reply

why does it get closer to -1?

Mr. Gang Banger

Oct 10, 2019, 11:43 am Reply

P=NP guys! Now Homer is a millionaire!

TallWaters

Oct 10, 2019, 1:20 pm Reply

This is very very good, thanks a lot.

Dave Rocket

Oct 10, 2019, 3:40 am Reply

Lamentably, Homer still wouldn't understand.

Rojasnetor

Oct 10, 2019, 6:08 pm Reply

I really enjoyed this video, I now love maths more than ever!

Alexander Mizzi

Oct 10, 2019, 7:23 pm Reply

Why underestimate Homer.

mario mario

Oct 10, 2019, 7:24 pm Reply

Why underestimate Homer.

mario mario

Oct 10, 2019, 7:26 pm Reply

i^2=1 you mean -1

WeGoTSkiLL

Oct 10, 2019, 10:38 am Reply

This might be a stupid question, but why doesn't this work with any number besides pi? I know it doesn't work, but why? If it's just important to get both sides of the triangle the same, any number divided by a huge m would get that sorted, wouldn't it? What role does the pi play? I seem to be missing a part to fix this 🙂

Márton Bálint

Oct 10, 2019, 12:46 pm Reply

…………………………………………

S

Oct 10, 2019, 7:52 pm Reply

That "shell"like shape made from triangles awfully looks like the golden ratio spiral shape

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