Euler’s formula about e to the i pi, explained with velocities to positions.
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Not familiar with the calculus referenced in this video? Try taking a look at this one:
Another perspective on this formula, from Mathologer:
Another perspective from this channel:
And yet another from the blog Better Explained:
I’m not sure where the perspective shown in this video originates. I do know you can find it in Tristan Needham’s excellent book “Visual Complex Analysis”, but if anyone has a sense of the first occurrence of this intuition do feel free to share. It’s simple and natural enough, though, that it’s probably a view which has been independently thought up many times over.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811
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If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
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Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.
As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!
Splendid
i is not sqrt(-1). i is defined as i^2 = -1, which is not the same thing. Defining i = sqrt(-1) gives inconsistent results in many equations
Why does i make something turn 90°.. why because let’s say a ray is pointing in the direction 1, ixi=-1 so you turned 180 degrees.. so if ixi is 180° the i must be 90°
π Ττ
> “3.14 minutes”
> looks inside
> 4 minutes
3.14 = 4?
and here I am watching because of Project Hail Mary 😭
Wow you explain it so well
this is incredible
i think of it as a graphics program.
e is a program. the symbols in the exponent are the argument. i is a program as well. rotates the way the unit is incremented. the taylor series is the steps of the program.
exponents really iterate with integers. i isn't an integer. but when you run it through the taylor series, it isn't iterated by i. its iterated in infinite integer steps. each step being a smaller increment after you do the calculation.
so… you see e^ipi. just open up the taylor serires. put i – which is a program that basically rotates- and pi – scales to radians- and build out the tayler series. and when you compute it… it returns a final position that is within precision tolerance of -1 + i0. then you add 1 + i0. and you get a finale returned position of 0 + i0.
that's all math ever is really…
someone said, if you think the symbols are static objects, you are screwed. think of them as programs that always start at 0 and increment. the solution is the final position.
u also need to know you're running the program on a 2d coordinate system.
they show this to new students just to watch their brains explode. kind of scammy
مچکرم ❤😂❤عالی بود ❤😂❤
I love how it doesn’t at all matter what the values of each of the numbers is. This equation is purely concerned with their properties, and I think that is beautiful.
e uses its property of its position equalling its velocity;
π uses its property of functioning in radians in a unit circle;
i uses its property of rotating values into the complex plane.
This is easily my favourite equation in mathematics. Three of the most important, fundamental constants, combining their properties in the absurd manner of raising a transcendental to a complex transcendental, to get not just a real, rational number, but an integer.
Simply beautiful.
Thanks so much for this video and other magnificent videos in your channel. You make me love maths more and more everyday.
btw, I think it would be the best if you can have another video about why multiplying by i has the effect of rotating numbers by 90 degree.
For me, I watched this video the first time and did not understand that point.
Then I watched other Youtube videos to understand that rotation effect, then came back to this to understand it all.
I’m so happy you used tau!
The audio track is perfect
maybe one of the best videos in the history of mankind
Tau is exactly double of pi bcuz it's the circumference over the radius(half of the diameter)
Thank you for this, awesome explanation.
3,14? PI REFERENCEE?!🥹
Finally brought it to a conclusive round wrap. e to the i.pi as a cross product thinking in terms of matrices and exponents is pi on one axis, i is the radius and it represents mapping points on e to every other point on pi. That finally explains e to the e and why the derivative is itself. Mwicho. 3b1b affiliatees brought me(kept bringing) here. Transformers embeddings is what finally made that nudge and tipped the scale.
What a brilliant intuitive explanation. Thank you so much!!!
so its basically the polar form of -1
I had to watch 2:47 possibly 20 times to understand what he means lmao
"at 90 degrees" is the key phrase
like only when things are radius ` with speed 1 does it make a circle and not a spiral of some sort (thus making 90 deg angle between pos and vel)
Thanks for creating and sharing this. This was very helpful in visualization.
When first time I saw this equation, I thought this is the nature’s way of telling the law of conservation or law of nothingness or a home coming equation.
I am sure GOD / infinite intelligence did not design maths first and then created universe .. so if I take this equation not considering maths .. and this is how I see it :
e is natures way of telling or doing continuous growth / decay.
i denotes unseen world ( not imaginary numbers .. or we can say numbers that exist in unseen world )
Pi is half life of any existence ( other half is manifested in the world )
e raised to power pi multiplied by i is
Continuous growth / decay whatever we call it in unseen world ..
When unseen world (quantum field)manifest .. in unseen world it becomes -1 .. if we consider seen / experienced world as +1
So any existence in unseen world + that existence after manifestation in seen world = nothing ( if 0 is nothing )
When I was thinking about it .. I realized we got the right interpretation (I mean it fits nature’s math and this formula shows it) of sin , cos, e formula, 0 and 1 and the “i” takes us to unseen world from seen world (90 degree rotation ) is also fascinating
Everyone should like and subscribe just for adoption of tau.
Why aren't you placing the velocity vector just above the postion vector
The study.of how e came about it's a complex calculus question
The derivative always returns the derivative plus an amplification factor in the complex plane
"√-1 = i " is not a number, it's part of an an operator like "negate", but the operation rotates the direction ("sign" ) of a real or complex number.
Examples:
1 × i = 1i means "rotate +1 anticlockwise 90°".
1 × i² = -1 means "rotate +1 anticlockwise 180°".
negate(1) = -1 = i² × 1 = (e ͥ ⷫ⸍²)² ×1 = (e ͥ ⷫ) ×1
3:37 I feel like the concept of repeated multiplication still applies here. In rectangular coordinates, repeated multiplication continuously "stretches" the number and it blows up to infinity. In the imaginary axis, repeated multiplication continuously ROTATES the number. 4x4x4x4 = 64, but i x i x i x i brings you back to 1. Repeated multiplication of magnitudes increases to infinity, but repeated multiplication of directions turns it the same amount each time. This is why it rotates at a constant rate. e^t is a function that increases by it's own magnitude, but the magnitude of e^(i t) never changes, it just rotates. Functionally, it's the exact same.
I understood the imaginary numbers so much better when I started seeing them as another form of negative numbers, because really that's all they are. You can stretch a thing by 9, or you can stretch it be 3, twice. Similarly you can rotate a thing pi radians using -1, or you can rotate it pi/2 radians, twice, using i. Negative signs signify direction, and the imaginary plane is all the inbetweens.
That's every intuitive