I love e. My favorite is this stochastic definition: Let X_i ~ Uniform(0,1) for integers i. If N = min { n | X_1+X_2+...+X_n > 1 }, then E[N] = e. In words, if you repeatedly draw uniform random numbers between 0 and 1 and keep a running total of the sum, then on average, it will take e draws for the sum to exceed 1.
Unrelated, but the video made me realize this: If Φ = (1+√5)/2> 1 is the positive root of x² -x -1 = 0, then Φ² = Φ +1, so that 1 = Φ² -Φ = Φ(Φ-1), but then, 1 = 1/Φ(Φ-1) = 1/Φ² × 1/(1-1/Φ). Since Φ > 1, we have 0 < 1/Φ < 1, thus *1 = 1/Φ² +1/Φ³ +1/Φ⁴ +1/Φ⁵ +1/Φ⁶ +…*
U reli LOVE using Dominated Convergence Theorem to switch the order of summation and integration in proving summation identities. u've already shot a video on when Feynman's trick doesn't work. When will u shoot a video when this Theorem doesn't work? 🤩🤩
Hi, I suppose you mean : "our favorit identity involving e", but also some way : "defining" e . So mine is : lim_{n->infty} (1 + 1/n)^n . But if you mean only "involving", my prefered indentity is e^ix = cos x + i sin x .
Why should numbertheoretical functions like these two (especially my(n)) combined with an geometric (classic sense, ratio of certain lengths) defined number be in any relation to a trancendental number? This is mathemaGics. :D Would be nice to see following scenario. Use this expression as a Definition for exp(1). Extend the definition to exp(x) for real x and assuming exp(ix) = cos(x)+isin(x) find similar expressions for sin(x) and cos(x).
There is a yt-video titled "Unveiling Connections between Mathematical Constants: The Conservative Matrix Field". That would be a nice topic in this context.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
I'm always amazed when somebody discovers something this obscure. Why would anybody ever think to look for this?
It's the same old answer for Maths and Physics : cocaine
@@MasterChakra7amphetamine, I reckon
@@donach9 Walter Weiss, the competitor of Leonard Euler
How far fetched can an identity be?
Michael: Yes.
What a crossover
I love e. My favorite is this stochastic definition:
Let X_i ~ Uniform(0,1) for integers i.
If N = min { n | X_1+X_2+...+X_n > 1 },
then E[N] = e.
In words, if you repeatedly draw uniform random numbers between 0 and 1 and keep a running total of the sum, then on average, it will take e draws for the sum to exceed 1.
Unrelated, but the video made me realize this:
If
Φ = (1+√5)/2> 1
is the positive root of
x² -x -1 = 0,
then
Φ² = Φ +1,
so that
1 = Φ² -Φ = Φ(Φ-1),
but then,
1 = 1/Φ(Φ-1) = 1/Φ² × 1/(1-1/Φ).
Since Φ > 1, we have 0 < 1/Φ < 1, thus
*1 = 1/Φ² +1/Φ³ +1/Φ⁴ +1/Φ⁵ +1/Φ⁶ +…*
U reli LOVE using Dominated Convergence Theorem to switch the order of summation and integration in proving summation identities.
u've already shot a video on when Feynman's trick doesn't work. When will u shoot a video when this Theorem doesn't work? 🤩🤩
I love the magical tapping on the board.
What the hell is up with that amazing thumbnail 😳
16:50
Haven't checked out Michael in a while, glad you're still here
Interesting. You could plug in other values than 1/ψ. For example, for x=1/2 one gets e^0.5 = replacing ψ with 2 in RHS.
Thanks for that! Indeed, for any 0
10:04 is this wizardry?
Michael is the math teacher at Hogwarts
I love the magic knock
رائع جدا كالعادة.
الطريق شاقة و الوصول ممتع
8:30 The terms are all positive so you can always exchange the two summations.
Hi,
I suppose you mean : "our favorit identity involving e", but also some way : "defining" e . So mine is : lim_{n->infty} (1 + 1/n)^n .
But if you mean only "involving", my prefered indentity is e^ix = cos x + i sin x .
Woouh! That's hard to grasp, but very nice!🤣
Damn, that's beautiful.
Sweet! Decomposition of unity meets up with friends...
Why should numbertheoretical functions like these two (especially my(n)) combined with an geometric (classic sense, ratio of certain lengths) defined number be in any relation to a trancendental number?
This is mathemaGics. :D
Would be nice to see following scenario.
Use this expression as a Definition for exp(1).
Extend the definition to exp(x) for real x and assuming exp(ix) = cos(x)+isin(x) find similar expressions for sin(x) and cos(x).
What. In. The. Actual. Phi.
There is a yt-video titled "Unveiling Connections between Mathematical Constants: The Conservative Matrix Field". That would be a nice topic in this context.
Didn't realize R. Schneider is in the band "the Apples in Stereo!"
e is everywhere /\ e is golden ---> means : gold is everywhere.😀
This topic under what branch in mathematics
Almost e hundred thousand subs!
Very interesting identity! I am a simple man and my favorite identity is Euler’s identity e=cosθ+isinθ
Thats e^(iθ) but yeah i see your point
? Something about pincers and prongs ?
Amazing - it has a poetic beauty all of its own. And I suppose it emphasizes: all of math is really a human construct or does it? 🙂
Nice
too abstract ....
Indeed m8
But you can't deny the coolness of this identity
It seems that there was a mistyping in #\delta$ definition: $\mu
e0$ is to be replaced by $\mu
e1$.