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/sci/ - Science & Math

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5293988 No.5293988 [Reply] [Original]

Doesn't it blow your fucking mind that the sum <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{P_k}\right)[/spoiler], where <span class="math">P_k[/spoiler] is the kth prime, diverges but <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k^{1+\epsilon}}\right)[/spoiler] converges for ANY <span class="math">\epsilon>0[/spoiler], no matter how small of an epsilon.
Topic of thread: Post mathematical "stuff" that still blow your mind even after you proved them
and understood them.

>> No.5294008

bump

>> No.5294025
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5294025

wow.

fundamental theorem of calculus still kinda blows my mind haha

>> No.5294033

>>5294025
that's because you just learned it and you are 16.

>> No.5294068

>>5293988
Interesting thread idea, OP.

Here's one: the fact that ZF set theory can't prove the statement, "the cartesian product of a family of nonempty sets, is nonempty."

>> No.5294081 [DELETED] 

c^0=1

This one has been bugging me for the longest time.
In my mind it should be 0

>> No.5294084

x^0=1

This one has been bugging me for a long time.
In my mind it should be equal to 0.

>> No.5294085

I never really have my mind blown by individual parts, but occasionally while I'm doing simple math then switch to calculus I have to pause for a second and consider how we got so far starting with so little.

>> No.5294090

>>5294084
this is a sign of an incredibly small mind.

x^0=1 is just its definition. it doesn't even have a proof (inb4 people try to "prove" it by the laws of exponents. the laws of exponents are what one proves from the definition, not vice versa)

If you want a genuinely astonishing fact about exponentials, look up Stirling's formula

>> No.5294107
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5294107

The idea that most of science, math, language, etc. could be completely redesigned if we ever had a few centuries to burn.

>> No.5294114

Mostly the way that we've come so fucking far
I mean, it's only the past 800 or so years that we've had the decimal system. Even less time where we've had calculus.

>> No.5294116

>>5294090
you wot?
let x, a be arbitrary
x^0 = x^(a-a) = x^a/x^a = 1

the x^(a-a) = x^a/x^a part is an axiom, but this definition has a proof.

>> No.5294126

cantor's diagonal proof.

>> No.5294160

>>5294116
>x^(a-a) = x^a/x^a
no it's not. i defy you to find a reputable text where that's listed as an axiom.

http://planetmath.org/encyclopedia/AxiomaticDefinitionOfTheRealNumbers.html
>no exponents anywhere in sight

>> No.5294163

>>5294160
meant to quote this line here
the x^(a-a) = x^a/x^a part is an axiom

>> No.5294170

>>5293988
no

<span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k*ln(k)*ln(ln(k))*.....*ln(ln(...
.ln(n) ))^{1+\epsilon}}\right)[/spoiler] converges for ONLY <span class="math">\epsilon>0[/spoiler] is mind blowing

>> No.5294179

WTF OP where is the proof for this

>> No.5294186

>>5294170

what is <span class="math">n[/spoiler]

>> No.5294203

>Distance between most primes is 2, even for a fucking big ass k
>Distance between exp numbers is always longer and longer.

I don't really see the mindblowing thing here, maybe a misconception of what infinity really is.

>> No.5294220

>>5294203
>Distance between most primes is 2

My sides!!! lrn2twinprimesconjecture

(but even if the TPC is true, it's still very badly false to say the distance between most primes is 2!)

>> No.5294227

>>5294220
Yeah I know, I just phrased it wrong, what I'm saying is that you have twin primes even with the biggest k and that just throws away everything mindblowing about those series.

>> No.5294231

>>5294227
No, we do not know that you have twin primes even with the biggest k. We do not know whether there are finitely many or infinitely many twin primes. If YOU know, you should publish a proof post-haste, and be hailed as one of the greatest mathematicians of the 21st century

>> No.5294236

>>5294231
I'm just trying to point out that is not mindblowing, I know that TPC is not proved yet.

>> No.5294304

I mean, I still think the unit circle is really cool.

>> No.5294316

(e^8-10)/the golden ratio = proton to electron mass ratio.
wtf nature.

>> No.5294323

>>5293988

http://en.wikipedia.org/wiki/Residue_Theorem

>> No.5294329

>>5294325
Squares don't end either, nigger.

>> No.5294325

I think the idea of the circle is pretty mindblowing, like how it doesn't end, and how practical it is.

>> No.5294331

>>5294316

Another good example of WTF nature;

http://web.williams.edu/go/math/sjmiller/public_html/RH/Hayes_spectrum_riemannium.pdf

>> No.5294342

>>5294329
That whole shapes thing in general really, likeb seriously! It was all likeyeah this is fine & cool & then it musta been like "but wait, what if there were boundries"
I wonder if that was a 'sad' moment :s

>> No.5294345

Anyone know anything about sacred geometry?

It's very cool stuff. It's about how perfect circles are generated naturally throughout the universe.

>> No.5294348

>>5294342
You're an interesting fellow.

>> No.5294418

>>5294186
>n
k

>> No.5294450

>>5294163
OK what is 0^0

>> No.5294463

This is my favourite mindblow:

Polynomial of degree 2: Use quadratic formula to find both roots in C
Polynomial of degree 3: Use cubic formula to find all three roots in C
Polynomial of degree 4: Use quartic formula to find all four roots in C
Polynomial of degree 5 or larger: There is no formula and there will never be any formula to find the roots. It is completely impossible to manipulate the coefficients in any way to factorise the polynomial.

CAPTCHA: Beer ismajob

>> No.5294469

>>5294345
Try
http://www.youtube.com/watch?v=ahXIMUkSXX0

>> No.5294474

>>5294463
we've barely touched on how this was proven using groups/Galois theory in my abstract algebra class, it definitely blows my mind.

>> No.5294495

>>5294474
It's so awesome, ties in nicely with the classification of finite simple groups - in a nutshell, the (normal!) subgroup A5 of S5 is the smallest finite non-abelian simple group, and it fucks everything up.

>> No.5294615

>>5294107
And what would you suggest we change?

>> No.5294754

>>5294450
(Not the guy you're asking by the way)
Can't you do it on your fucking own?
<span class="math">0^0=0^{a-a}=\displaystyle\frac{0^a}{0^a}=\displaystyle\frac{0}{0}=undef[/spoiler]
The limit of x^x when x approaches 0 is defined though and is 1.

>> No.5294885

Bump for interest.

>> No.5294904

>>5294754
Then what is <span class="math">\displaystyle \sum_{n=0}^\infty \frac{0^n}{n!}[/spoiler]?

>> No.5294986

>>5294754
limit of 0^x as x approaches 0 is 0
limit of x^x as x approaches 0 is 1

>> No.5294996

>>5294615
I think he means we would learn some important new things about the universe that changes what we know on a fundamental level.