masterofspacetime

[masterofspacetime]

Unendlich viele Primzahlen

Euklid zeigte seeeehr lange vor der unsrigen Mathematik, dass ein fundamentaler Fakt bewiesen werden kann. Er zeigte, dass die Primzahlen unendlich häufig vorkommen, es gibt unendlich viele davon. 

Um dies vorzuführen, bedarf es nichts weiter als ein wenig Konstruktion: Stelle dir vor, dass es endlich viele Primzahlen gibt.  Nun kann man eine neue Zahl konstruieren, die entsteht, wenn man alle endlichen Primzahlen multipliziert. Beispiel: angenommen es gäbe nur 4 Primzahlen. Dann wäre meine neue Primzahl: 2*3*5*7 = 210. Diese ist offensichtlich durch alle dortigen Primzahlen teilbar... Nun kommt die Magie: Nun addiere auf deine jetzt entstandene Zahl n eins dazu. Betrachte nun n+1. N+1 ist nun durch keine der Primzahlen teilbar. Also ist N+1 selber eine Primzahl. verwunderlich? Nein, denn jede Zahl ist in seine Primfaktoren zerlegbar, damit wäre ein Widerspruch erzeugt und damit gibt es unendlich viele Primzahlen, denn endlich können sie mit obiger Konstruktion nicht sein. q.e.d

[masterofspacetime]

Reblog if you would be devastated if you found out one of your followers committed suicide.

[masterofspacetime]

Don't kill yourself, please.

If you’re suffering from depression and are looking for a sign to not go through with ending your life, this is it. This is the sign. We care.

If you see this on your dash, reblog it. You could save a life.

[masterofspacetime]

[masterofspacetime]

What happens once you kill yourself? Because I'm ready to go.

You wanna know what happens once you kill yourself? Your mother comes home from work and finds her baby dead and she screams and runs over to you and tries to get you to wake up but you won’t and she keeps screaming and shaking you and her tears are dripping onto your face and your dad hears all the screaming and runs into the room and he can’t even speak because the child that he loved and the child that he watched grow up is gone forever and finally your little sister runs into the room to see what all the fuss is about and she sees you dead. The person she looked up to and loved. The person she bragged about to  her friends, the person she wanted to be just like when she grew up, the person that made her feel safe. But she’s never really going to get to grow up and smile and laugh and love because she’ll always be consumed with this feeling of missing you. And now there’s something missing from your family and they can barely look at each other anymore because everything reminds them of you but you’re gone and hurts more than anything. and you think that your mom never cared because she was always busy and yelling at you to finish your homework and clean your room and forgot to say I love you sometimes but really, she loved you more than anything and she doesn’t leave the house anymore, she can’t even get out of bed and she’s getting thinner and thinner because it’s too hard to eat. Your father had to quit his job and he doesn’t sleep anymore, every time he closes his eyes he sees his baby dead, and the image never goes away no matter how much alcohol he drinks. And at school your best friend sees that your seat is empty and she gets this sick feeling in her stomach and that’s when she hears the announcement. You killed yourself. And suddenly she’s screaming and crying in the middle of class and no one even bothers comforting because they’re all  busy sitting there staring at your empty seat with tears dripping down their cheeks and all she wants is for you to hug her and tell her it’s gonna be okay like you always did, but this time, you’re not there to do it, everything is dark now that you’re gone and her grades are slipping, she barely goes to school anymore and she ended up in hospital after taking too many pills because she wanted to see you again. the girls who used to make fun of the way you dressed feel their throats get tight, they don’t talk to each other anymore, they don’t talk to anyone, they’re all in therapy trying so hard not to blame themselves but nothing works. and your teacher who always gave you a hard time stares blankly at the wall, she quits her job a few days later. And then your boyfriend hears the news and he can’t breathe, he still calls you a lot just to hear your voice and he talks to you on facebook but you never message him back, he can’t fall in love again because every girl he meets reminds him of you, he’s never going to get over you, he loved you and he cries himself to sleep every night, hating himself and slicing his skin because he couldn’t save you and he’s never going to hold you in his arms or hear you laugh again. Now everyone who knew you, whether they were a big part of your life or someone you passed in the hallway a few times a week, they carry this aching feeling around inside them because you’re gone, and they miss you, and they don’t know why you left but it must’ve been their fault and they should’ve stopped you and they should’ve told you they loved you more and that feeling is never going to go away. And so you killed yourself

but you killed everyone else around you too. 

[masterofspacetime]

http://www.psyweb.com/

[masterofspacetime]

"Denn Angst, Immanuel, besiegt dich, ohne dich anzugreifen

Professor Kurt Kant - Illusio (Die Geschichten des alten Mannes)

[masterofspacetime]

"Was ist Glück?" fragte der Hase. Der alte Mann blickte ihn an und lächelte: "Sehnsucht nach dem Wahren."

Beyond - die letzten Geschichten des Weisen

[masterofspacetime]

Every time !!

Have you ever wanted to end a proof not with an unfilled square or “Q.E.D.,” but rather with “#get rekt”?

[masterofspacetime]

IMPOSSIBLE! Right? You may have heard “the interior angles of a triangle always add up to 180 degrees”. This is not always true. Check out the second image, it shows a triangle with 3 right angles for a total of 270 degrees! 

It is true in flat Euclidean geometry (the geometry you probably learned in school) however. But there are so many other geometries out there! You may be thinking, are other geometries real though? A mathematician would argue they are just as real as the typical flat geometry you know and love (or hate). These alternative geometries can be practically useful too!

The images above show triangles in spherical geometry. Those aren’t triangles though! Oh but they are! A triangle is just a polygon enclosed by three lines. Looks like it fits the criteria. Wait but those aren’t lines, they are curved! Ah yes. I argue that these are, for all intents and purposes, just as good as lines. We need to ask: What is a line? A line is so basic to us we may not know how to describe it. I offer this definition: A line is the shortest path between 2 points. The 3 curves that make the triangle above are in fact the shortest paths from one vertex to the other on the surface of the sphere (they just so happen to be on circumferences of the sphere, which are often referred to as great circles). So it may be more useful to think of lines, in general, as length minimizing curves. In conclusion, we would consider the shape above to be a triangle as it is enclosed by 3 length minimizing curves on a surface.

Spherical geometry can be very useful; think about the Earth. To reduce travel time, airplanes would want to travel along great circles as they are the shortest paths from one place to another. Additionally, this type of thinking (rethinking straight lines as length minimizing curves) is central to Albert Einstein’s general theory of relativity.

read more at http://staffrm.io/@missnorledge/35H6cS1T52

[masterofspacetime]

Erinnert mich stark an Instant Insanity ;D

Properties of the Symmetric group

The group of permutations of a finite set is also called the Symmetric group on n elements, and notated Sn.

|Sn| = n!

This means: the number of elements of the symmetric group on n elements is n! (n factorial), which is n·(n-1)·(n-2)·…·3·2·1

Proof, by induction:

Base case: |S1| = 1, because the only way to put one object into one slot, is keep it in the same place (the only permutation is the identity)

Inductive step: fix n, and assume that for r<n, |Sr| = r!.

You can put any of the n elements in the first slot, and now you have n-1 elements to put in n-1 slots, which there are |Sn-1| ways to do

In equation terms, |Sn| = n·|Sn-1|.

By the inductive hypothesis, |Sn-1| = (n-1)!, so we have shown that |Sn| = n·(n-1)!= n!

If m<n, Sm is a subgroup of Sn

Only have the permutations that affect the first n elements, and keep the ones after it in place, and this is (isomorphic to) Sn.

If n>2, Sn is not Abelian

If every element of a group G commutes with every other element of G, or a·b=b·a for all a,b in G, then we say it is Abelian. If at least one pair does not commute, a·b≠b·a, then the group is not Abelian

We just need to show if there are at least 3 slots, we can use the two permutations:

(1 2)(1 2 3) ≠ (1 2 3)(1 2)

Read this as “cycling the first three elements to the right (or to the start if its at the end) flipping the first two elements” is not the same as “flipping then cycling”

left side: (1 2)(1 2 3) = (2 3) right side: (1 2 3)(1 2) = (1 3)

[masterofspacetime]

Unglaublich, was man mit Mathematik alles anstellen kann. 

[masterofspacetime]

“Fascinating”

-- Mr. Spock

Black Hole’s Singularity. In the centre of a Black Hole is a Gravitational Singularity, a one-dimensional point which contains a huge mass in an infinitely small space, where density and gravity become infinite and Space-Time curves infinitely, and where the laws of physics as we know them cease to operate.  Current theory suggests that, as an object falls into a Black Hole and approaches the Singularity at the centre, it will become stretched out spaghettified due to the increasing differential in gravitational attraction on different parts of it, before presumably losing dimensionality completely and disappearing irrevocably into the Singularity. An observer watching from a safe distance outside, though, would have a different view of the event. According to relativity theory, they would see the object moving slower and slower as it approaches the Black Hole until it comes to a complete halt at the event horizon, never actually falling into the Black Hole. According to the “Cosmic Censorship” hypothesis, a Black Hole’s Singularity remains hidden behind its event horizon, in that it is always surrounded by an area which does not allow Light to escape, and therefore cannot be directly observed. The only exception the hypothesis allows (known as a Naked Singularity) is the initial Big Bang itself. It seems likely, then, that, by its very nature, we will never be able to fully describe or even understand the Singularity at the centre of a Black Hole. Although an observer can send signals into a Black Hole, nothing inside the Black Hole can ever communicate with anything outside it, so its secrets would seem to be safe forever.