Joseph Liouville and his Numbers (especially his constant)

Number theory. Whoo!

In number theory, let there be a number x where, for any positive integer n, there are infinitely many number pairs (p, q) where q > 1 that:

0 < |x – p/q| < 1/q^n

x is the Liouville number.

Liouville numbers can be described as “almost rational”, and can be approximated by sequences of rational numbers. In other words, Liouville numbers are transcendentals that can be more closely approximated than any other algebraic irrational.

In 1844, Joseph Liouville proved that all Liouville numbers were transcendental. Maybe, if I can find the proof, I might post why, but not now.

One Liouville number is the Liouville Constant, which takes the form of:

 L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001...
0.1100010000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001… I’m not sure if that’s 120 decimal digits.

One thing: Liouville’s Constant (call it L; TapL hahaha) almost satisfies 10x^6 -75x^3 – 190x + 21 = 0, if the solution was rounded, then maybe.

If my mind comes to it, I might go into more detail

P/NP?

Back to another Millennium Prize problem!

There are two types of questions: one, where an algorithm can solve the question in polynomial time (P), and ones, where there is no way to find the answer quickly, but if information is given on showing what the answer is, it can be quickly verified, which is NP, or nondeterministic polynomial time.

The question, stated by Wikipedia, is “if the solution to a problem is easy to check for correctness, must the problem be easy to solve?” or in other words, is P = NP?

If P ≠ NP, then both P questions and NP-complete questions are within the class of NP questions. NP-complete questions are a set of problems where to each of which any other NP-problem can be reduced in polynomial time and the solution can be verified in polynomial time as well. NP problems, therefore, can transform into any NP-complete problems.

Now, if you want, go visit the Wikipedia page on it to understand better about P vs NP.

Poincaré Conjecture… with less detail.

Yes, this was already solved by Grigory Perelman, but I have nothing much, so…

Imagine a spherical item. And, for the easiest solution, let’s use an apple. Wrap the rubber band around the apple. Without tearing it or letting it leave the surface, we can slowly shrink the rubber band until it becomes a single point.

Now, think of a doughnut. Before you get hungry, quickly wrap a rubber band around the doughnut. Now, no matter how hard you try, you can’t shrink the rubber band down to a point without breaking it or making it leave the surface.

The surface of the apple is “simply connected”, while the surface of the doughnut isn’t. Poincaré, about a hundred years ago, knew that a two-dimensional sphere is essentially characterised by this property of simple connectivity, but questioned that of the three-dimensional sphere.

Grigory Perelman already solved it, and the link to the arXiv is here. The upload is by Terence Tao.

Until next time…

Riemann Hypothesis

Yesterday, I said that I would do a post on the Riemann Hypothesis, so here we are now.

There is a function with the name of the Riemann Zeta Function, named after Bernhard Riemann. The Riemann Hypothesis states that the Riemann zeta function has zeroes only at the negative even integers and complex numbers with the real part of 1/2. So, in explanation, ζ (s) is a function where the argument s may be any complex number not equal to one, and its zeroes are at points where s is a negative even integer, like -2, -4, -6, -8 and so on. These are trivial zeroes.

The non-trivial zeroes of the Riemann Zeta Function have a real part of 1/2. So if the hypothesis is true, then all non-trivial zeroes take a form of 1/2 + it, where t is a real number and i is the imaginary number.

There’s actually a formula for the Riemann Zeta Function:

{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }
Ye

For more info, visit this website, Wikipedia.

Maybe, I’ll go more in depth about the Riemann Hypothesis, but not now.

Updates.

Dante was elected as mayor in Hypixel Skyblock. The admins warned us, telling us to not vote for him. We all thought, “shut up admins! New update! NEW MAYOR! VOTE FOR THE NEW MAYOR DANTE!!!!!” But the admins were right. We should not have voted for him.

How strong are Dante’s debuffs?

First debuff: More tax!

The auction house and bazaar taxes have been doubled. The NPC buy prices have been raised by 10%. Five coins/minute for talking, two coins for travel. WTF? That means bazaar flipping is 9% less effective. 9% might not seem like a lot, but when it comes to bazaar flipping, it’s lots of money. Kinda.

Second debuff: Official buildings!

“Official buildings”, like AH, Bazaar Alley, THE BANK and more cannot be accessed without either a Happy Mask or a Dante Talisman, which the Happy Mask is the cheaper option as it is only 100 coins, but takes up the helmet slot. Dante Talisman is not only common, it is one million coins. ONE MILLION! Plus, how are the new players supposed to progress now?

Third debuff: Cancelled events!

Oringo, Winter Island and Spooky Festivals have been cancelled. WTF WHY ARE THEY CANCELLED THEY’RE LITERALLY SOME OF THE BEST MONEY-MAKING METHODS!!!!! What’s dangerous about Oringo? Literally all he does is give out pets that buff you, it’s not like they end up killing you.

Extra debuffs: Mayor elections no more!

Dante elections have replaced the normal mayor election. That means that Dante will be around for a long time, unless the Resistance does something about it. I miss doing slayers while Aatrox was mayor, or waiting for that moment when Diana was mayor so that I could finally start doing mythological events. When am I going to be able to do that?

Literally, Dante is so useless, Barry is better than him, and Barry’s never been elected once. Neither has Diaz, and Diaz’s perks are actually 100x better than Dante’s perks. I swear, I want to join the Resistance and be the one to behead him. Period.

Second update: “You’re clicking too fast!”

This one I personally hate. I was doing an experiment, when suddenly, bam! I was kicked out of the experiment menu and in the chat, there was this message: “You’re clicking too fast!” Many people have been annoyed about this, complained in the forums, and Donpireso has said that they are working on a fix, but it’s unclear when the fix is. It’s made people lose so much money, it’s gone into the millions. One guy had very OP loot from experiments then lost it all because of “You’re clicking too fast!” This is all because of autoclickers that use them for some reason, but please disable this during experiments, starring dungeon items, and other clicking-related things.

Why am I writing about this? I dunno… content drought.

Also, the nerfs that Hypixel admins made doesn’t impact me much, but other people must be mad about it. Bye!

Sigma proof… explained

Unrelated, but a GD level called Sigma: https://geometry-dash-fan.fandom.com/wiki/Sigma

Yesterday, I posted a proof on why the sigma sum for phi was true, but I kind of rushed it, so I will be going into details.

If you can do a little bit of “hard maths”, then it is easy to prove that:

{\displaystyle \varphi ^{2}=1-\varphi }
is true.

is true.

\sum _{{n=2}}^{{\infty }}\varphi ^{n}=1
Right?

That means that the sum is Φ^2 + Φ^3 + Φ^4 + Φ^5 + Φ^6 + …, but since Φ^2 = 1 – Φ, that means that Φ^3 = Φ^2 * Φ, or Φ-Φ^2. If we keep multiplying, we begin to see a pattern here, where the exponent keeps going up by one, and in the full sum, there is, for any positive integer k, a Φ^k and a -Φ^k, which implies that they can be cancelled out. But there is no term to cancel the one in Φ^2, so one is left, and Φ^∞, since Φ is smaller than one, the limit as x approaches infinity of Φ^x converges to 0. That means the final sum is 1-0, or just 1.

QED.

The Tangram

DUN DUN

DUUUUUUUN!!!!!

Okay, okay, the Tangram is not a monster, you are assured of that. It’s a kind of puzzle, but also a toy.

It is:

A tangram

And you can split the seven parts up and put them together in fun ways. Like:

The number 4

You can go ahead and search it up online!

Topology = potatoes

Say WHAT?!?!

You probably were freaked out by that. There hasn’t been a post on topology and I want to do one. So … here comes nothing.

To explain the topology and potatoes, we have to use topology, but before that, what even is topology?

Topology is an area of mathematics where a pencil is basically the same thing as a book. That is, you can bend and stretch and enlarge and shrink stuff, but NEVER, EVER,

EVER!!!!!

cut or glue stuff. So a cup with a handle is not equivalent to a water bottle. YES, EVEN WITH THE CAP OFF. Let me explain.

This is a cap-less water bottle.

Image result for plastic water bottle without lid
Well, not a WATER BOTTLE, but it will do for now.

Link: https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTr_A-bhzQ4_lRyWvA_mV4OUcureOsTx4NUYMiZtGTg726GbLOE

Make the hole 5 times as big as it is now, reach in, and grab the bottom. Then PULL YOUR HAND OUT OF THE BOTTLE! And you are left with a cylinder. MAGIC!!!!!

The best-known topology example is the doughnut and the coffee cup. GIF:

Image result for doughnut and coffee cup topology
Coffee cup and the doughnut

Link: https://upload.wikimedia.org/wikipedia/commons/2/26/Mug_and_Torus_morph.gif

The Part 2 will show up next with explanations of the topological potato equation. BYE!