The Golden Number

When someone says "Maths", maybe the most iconic number they can come up is pi, or rather the e number. But what if there's another number, mucho more important than this other two? Well, it actually exists, and it's called the phi number, also known as the golden ratio.

The value of this number is 1.6180339887 and it represents a proportion between two segments:

The usage of this numbers has been developed by years. Euclid was the first Mathematician who studied the divinal proportion in a regular pentagon and a golden rectangle. By cutting the gold rectangle, it creates a square and a smaller rectangle with the same aspect radio. The most remarkable ones in Mathemathics are Michael Maestlin, who published in 1597 a decimal approximation of the golden ratio, Fibonacci mentioned the numerical series, Johannnes Kepler created the Kepler triangles, a combination of phi along with the Theorem of Pythagoras, and Édouard Lucas gave us the Fibonacci sequence, a numerical sequence.

There are several applications for this peculiar number. Some of them are related to Architecture, painting, design(bookdesign, postcards, playing cards, posters, wide-screen televisions...), music, narute and optimization.

Let's go deeper into the number. This is the negative root of the number, known as the conjugate root

Taking it's absolute value and dividing one by it, we have a new number called golden ratio conjugate.

The golden ratio and the golden ratio conjugate can be expressed like this

Those two expression are used as a scale for every proportion. For example:

The major usages of the number is seen in triangles and pyramids, containing different symmetries.

Monty Hall

Imagine a game show where you’re the participant. You see in front of you three doors: One of them contains a car and the other two, a goat (I think is obvious what the prize is…). The presenter knows which door leads to the car and because he’s a generous guy, he offers you to choose a door. So, you choose one of the doors, but wait, don’t open it yet! The presenter seems to be more generous and opens one of the remaining doors, revealing a goat. Furthermore, you have the possibility of keeping your door, or choose the other one. At this point, is obvious to think the probability of having the car is 50 %, isn’t it? Well…. What if I told you… That statement is wrong?

This problem is known as the “Monty Hall” problem, asked in Parade Magazine in 1990, where Craig F. Whitaker sent a letter to Marilyn vos Savant’s column. The TV program problem’s is a reference to Let’s Make a Deal, a game show where they can’t change their boxes.

So here’s the thing: The presenter knows exactly where the car and the goats are, so he can’t eliminate the door which leads to the car. If the presenter wouldn’t know anything, then we have another probabilistic problem, but that’s not what we’re talking about. We’re talking about the presenter making his decision AFTER the player chooses a door.

When you have the three doors and choose one of them, the probabilities are 1/3 for the car and 2/3 for the goat. So it’s more probable that you first choose a goat and not the car. But either choosing the car or a goat the first time, there will be always a goat that can be eliminated. In that case, the probability of eliminating the car is 0.

So now, we have two doors, one leading to the prize and the one to the failure. Considering the fact that it’s more possible to choose a goat in the first choice, what happens when you change it? That your chance of obtaining the car is higher!

You have a simulator here in this link to contribute the Mathemathics. Keep calm and use logic!

Carlos Beltrán, the resolver of problem 17 from Smale's list

Steven Smale, one of the most recognized mathematicians of the XXI century, left 18 problems known as Smale's list, which his creator hismself considered as the biggest problems of this century. And one of those problems was solved by a spanish mathematician called Carlos Beltrán.

Carlos Beltrán received the Jose Luis Rubio de Francia for resolving the 17th problem of the list. Despite his thesis director affirmed that Beltrán has enormous skills in a lot of math areas, his pupil stated that the problem was less harder than other problems from the list than aren't unresolved yet. But even that, the truth is that this problem never had a progress aside from Smale's contribution.

The problem number 17 consist of solving polynomial equations in polynomal time in the average case. But we're not talking about a second degree ecuation, we're talking about this kind of ecuation:

Normally, this kind of ecuation always has many ways to be solved, and has lots of solutions, but the problem is that solving this must be efficient (that's what ti means to solve the problem "in polynomal time"). Resolving this has caused quite a change in Maths, particulary in programming, because with this resolution, now someone who solves an ecuation on a math program knows if the program is going to take a long time or not, or even success on all the possible solutions. There are different ways on solving an ecuation, but with Beltran's method, a uniform probabilistic algorithm called Las Vegas algorirthm, will give always the correct results and will only gamble on the resources used for the computation. And it's no secret that on Maths, making sure the result is correct has to be checked with other ways.

"There are lots of ways to caculate things, but it's no easy to prove that the way you're doing if the right one" Carlos Beltrán.

How to solve a double integration

Hello everyone, welcome to my first post. Today I'm going to solve a double integration.

Find the volume of the solid bounded above by the plane z = 4 − x − y and below

by the rectangle R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2}.

Solution

The volume under any surface z = f(x, y) and above a region R is given by

f(x,y) is, in our case, 4-x-y. And the limits on the integration are very simple: in the variable x, the limits are 0 and 1, and the y is from 0 to 2. We resolution is simple: You solve one of the integrals(in this problem, we'll solve the x first), considering the other variable a constant. With the first integral done, just solve the area left with the limits.

Repeat to solve the integral of y and the solution is done.

This double integral  are the easiest types to evaluate because all four limits of integration are constants. This happens when the region of integration is rectangular in shape. In non-rectangular regions of integration the limits are not all constant so we have to get used to dealing with non-constant limits.

The 3d plot has been done by Wolfram Alpha. It's a good site to see the solutions if you're stuck doing a math problem.

First post

Hello and welcome to my blog! Here, I'll write several posts about Calculus and other interesting things during these months. I come from the "Universidad Europea de Madrid" and this is my first year academy. I hope you enjoy this blog and don't be too nitpick, because this is the first time I'm doing one of those...

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