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endless and the search for beauty and recursion

The other day I suggested a way of showing that the sum of the first odd numbers is always a perfect square. Use chickpeas, remember?
In mathematics, sometimes the obvious is not always easy. When faced with a problem, often, the hardest part is selecting the right tool. Once this is chosen (well chosen, that is), the resolution becomes trivial.
Today I propose a problem to solve using a common tool but most of the fans do not know math: the ralación between algebra, geometry and arithmetic.
The approach is simple: How much is this infinite sum?
adding What is the infinite things that, as the mathematicians love. If you want to try on their own to stop reading and get back in time with the solution (1 / 3, in case you are curious and want to stop reading right now).

methods exist for adding this series easily, anyone who has made a first course in a Bachelor of Science known, but since this is a blog for men of letters I I save that way.

The first impulse is brute force: to make all these fractions in decimal numbers and put them into a calculator (better in Excel) to see what comes out. The sum of the top five is: 0.25

+0'0625 +0'015625 +0'00390625 +0,00390625 = 0.3330078125

if we add fractions, we see that the first decimal always 3, so if one is sufficiently clever, it can venture and hit the correct solution (1 / 3 = 0'333333 ...)

But in mathematics, as in justice intuition is not enough to solve. It must be demonstrated.
And brute force can be very effective, but is not elegant, and mathematics, as important as the rigor is the pursuit of beauty: Look at this picture

:
is a triangle. Let's put another triangle inside it:
the triangle we have divided into 4 equal parts (what he seems to have put a thong?). We are left with the central and painted blue. Blue triangle that represents a quarter of the large triangle.

We'll repeat the process in the upper triangle:
That blue triangle above represents a quarter of a quarter of the big triangle, that is 1 / 4 ²
can repeat as many times as you want:
To sum just have to keep in mind that each tiángulo blue is the fourth of which is directly below. Therefore, the blue triangles chain represents the infinite sum we want to calculate.
Now watch the original triangle has been divided into three pieces exactly the same (I have painted different colors to distinguish them), and if the three pieces are identical, then each of them represents 1 / 3 of the triangle.

Therefore our infinite sum of fractions (or triangles) is 1 / 3.
You see, math is not enough that all things are correct, it is also important that they are beautiful. I know that for one who had bad experiences in mathematics at school, talking about beauty in a problem is almost a provocation, but I do not deny that, at least, is smart. It seems to me wonderful.

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Google

This story today they appeared as national news a few days ago.
Let's do an experiment. Open another browser window Google page and type Varcelona well, with V of Warsaw. get a lot of search results , but what we are is what comes up: If Google detects that the search word is misspelled we stand corrected.
Did you mean: Barcelona.
And if we click on the suggestion, it disappears.

Now repeat the same with the word Recursion. Google is proposing to do the search for the term recursion, but when we click on the suggestion, the same suggestion back out, over and over again, ad infinitum.

This is a programmer's joke from Google that shows the fine sense of humor they spend. Let's try to understand the joke.

In mathematics called recursion recursion or (two palabros que no existen en español, por cierto. El término correcto es recurrencia ) a una manera de escribir sucesiones de números en la que cada término se calcula a partir de los anteriores.
Por ejemplo, la famosísima sucesión de Fibonacci:
1, 1, 2, 3, 5, 8, 13, 21, ...
en la que los dos primeros números son 1 y 1 y a partir de ahí cada uno de los elementos se calcula sumando los dos que tiene detrás.

1+1=2
1+2=3
2+3=5
3+5=8
5+8=13

La manera de escribir esto usando la recursión es:

F ₀ =1
F ₁ = 1
₂ F = 0 F + F 1 = 1 +1 = 2
F ₃ = 1 F + F _{2 = 1 +2 = 3}
... F
n = F n-2 + F n-1

words, to define an element refers to the definition of the element in smaller cases, and until the case but simple, which is defined numerically.
For example, we calculate F ₅ , 5 th term of the sequence Fibonacci:

F ₅ = F ₃ + F ₄

therefore, must first calculate the 3 rd and 4 th (ie F ₃ and F ₄ )

F ₄ = F ₂ + F ₃
F ₃ = F ₁ + F ₂

Whereupon the 5 th term would be (the parentheses are only to separate and not tiing):

5 F = (F 3) + (F ₄ ) =
= (F + F ₁ 2) + (F ₂ + [F 3]) =
= (F + F ₁ 2) + (F ₂ + [F + F ₁ 2])

But F 1 = 1 and F 2 = 1, so that to calculate F ₅ just replace the previous formula: F
₅ = (F + F ₁ 2) + (F ₂ + [F + F ₁ 2])
5 F = (1 +1) + (1 + [1 +1]) = 5

other words, to calculate a number use your own definition as many times as do need to get to the simplest case. You try to calculate other terms for themselves, help them understand.

The Google search does the same thing: If you are asked to find recursion recursion is proposing that we seek, and if we accept the suggestion we again make the same suggestion, and so on to infinity.
For Google, recursion is an endless recursion.

Note: This form of calculation is used by computers to perform complex tasks, split the task into smaller ones recursively up to a simple case easily solved. Two examples:

1. Management de un conjunto de números:
Para ordenar una lista de números se divide la lista en dos trozos, se ordenan cada uno por separado y luego se mezclan. Para ordenar cada uno de los trozos se utiliza el mismo procedimiento recursivo.
2. Buscar un número (o una palabra) en una lista no ordenada:
Se divide la lista en dos partes y se busca en una de ellas. Sino está se busca en la otra mitad. Cada una de las búsquedas se hace usando el mismo procedimiento. Aunque parezca mentira los ordenadores encuentran las cosas antes usando este método que mirando uno a uno los elementos de la lista.

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perfect squares and odd numbers (post 3 chapters)

CHAPTER 1. The anecdote.
Jorge tells me a curiosity about the square roots of some numbers and their relationship with the odd numbers:
1 +3 = 4. Its square root is 2 (we added two odd numbers)
1 +3 +5 = 9. Its square root is 3 (we scored the first three odd)
1 +3 +5 +7 = 16. Its square root is 4 (we scored the first four odd numbers)
1 +3 +5 +7 +9 +11 +13 +15 = 64. Its square root is 8 (we scored the first eight odd)
etc.

CHAPTER 2. The theoretical rollazo .
This property of numbers is known from Tartaglia times (s. XVI) and even before:

The sum of the first n odd numbers is always n 2.

or also:

1 +3 +5 +...+( 2n-1) = n 2

Therefore, it is logical that if we add 8 consecutive odd numbers, square root result is 8.

Note: (2n-1) is how to represent an odd number either: if we take a number n is multiplied by 2 we get a number, and if you subtract 1 what remains sure is odd

CHAPTER 3. Demonstration practice.
This has a very simple demonstration using the known method of chickpeas: Take a humble
Thumb:

1 = 1 2

We will form a square by adding chickpeas:

1 +3 = 4 = 2 2

Note that we have added 3 chickpeas, and so we now have four chickpeas. 4 is a perfect square (2 2).
Note: The perfect squares are the numbers that can be put into a square using chickpeas. Let
. Now I will put more beans to form a square:

1 +3 +5 = 9 = 32
5

I added chickpeas. Now my square has 9 peas (3 ² )
Next step: Build a square a little larger.

1 +3 +5 +7 = 16 = 42

I added 7 chickpeas. Now my square is 16 beans (4 ² ).
Let's do it again. A square slightly larger:

1 +3 +5 +7 +9 = 25 = 5 2

can continue this procedure until you achieve the necessary number of chickpeas to make a stew, although from here strongly recommend to leave it after a suitable date.
And this is demonstrated and explained the fact that adding odd numbers, the result is always a perfect square.

Note: If beans are not available, you can use lentils or other legumes, but I assure you somo best understand this is to use slices of chorizo. In this case it is recommended practice until it is understood and beyond.