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
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:
I added chickpeas. Now my square has 9 peas (3 2 )
Next step: Build a square a little larger.
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.
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 
1 +3 = 4 = 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
5
Next step: Build a square a little larger.
1 +3 +5 +7 = 16 = 42
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.
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