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[Solved] Pythagorean arithmetic mean in the first circle.  

 
doubop
(@doubop)
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One of the discoveries of Pythagoras was to have established the formula of the arithmetic mean. This formula we use to calculate our quarterly scores. As part of the number line we can reformulate it. From the first circle of the number line. Assume first of all that the number line corresponds to the dimension one as it is defined in a system of Cartesian coordinates. Each point of the line can be represented by its coordinates of abscissa which represents at the same time its position compared to the origin. Here in fact several concepts relating to numbers are also superimposed: quantity and the concept of order. The pythagorean formula for the medium term is as follows. Let two whole numbers a and b. Let us put a < b, so as not to have any sign problem at this stage.

 

c = (a + b)/2

 

The number c is halfway between the two. a and b are symmetrical with respect to c. we can demonstrate that c is unique by calculating:

 

a + (a + b) / 2 and b- (a + b) / 2

 

we get in both cases:

 

a + (a + b)/2 = a + c

and

 

b − (a + b)/2 = b − c

 

a + (a + b)/2 = a + a/2 + b/2 = 3a/2 + b/2 = (3a + b)/2

but

a + b = 2c

 

so

(3a + b)/2 = (2a + a + b)/2 = (2a + 2c)/2 = a + c

(3a + b)/2 = a + c

3a/2 + b/2 = a + c

3a/2 − a = c − b/2

a/2 + b/2 = c

In the case of the first circle we have a = 2, b = 3, and c = 5/2. The centers of the circles are the arithmetic mean of two consecutive prime numbers. Notice that the diameter is 1. And the radius 1/2. It gives a kind of similarity with half-step one step, in the spelling of intervals. As radius is not diameter, we have to consider both. it's like in music. And a relation with Pi that will examine in pi section.

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Posted : 30/05/2020 1:22 am