Pythagoras is not the inventor of the theorem that bears his name: the properties of the theorem were used, especially for the measurement of agricultural and real estate properties, in China and Mesopotamia already a thousand years earlier. He is ascribed to Pythagoras only because he was the first to prove that the theorem was valid in "absolute" (and/or only in relation to "finite" values of the sides of the right triangle). This is because the demonstration of its validity was obtained by him with a "geometric" and not "arithmetic" process, i.e. starting from "real" data (and, therefore, finite as all those of geometric figures are) and not "abstract" (as are those described by numbers, which can be both finite and infinite).

     Pythagoras starts from the underlying geometric figure and arrives at the determination of the value of c through a simple geometric reasoning that works "always", whatever the values of a and b are (and it works "always" whether we approach the problem in a geometric or arithmetic key):

     This is the geometric figure from which Pythagoras starts: a square whose side is equal to the sum of the sides a and b of the right triangle (of which the numerical value is known).

     If we construct a square inside it whose side has a value c (i.e., the hypotenuse of the starting triangle, of unknown numerical value), the figure is composed of this square and 4 triangles with a base and height equal to a and b.


     Translated into an arithmetic formula, this set can be described as follows:
                   c2 = (a + b) 2 – 4 x [(a x b) : 2]
     If we give any numerical value to a and b, the structure of the resulting geometric figure will always be the same and will lead us to the determination of the value of c.

Given a=3 and b=4 the formula becomes:
     c2 = (3 + 4)2 – 4 x [(3 x 4) : 2]
or:     

     c2 = 49 – 4 x [12 : 2]
or:            

       c2 = 49 – 24
therefore: 

     c2 = 25 and c = 5

     Arithmetic and mathematics (*) were born to solve practical problems. We face daily problems of a practical nature with which we interface through geometry (i.e.: deducing them and/or placing them in space). Arithmetic and mathematics are an abstraction of the practical problems that help solve them, but once solved, we bring them back to practicality.

     For this reason, the "imperial" units of measurement (foot, inch, yard, mile), so hated by Europeans, work better in the conversion from arithmetic to geometry (i.e. from theory to practice) work better: the foot can be divided by 2, 3, 4, 5, 6, 8, always obtaining finite measurements; 10 gives finite measures only if you divide it by 2 and by 5. Already if you divide it by 3 you get an inaccurate measurement, i.e. 3.3 periodic, therefore infinite.

     Yet, at first glance, almost all of us can perceive that in a room wall A measures a third of the wall B. But, if wall B measures 10 meters, using the decimal system, wall A will have an infinite value (3.3 meters periodic).

(*) Arithmetic focuses primarily on basic numerical operations such as addition, subtraction, multiplication, and division. Mathematics, on the other hand, encompasses a wide range of concepts, including algebra, calculus, number theory, statistics, and so on.