Friday, February 5, 2021

3021 - MATH - a learned discipline?

 -  3021 -  MATH  -  a learned discipline?  -  Mathematics is so amazing because it recreates and communicates so much of nature in its language and logic.  A simple “pi” can represent all the perfect circles in the Universe.  Their circumference is always 3.14 times larger than their diameter. 

--------------  3021  -  MATH  -  a learned discipline?

-   It is one of the laws of mathematics that “pi” is an irrational number that can not be formed as a fraction and as a decimal the numbers continue to infinity, never repeating,  pi  =  3.14176254254.…… etc

Yes, this is very irrational.

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-  Pythagoras (572-497 B.C.) introduced the term “ mathematics”.  He called it a learned discipline.  From that moment it has been controversial as to whether the discipline invents  or discovers the laws of mathematics, the laws of Nature.

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-  Here are some discoveries.  Take 365 days in a year.  You can get the number 365 by taking the sum of the  squares of 10, 11, and 12. 

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---------------   10^2 +  11^2  + 12^2  =  100 + 121 + 144  =  365

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-  Or, you can get the number 365 by taking the sum of the squares of 13 and 14.

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---------------  13^2  +  14^2  =  169  +  196  =  365

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-  How about the number 100.  You can get that number by summing the cubes of the first four numbers, 1 ,2, 3, and 4.

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---------------   1^3 + 2^3  +  3^3  +  4^3  =  1 + 8 + 27 + 64  =  100

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-  Then someone discovered that the sum of odd consecutive numbers always equals a number squared:

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---------------  1  =  1^2

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--------------- 1 + 3  =  2^2

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---------------  1+3+5  =  3^2

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---------------  1 + 3 + 5 + 7  =  4^2

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---------------  1 + 3 + 5 + 7 + 9  =  5^2

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-----------------  etc. …………….

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-----------------  Why is that? 

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-  Of course Pythagoras discovered that the square of the hypotenuse is always and everywhere equal to the sum of the squares of the other two sides.

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----------------  c^2  =  a^2  +  b^2

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-  But, also the discover was made that for any triangle, not just a right triangle, with the angles A, B, C and sides opposite a, b, c respectively.  

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--------------  a^2  =   b^2  +  c^2  - 2bc cosA

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-  The square of the side opposite the cosine angle is equal to the sum of the squares of the other two sides minus 2 times the product of these two sides and the cosine of the angle between them.  Another law of mathematics.

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-  The area of a triangle is ½ the base times the height.   A  =  ½ b*h

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-  But, the area of any triangle is    A  =  ½ a*b sinC


-  This ideas of areas bets even stranger for polygons.  The area of a polygon is equal to the number of points (vertices) plus ½ the number of boundary edges minus 1

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-------------  A  =  i  +  b/2 - 1

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-  “Epistemology” is the investigation into the limits of human knowledge.  How far can mathematics take us into our understanding of the Universe?

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-  Every even integer greater than 2 is the sum of 2 prime numbers:

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--------------  4  =  2 + 2

---------------  6  =  3 + 3

---------------- 8  =  3 + 5

---------------  10  =  3 + 7

---------------- 12  =  5 + 7

---------------  14  =  3 + 11

---------------- 16  =  5 + 11

---------------  etc.  ………………..

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----------------  Why is that ?

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-  Books have been written on each of these numbers, pi,  “e” the natural logarithm ,  the golden ratio 1.618  See the footnote for some of the book reviews.

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--------------  the Golden Ratio  =  (1 + 5^12) / 2  =  1.618 …….

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-  The Golden Ratio is the sum of 2 n umbers divided by the larger number:

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 ---------------  (a + b) / b  = 1.618

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----------------  (21  +  34  )  /  34  =    55 / 34  =  1.617

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----------------  10,946  +  6,675 ) / 10,946  =  17,711 / 10,946  =  1.618 

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-  “e”  the natural logarithm is in the most famous mathematical equation:

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----------------  e ^i*pi + 1 = 0

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-  The natural logarithm raised to the imaginary “pi” is equal to -1.  You can explore the limits of mathematical knowledge, it goes on forever:

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-----------------------------  Other reviews available:

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(1)  #1086  “  Formulas for space, time and velocity “

(2)  #1087  “  Formulas for space, time and velocity “

(3)  #1088  “  Formulas for space, time and velocity “

(4)  #1089  “  Formulas for space, time and velocity “

(5)  #1090  “  Formulas for space, time and velocity “

(6)  #1042  “  Calculating areas by counting dots“

(7)  #1041  “  The star with the golden ratio “

(8)  #745  “  The calculus of a circle “

(9)  #853  “  Math, the golden ratio “

(10)  #803  “  Transcendental numbers “e” and “pi” “

(11)  #649  “  The Greeks invented numbers “

(12)  #805 “  Gambling “

(13)  #796  “  Science and math “

(14)  #798  “  Science and math, Part II  “

(15)  #799  “  Science and math , Part III“

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February 4, 2021      MATH  -  a learned discipline?               1095      3021                                                                                                                                                           

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--------------------- ---  Friday, February 5, 2021  ---------------------------






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