Archimedes Plutonium

2017-06-14 14:57:01 UTC

Now I need to do the Fundamental Theorem of Array, using Conservation Principle

....,

....,

Fundamental Theorem of Arithmetic

Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.

Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.

....,

Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.

....,

Fundamental Theorem of Calculus

...,

Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.

....,

.....,

Proof of FTC::

From this:

/|

/ |

/----|

/ |

/ |

_____

To this:

______

| |

| |

| |

----------

You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.

....,

Comment::

Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.

.....,

.....,

Fundamental Theorem of Algebra

...,

Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.

....,

Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).

Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.

Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.

Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.

....,

S_m

S_m+1

....,

- hide quoted text -

Comment:: now the order of the ARRAY is desirable to be of a history order. But, in many cases, we can lump and repeat statements. Remember, the ARRAY is going to be huge. I am talking of volumes that would fill a entire library, just on math ARRAY. And of course, in our computer age, we have the ARRAY accessible by computer.

.....,

AP

Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY (History of Mathematics) by AP

Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY

ARRAY of MATHEMATICS, governed by Conservation of Proof Principle

Each line is a data, or fact or theorem of mathematics. Some lines are Comments

Comment_p_i::

Comment_p_j::

Fact_k::

Statement_r_u::

Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.

Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.

....,

....,

S_i

S_i+1

....,

....,

....,

S_j

S_j+1

....,

Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have

a^2 +b^2 = c^2

....,

....,

Proof of PT::

Picture proof

a b

L 1 K

c

c

2

4

c c

I 3 J

b a

then

L a K

b b a

I b J

Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.

Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter

Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.

Proof-Statement:: Connect PB thus two triangles ABP and PBC. Since AP, PB, PC are all equal as radii, the two triangles are isosceles. The base angles are equal. Picture::

B

v u

A v P u C

v + (v+u) + u = 180

2v + 2u = 180

v+u = 90

Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.

Data:: A history timeline goes like this::

Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.

Pythagoras of Samos 570-495 BC

Democritus, Abdera Thrace, 460 - 370 BC Atomic theory

Theaetetus Athens 417- 369 BC

Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron

Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.

Archimedes 287 - 212 BC

(sources mostly Wikipedia)

According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.

.....,

Theorem Statement:: Angle Sum of Triangle interior angles is 180 degrees.

.....,

Comment:: the Angle Sum theorem is reportedly proven by the Pythagoreans as perhaps one of their few actual proofs given in a deductive method. I give it. And later I give my own new updated version that uses polygons rather than a parallel line argument.

.....,

......,

Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.

Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.

....,

....,

....,

Statement-Theorem:: There can Only Exist 5 Regular Polyhedra-- tetrahedron, octahedron, cube, icosahedron, dodecahedron. A Regular-Polyhedron is a 3rd dimensional object with surfaces of regular-polygons. All of its surfaces, edges and vertices are the same. Tetrahedron has 4 equilateral triangles, where 3 triangles meet at each vertex. The octahedron has 8 equilateral triangles, where 4 triangles meet at each vertex. The cube or hexahedron has 6 squares, where each vertex has 3 squares meeting.

The icosahedron has 20 surfaces of equilateral triangles and each vertex has 5 triangles meeting. And finally the dodecahedron has 12 surfaces of pentagons where each vertex has 3 pentagons meeting.

Proof-Statement ;; total angles at each vertex must be less than 360 degrees. The vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.

Otherwise, the surfaces would lie flat or disassemble or not form a solid in third dimension.

For Surfaces:

3 triangles at each vertex: Tetrahedron

4 triangles at each vertex: Octahedron

5 triangles at each vertex: Icosahedron

6 triangles at each vertex would be 360 degrees, no figure possible

3 squares at each vertex: Cube

4 squares at each vertex would be again 360 degrees, no figure possible

3 pentagons at each vertex is 3*108 degrees: Dodecahedron

4 pentagons at each vertex is 4*108 degrees and no way possible.

Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.

Fact:: Theaetetus proved Only 5 Regular Polyhedra Exist and Plato was so enamored of the regular polyhedra they are often called the Platonic Solids. Plato thought the 4 of the solids were the atoms of Democritus Atomic theory where tetrahedron was fire, cube was Earth, octahedron was air, and icosahedron was water. The dodecahedron was so special to Plato he made it the Universe.

Comment:: It would be fitting, if Plato made the Universe the dodecahedron, in other words God as atom as Dodecahedron, fitting that the final words in the Euclid Elements is words of God, the 5 Regular Polyhedra are the Only ones to exist.

....,

S_k

S_k+1

....,

....,

Fundamental Theorem of Arithmetic

Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.

Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.

....,

Comment:: Here we have the fine example of where the proof statement is smaller in length than the theorem-statement. But it only goes to show, that if you have a valid proof, you cut away the fat, and that is ironic since FTArithmetic is UPFAT, unique prime factorization theorem.

....,

Fundamental Theorem of Calculus

...,

Statement of Fundamental Theorem of Calculus: The integral of calculus is the area under the function graph, and the derivative of calculus is the dy/dx of a point (x_1, y_1) to the next successor point (x_2, y_2). The integral is area of the rectangle involved with (x_1, y_1) and (x_2, y_2), and the derivative is the dy/dx of y_2 - y_1 / x_2 - x_1. Prove that the integral and derivative are inverses of one another, meaning that given one, you can get the other, they are reversible.

....,

.....,

Proof of FTC::

From this:

/|

/ |

/----|

/ |

/ |

_____

To this:

______

| |

| |

| |

----------

You can always go from a trapezoid with slanted roof to being a rectangle, by merely a midpoint that etches out a right triangle tip which when folded down becomes a rectangle.

....,

Comment::

Here we see that a geometry diagram is ample proof of Fundamental Theorem of Calculus. And a geometry diagram is preferred for its simplicity and brevity, as the Pythagorean theorem started to do with mathematics in Ancient times. Other pieces and parts of the proof are scattered among the 10^60 facts and data of the Array of math. Here we show that a Statement is proven by a kernel of math knowledge-- you take a rectangle and procure a right triangle from the midpoint of the width and form a trapezoid for derivative and then reform back to a rectangle for integral. The derivative and integral pivot back and forth on a hinge at the midpoint of the rectangle width. The two operatives of integration and differentiation are reversible operators.

.....,

.....,

Fundamental Theorem of Algebra

...,

Old Math statement of FTA:: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number A is a complex number with an imaginary part equal to zero.

....,

Comment:: New Math statement theorem of FTA need not be so long, because in New Math we recognize that sqrt-1 is simply A/0 where A is a real-number (New Real Number).

Theorem-Statement of FTA:: given any polynomial with New Real Numbers, that there always exists at least one New Real Number, call it A, as a solution.

Proof of FTA: x^2 +1 = 0, goes to 1/x^2 = -1, with solution x=0 as 1/0 = sqrt-1, thus any polynomial is 1/polynomial = +-k has at least, one solution for A.

Comment:: Notice the beauty of the Statement of Theorem then Proof are almost identical in length of words. This is what happens when you have a true proof of a statement in mathematics.

....,

S_m

S_m+1

....,

- hide quoted text -

Comment:: now the order of the ARRAY is desirable to be of a history order. But, in many cases, we can lump and repeat statements. Remember, the ARRAY is going to be huge. I am talking of volumes that would fill a entire library, just on math ARRAY. And of course, in our computer age, we have the ARRAY accessible by computer.

.....,

AP

Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY (History of Mathematics) by AP

Ancient Greeks, it was Euclid's Elements, by 2017 it is AP's ARRAY

ARRAY of MATHEMATICS, governed by Conservation of Proof Principle

Each line is a data, or fact or theorem of mathematics. Some lines are Comments

Comment_p_i::

Comment_p_j::

Fact_k::

Statement_r_u::

Fact:: First known use of Pythagorean theorem was with Babylonian and Egypt math using 3,4,5 as a tool in building.

Comment:: Euclid was Ancient Greek but now in 2017 we start a new fresh encyclopedia on mathematics and I chose to call it the ARRAY, where not one theorem is a guiding principle, but rather a Conservation Principle as the guiding light for the Array. This means that proofs are of almost equal size as the theorem statement.

....,

....,

S_i

S_i+1

....,

....,

....,

S_j

S_j+1

....,

Statement of Pythagorean Theorem, given any right-triangle with sides a, b, c, that we have

a^2 +b^2 = c^2

....,

....,

Proof of PT::

Picture proof

a b

L 1 K

c

c

2

4

c c

I 3 J

b a

then

L a K

b b a

I b J

Comment:: Notice the length of the statement of theorem is about equal length as the proof-statement.

Fact: the Thales theorem or called Inscribed Angle theorem is that the triangle inside a circle with one side as diameter of circle always generates a right triangle. This theorem is one of the oldest proven theorems in math history. We have history data of its proof by Thales. And is a theorem that forges a path to the trigonometry from the unit circle with right triangles whirling around inside the unit circle. When we adapt the theorem to the radius instead of diameter

Statement-theorem:: You have a circle O with center at P and points A, B, C where AC is the diameter, then the angle at ABC as seen in picture below as v+u, is a 90 degree angle.

Proof-Statement:: Connect PB thus two triangles ABP and PBC. Since AP, PB, PC are all equal as radii, the two triangles are isosceles. The base angles are equal. Picture::

B

v u

A v P u C

v + (v+u) + u = 180

2v + 2u = 180

v+u = 90

Comment:: In the early history of mathematics, theorems of math with a deductive proof show up at the time of the Ancient Greeks, Thales being one of the earliest.

Data:: A history timeline goes like this::

Thales of Miletus 624-546 BC magnetism and Thales theorem about inscribed triangle in circle, and a precursor of Atomic theory only with arche-- water.

Pythagoras of Samos 570-495 BC

Democritus, Abdera Thrace, 460 - 370 BC Atomic theory

Theaetetus Athens 417- 369 BC

Plato Athens 426-347 BC theory that four solids of matter correspond to four elements and 5th is the universe-ether as a dodecahedron

Euclid Alexandria 360- 270 ?? BC his textbook Elements would become history's first Deductive and Logical Science and based on the 5 only existing regular polyhedra as a model. The term "Elements" comes from the Greek meaning guiding principle.

Archimedes 287 - 212 BC

(sources mostly Wikipedia)

According to Proclus, the term "element" was used to describe a theorem that is all pervading and helps furnishing proofs of many other theorems.

.....,

Theorem Statement:: Angle Sum of Triangle interior angles is 180 degrees.

.....,

Comment:: the Angle Sum theorem is reportedly proven by the Pythagoreans as perhaps one of their few actual proofs given in a deductive method. I give it. And later I give my own new updated version that uses polygons rather than a parallel line argument.

.....,

......,

Theorem Statement of FTArith.:: Also called the Unique Prime Factorization Theorem, for it is about Natural numbers, the set 1, 2, 3, 4, etc etc and that they can be expressed each, beyond the number 1, as a unique product of prime numbers. For example 7 = 7, and 10 = 2*5 and 24 = 2*2*2*3. We leave out 1 for it is unit, and leave it out for then we have no uniqueness since 1 = 1*1*..*1 any number of 1s.

Proof-Statement of FTArith:: If N is prime end of job. If N is composite divide until only primes remain. Are these remaining primes unique? Primes p1, p2, ..p_k if not unique and another collection of primes q1,q2, . .q_j different from p's also equals N would violate the prior proved theorem that if prime r divides uv then r divides u or v.

....,

....,

....,

Statement-Theorem:: There can Only Exist 5 Regular Polyhedra-- tetrahedron, octahedron, cube, icosahedron, dodecahedron. A Regular-Polyhedron is a 3rd dimensional object with surfaces of regular-polygons. All of its surfaces, edges and vertices are the same. Tetrahedron has 4 equilateral triangles, where 3 triangles meet at each vertex. The octahedron has 8 equilateral triangles, where 4 triangles meet at each vertex. The cube or hexahedron has 6 squares, where each vertex has 3 squares meeting.

The icosahedron has 20 surfaces of equilateral triangles and each vertex has 5 triangles meeting. And finally the dodecahedron has 12 surfaces of pentagons where each vertex has 3 pentagons meeting.

Proof-Statement ;; total angles at each vertex must be less than 360 degrees. The vertex angle summation has to be always less than 360 degrees and always have more than two regular polygons participating.

Otherwise, the surfaces would lie flat or disassemble or not form a solid in third dimension.

For Surfaces:

3 triangles at each vertex: Tetrahedron

4 triangles at each vertex: Octahedron

5 triangles at each vertex: Icosahedron

6 triangles at each vertex would be 360 degrees, no figure possible

3 squares at each vertex: Cube

4 squares at each vertex would be again 360 degrees, no figure possible

3 pentagons at each vertex is 3*108 degrees: Dodecahedron

4 pentagons at each vertex is 4*108 degrees and no way possible.

Comment:: When you fully define these regular Polyhedra, that takes up a lot of space. And the proof of that statement will be of approx. equal size.

Fact:: Theaetetus proved Only 5 Regular Polyhedra Exist and Plato was so enamored of the regular polyhedra they are often called the Platonic Solids. Plato thought the 4 of the solids were the atoms of Democritus Atomic theory where tetrahedron was fire, cube was Earth, octahedron was air, and icosahedron was water. The dodecahedron was so special to Plato he made it the Universe.

Comment:: It would be fitting, if Plato made the Universe the dodecahedron, in other words God as atom as Dodecahedron, fitting that the final words in the Euclid Elements is words of God, the 5 Regular Polyhedra are the Only ones to exist.

....,

S_k

S_k+1