1. ## Re: Post 664 Euclid's Formulae

Originally Posted by BuleriaChk
Actually, the equation

, where the null vector means the dimensions don't exist for m and n at the point of intersection of the legs of the right triangle. (that is, the unit vectors and are orthogonal (independent); there is no component of projected onto and vice versa.)

The Binomial Theorem is as affirmation that Descartes was correct; that is, that there are independent dimensions, and geometric objects depend on them. John Gabriel at least (sort of) recognized the existence of two dimensions, since he did mention secants, but not curvature and continuity in functions. Grapes (and, I suspect, Wiles) don't even seem to recognize the existence of independent dimensions, effectively saying Descartes was wrong (and thus Newton and the Binomial Theorem), and that number theory is complete and consistent on a single number line (i.e., one dimension)

Some wannabe number theorists ignore Descartes altogether ( and thus geometry, and the concepts of areas, volumes, vectors, hyper-volumes, etc.) so they can concentrate on basic arithmetic operations that don't involve division, since fractions require two dimensions (a fraction is the slope of some straight line, and is a ratio that requires two independent dimensions (x,y) in the expression z = y/x - if x = 0, its dimension does not exist, so z is undefined.

(That is, if the dimension x has a metric (and as a real number, it must), then 0 can be defined as the midpoint of any length in that dimension. If there are no lengths, there is no dimension), and the fraction is undefined.

Assuming two dimensions exist, Fermat's proof boils down to:

1. For any three positive integers a,b, and c, they are all integers, or at least one of them is not.
(a) If one of them is not an integer, Fermat's Theorem doesn't apply to begin with.
2. If two dimensions exist, and at least two of the integers are on different dimensions, then there is a triangle.
3. The triangle is either a right triangle or it is not.
4. If the triangle is not a right triangle, then there will always be "rem(a,b,2)" elements in the calculation of its area, the Binomial Theorem holds, and Fermat's theorem is correct for n = 2.
5. If the triangle is a right triangle, then either it is Pythagorean or it is not.
(a)If the triangle is Pythagorean, it is the exception to Fermat's Theorem for the case n = 2, which is why Fermat restricted n to n > 2. Rotating the triangle around the point intersection of one of its legs and the hypotenuse defines a circle, whose area and circumference is a transcendental number involving
6. If the triangle is not Pythagorean, then it satisfies the Binomial Theorem for the case n=2, and rem(a,b,2) > 0
7. The Binomial Theorem is then easily extended to the case n > 2 to prove Fermat's Theorem, since rem(a,b,n) always exists and is greater than zero.

If the right triangle is considered to be one half of a rectangle (2A=ab) then the side of an equivalent square is
The above is mostly nonsense.

Below is a prime example of how dishonest you are with yourself:
(for a square, set a = b), so that amd the area is irrational, where S is the length of the square - equivalent side.
The area of a right triangle with sides 3 and 4 is 6, definitely not irrational.

The equation for area is , something most kids know before they're 12. It follows directly from your first equation. You would've realized that as well if you were honest with yourself.
For Pythagorean triples, the geometric object is a circle; , , and the area is transcendental, and thus the one-dimensional circumference.

The Special Theory of Relativity is actually a case of generating real numbers for the basic concepts of independence of (c,v) (where the relation is complete because they are related by scaling - i.e. ) and continuity, where the "energy" and "momentum" are independent (ct,vt'). However, it shows that if there is only one positive particle (conceived of as a real number), then (the ultimate "Black Hole", since there is no additional v/c where v > c. "Black Holes have no hair.

Philosophically, it means there is only The One... that is, me (the "wow" of physics).

Quantum mechanics assumes that one can't be sure about that, since one occasionally gets a sunburn (the "ow" of physics), so it is a statement ("There is me and thee, but I'm not absolutely sure of thee, especially if I am wearing sun-tan lotion

(The Egyptians may have been right about the sun god...

Wittgenstein declares there is no possibility of talking only to one's self, which is equivalent to asking "what language did you speak before you were born?"...

Goedel (and Einstein) assumes there are two observers, but if one says his language is complete, the other guy just adds an integer to the conversation ....

2. ## Re: Post 664 Euclid's Formulae

Originally Posted by grapes
The above is mostly nonsense.

Below is a prime example of how dishonest you are with yourself:

The area of a right triangle with sides 3 and 4 is 6, definitely not irrational.

The equation for area is , something most kids know before they're 12. It follows directly from your first equation. You would've realized that as well if you were honest with yourself.
You simply howl "nonsense" when you don't understand the concepts involved.

(In the reference to the area of the rectangle, the fact that the circle can't be squared), because the area of the equivalent square is irrational and the area of the circle is transcendental, and both require a geometry in two, count 'em, TWO dimensions.

Read the post again; that had already been corrected to reference the side S of the equivalent square before you posted, rather than the area of the rectangle by the time you posted (or if it hadn't, I hadn't read your post yet, but corrected it in the meantime).

And if you had understood the point I was making, you simply would have suggested I corrrect it, rather than calling it "nonsense".

It has nothing to do with me being dishonest, it has everything to do with your being, uh, challenged...

You simply do not understand the concept of area vis a vis independence in Cartesian coordiates. Rather than trying to uncderstand the foundation of the argument, you jump on trivial mistakes, which do not affect the relevance of the discussion, only ttat I was typing fast. I don't have a proof reader, and try not to make mistakes - I am only human. But the foundation of the argument is correct.

You, on the other hand, have not had a single oriiginal idea that I have read in thsi whole thread, and so have contributed nothing except proof reading (for which I thank you, while rejecting your cries of frustration that you can't contribute anything else.)

Like I said, I'm sure others are reading this and can judge for themselves.

And I'm outta here, if that's all Grapes has to say (a tale told by a mathematical mind at the third grade level), until I discuss the relation of the Euler identity to STR and the Pauli matrices...

3. ## Re: Post 664 Euclid's Formulae

And I'm outta here, if that's all Grapes has to say, until I discuss the relation of the Euler identity to STR and the Pauli matrices...

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