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Raider of the lost time
Join Date: Nov 2003 Posts: 6,025
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09-29-2005, 03:14 PM
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Originally Posted by GUILLE What do cos and sin ahve that tan hasn't? | The circular functions: sine, cosine, and tangent are all derivable through the validity of right triangle and the Pythagorean theorem. And the existence of the trigonometric identity sin²β+cos²β ≡1 justify the truth of right triangle and Pythagorean theorem.
__________________
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
| | | | The Thinker
Join Date: Mar 2005 Posts: 3,278
48  | |
09-29-2005, 03:26 PM
| | hi.
I have read your paper, but not attentive neough because I had other htings to do also. I'm going through it slowly now again.
I have atleast one question now:
What is meant by "dimensionless scalars"? What is a scalar?
I find it dificult to understand the concept of scalar, it's odd for me. More odd than human behaviour (and human behaviour is odd...oh yes...I can tell it's very odd). | | | | Raider of the lost time
Join Date: Nov 2003 Posts: 6,025
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09-29-2005, 03:45 PM
| Quote: |
Originally Posted by GUILLE What is meant by "dimensionless scalars"? What is a scalar? | A pure number. In this case, a pure rational number. By the way, you need to replace the word 'integral' and 'integer' with the word 'rational' for the paper to make more sense.
__________________
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
| | | | The Thinker
Join Date: Mar 2005 Posts: 3,278
48 | |
09-29-2005, 04:17 PM
| Quote: |
Originally Posted by AntonioLao A pure number. In this case, a pure rational number. By the way, you need to replace the word 'integral' and 'integer' with the word 'rational' for the paper to make more sense. | ok.
Now, what is a "pure number"? | | | | Raider of the lost time
Join Date: Nov 2003 Posts: 6,025
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09-29-2005, 04:59 PM
| Quote: |
Originally Posted by GUILLE what is a "pure number"? | A number that is not associated with any physical meaning like: force, energy, mass, charge, velocity, momentum, temperature, angular momentum, time, space, distance, weight, volume, area, etc.
__________________
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
| | | | The Thinker
Join Date: Mar 2005 Posts: 3,278
48 | |
09-30-2005, 03:00 AM
| Quote: |
Originally Posted by AntonioLao A number that is not associated with any physical meaning like: force, energy, mass, charge, velocity, momentum, temperature, angular momentum, time, space, distance, weight, volume, area, etc. | oh....I call those "fake numbers", for they are not onl real, or not only not related to reality indirectly, but not even related to properties of reality that humans have invented. | | | | Raider of the lost time
Join Date: Nov 2003 Posts: 6,025
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09-30-2005, 04:03 PM
| Quote: |
Originally Posted by GUILLE they are not onl real | Although all number systems are not considered physically real, the definition of real numbers has been done since the field of all rational and irrational numbers is called the real numbers, or simply the "reals," and denoted  . The set of real numbers is also called the continuum, denoted  . The set of reals is called Reals in Mathematica, and a number  can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real. The real numbers can be extended with the addition of the imaginary number i, equal to  . Numbers of the form  , where  and  are both real, are called complex numbers, which also form a field. Another extension which includes both the real numbers and the infinite ordinal numbers of Georg Cantor  is the surreal numbers. Plouffe's "Inverse Symbolic Calculator" includes a huge database of 54 million real numbers which are algebraically related to fundamental mathematical constants and functions. Almost all real numbers are lexicons, meaning that they do not obey probability laws such as the law of large numbers (Gruber 1991; Calude and Zamfirescu 1998; Trott 2004, p. 69).
__________________
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
| | | | The Thinker
Join Date: Mar 2005 Posts: 3,278
48 | |
09-30-2005, 04:44 PM
| | I know a bit about complex and real numbers. The term real is really unfortunate, but mathematicians did it on purpose to make themselves think that they have a place in reality, whiles, in reality, they don't.
What I don't really understand is what are hypercomplex and hyperreal numbers? | | | | Raider of the lost time
Join Date: Nov 2003 Posts: 6,025
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10-01-2005, 02:17 PM
| Quote: |
Originally Posted by GUILLE what are hypercomplex and hyperreal numbers? | In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions. Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 4 for the tessarines, 4 for the coquaternions, 8 for the octonions, 8 for the biquaternions, 16 for the sedenions). More precisely, they form finite-dimensional algebras over the real numbers. But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed — see fundamental theorem of algebra. In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R. An important property of *R is that it has infinitely large as well as infinitesimal numbers, where an infinitely large number is one that is larger than all numbers representable in the form 1+1+...+1.
__________________
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
| | | | The Thinker
Join Date: Mar 2005 Posts: 3,278
48 | |
10-01-2005, 04:18 PM
| Quote: |
Originally Posted by AntonioLao But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed — see fundamental theorem of algebra. |
What is the fundamental theorem of algebra?
Why does it impply that complex numbers are algebraicallt closed?
Why does the fact that the complex numbers are algebraically closed impply that the hypercomplex numbers don't create new number fields? Quote: |
Originally Posted by AntonioLao denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. | I didn't get this part. Can you re-explain it? | | | | | |
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