Number
Revolution
New Numerical Methods
The Rational Mean
New extremely simple high-order root-approximating arithmetical methods, that do not require either derivatives, geometry, decimals, or any Trial-&-Error tests, but just the simplest arithmetic.
The Rational Mean
and Brocot Sequences
Starting with the values:
0/1 1/0
Representing Zero and Infinite, Brocot Sequences are generated by successively calculating the Mediant; that way, all the reduced rational numbers are generated.
The Mediant is a particular case of the Rational mean that operates only with reduced rational numbers.
The New General
and unifying Concept
Description of the first video (English Subtitles)
The Fifth Arithmetical operation.The Missed Link:
*Time: 0:00 – 5:06 *
Brief description of the Mediant operation, and probable causes for which this operation has been ignored throughout math history.
*Time: 5:07 – 10:19*
The Rational Mean is a general and unifying operator that integrates fields that seem unrelated. The Mediant and all the known means as particular cases of the Rational mean. A new definition for rational and irrational numbers.
*Time: 10:20 – 15:16*
Elementary root-approximating rational processes –based on the Rational Mean– with a slow convergence rate. Daniel Bernoulli’s recurrence relations method and its connection to those rational processes.
*Time: 15:17 – 17:53*
The new Arithmonic Mean is a particular case of the Rational Mean and its role in generating root-approximating methods at high-order convergence rates.
*Time: 17:54 – 22:30*
New straightforward high-order root-approximating methods based on the particular case of the rational mean: The Arithmonic Mean, achieving any desired convergence rate.
*Time: 22:31 – 25:36*
A brief story on root-approximating methods, and the lack of precedents of these rational processes, from ancient times up to now.
The Rational Process: A new method based on the rational mean that can achieve any desired high-order convergence rate without using neither derivatives, geometry, or any trial-&-error tests but just the simplest arithmetic.
About the story on root-approximating methods: A brief analysis on the relevance of the existence of these elementary high-order methods when compared to the enormous structure that was required to create the infinitesimal calculus and.
*Time: 25:37 – 28:54*
The rational process is expressed in matrix form.
An extension of the rational process for the approximation of roots of the general algebraic equation.
*Time: 28:55 – 30:00*
The new Generalized Continued Fractions based on the Rational Mean. References on peer-reviewed journals, papers, and books.
*Time: 30:01 – 31:07 (End)*
The multiple properties of Number, far more than being just a representative of an equivalence class, I mean, just a decimal value.
End of Video