algorithmic rationals

[AntonioLao]

The first known procedure for finding the greatest common factor (GCF) of two positive integers is Euclid’s Algorithm. Euclid lived and died about twenty three hundreds years ago between the cities of Athens and Alexandria in ancient Greece. His published works, the thirteen books of the Elements earned him the honor as founder of a basic branch of mathematics now known as Euclidean geometry. However, his GCF algorithm could also have made him foremost among algebraists. Unfortunately his failures to solve by geometric means problems of Apollonius conic sections relinquished the title as founder of algebra to Arab mathematician Mohammed ibn-Musa al-Khowarizmi (Mohammed, son of Moses, Khowarezmite) and in fact the word ‘algebra’ is a Latin variant of the Arabic word ‘al-jabr’ or ‘al-jebr.’ He published a book ‘ilm al-jabr wa’l-mukābala ‘The Science of Restoring What Is Missing and Equating Like with Like.’ Nevertheless, Euclid’s Algorithm in its completed form became the foundation of several branches of modern mathematics including the theory of ideals. Moreover, the use of ideals finally helped and proved Fermat’s Last Theorem.

In its simplest form, the GCF algorithm starts with two positive integers A and B, where A is greater than B then proceeds by subtracting B from A as many times until the difference is less than the subtrahend. Repeat the procedure where the subtrahend now becomes the minuend and the difference becomes the next subtrahend getting an even smaller difference. If the final difference is zero then the previous one was the sought after GCF. If the final difference is unity then the GCF is also unity. Extending this procedure to rationals and as a special case then A – B = AB derive as product of two rational functions: A = B/(1-B) and B = A/(1+A). Discontinuities exist at -1 and 1 and when A vanishes then B is identically zero, vice versa. When A is the kinetic energy and B is the potential energy then AB is the square of energy and (A – B) is the Lagrangian function. Logically the square of energy cannot equal the difference of energies by dimensional analysis. However the variational principle of least action indicates that it is true only if both energies are rationals and that only the difference can be multiplied by a unit energy factor.

[mkirkpatrick]

Antonıo we need to from A To Z to solve thıs sucker.fusıon lıes where deep reason begıns
we are overlookıng the ABC and jumpıng straıght to D.

Glad to see you back postıng my frıend.am ın Turkey at the moment.very hot here 38c
need to fınd a shady palm tree?


warm regards mıchael.

[AntonioLao]

we need to from A To Z to solve thıs sucker.fusıon lıes where deep reason begıns we are overlookıng the ABC and jumpıng straıght to D....am ın Turkey at the moment.very hot here 38c need to fınd a shady palm tree

If it was as easy as ABC someone would have already solved it. As for the heat in Turkey drinking plenty of water is probably the only remedy.