Fermat’s Last Theorem can Decode Nazi military Ciphers
... This phrase was a simple switching positions of the alphabet letters, which was (n-1), therefore, it would be the previous alphabet letter. This above phrase translates to “An easy cipher to break!” Such success of a translation entices one to prepare for more challenging ciphers. Now let’s tackle a ...
... This phrase was a simple switching positions of the alphabet letters, which was (n-1), therefore, it would be the previous alphabet letter. This above phrase translates to “An easy cipher to break!” Such success of a translation entices one to prepare for more challenging ciphers. Now let’s tackle a ...
REDUCTIO AD ABSURDUM* (Proof by contradiction) Y.K. Leong
... recent vintage - if one may consider one hundred years ago as recent enough. The subject itself has only been recently introduced to elementary levels of education. We are referring to set theory. We are all aware of the controversy raised by the introduction of sets at the primary school level at t ...
... recent vintage - if one may consider one hundred years ago as recent enough. The subject itself has only been recently introduced to elementary levels of education. We are referring to set theory. We are all aware of the controversy raised by the introduction of sets at the primary school level at t ...
PDF
... is finite, how can we determine the exact number of elements that S contains? Basic principles and techniques of enumerative combinatorics include: • the addition principle; • the multiplication principle; • the involution principle; • the inclusion-exclusion principle; and • the use of generating f ...
... is finite, how can we determine the exact number of elements that S contains? Basic principles and techniques of enumerative combinatorics include: • the addition principle; • the multiplication principle; • the involution principle; • the inclusion-exclusion principle; and • the use of generating f ...
Standard Index Form
... Reduce your workload and cut down planning Enjoy a new teaching experience Watch your students interest and enjoyment grow Key concepts focused on and driven home Over 100 files available with many more to come 1000’s of slides with nice graphics and effects. ...
... Reduce your workload and cut down planning Enjoy a new teaching experience Watch your students interest and enjoyment grow Key concepts focused on and driven home Over 100 files available with many more to come 1000’s of slides with nice graphics and effects. ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".