COMPLEX NUMBERS AND VECTORS IN THE SECONDARY CURRICULUM INTRODUCTION Complex numbers and vectors are both important areas of study within the senior secondary mathematics curriculum. The rationale for the inclusion of complex numbers in the curriculum is often related to: arguments for completeness of algebraic analysis of polynomial functions and the solution of related equations consideration of certain types of transformations of the (complex) plane, in particular those involving combinations of dilations and rotations, as well as some curves and regions in the complex plane Students typically encounter complex numbers in the guise of some kind of special number that enables one to extend certain algebraic manipulations on quadratic functions of a real variable with real coefficients to ensure that the rule of any quadratic function q]g x = ax + bx + c can be expressed as a product of two linear factors, and the equations q]g x = 0 always has two (not necessarily distinct) roots. The best problems can appear deceptively simple, but can create the need to explore new ways of thinking about our world and how we choose to describe it. A small problem may allow an insight. Student Activity 2.1 can be used to illustrate this point to students.
Which is the smallest integer that can be expressed as a sum of consecutive integers in a given number of ways?
Alternating iterations of the Smarandache function and the Euler phi-function respectively the sum of divisors function. Some light is thrown on loops and invariants resulting from these iterations. An important question is resolved with the amazing involvement of the famous Fermat numbers.
A particularly interesting subject is the Smarandache partial perfect additive sequence, it has a simple definition and a strange behaviour.
Smarandache general continued fractions are treated in great detail and proof is given for the convergence under specified conditions.
Smarandache k-k additive relationships as well as subtractive relationships are treated with some observations on the occurrence of prime twins.
A substantial part is devoted to concatenation and deconcatenation problems. Some divisibilty properties of very large numbers is studied. In particular some questions raised on the Smarandache deconstructive sequence are resolved.
In elementary mathematics, there are many difficult and interesting problems not connected with the name of an individual, but rather possessing the character of “mathematical folklore”. Such problems are scattered throughout the wide literature of popular (or, simply, entertaining!) mathematics, and often it is very difficult to establish the source of a particular problem.
These problems often circulate in several versions. Sometimes several such problems combine into a single more complex one; sometimes the opposite happens and one problem splits up into several simple ones. Thus it is often difficult to distinguish between the end of one problem and the beginning of another. We should consider that
in each of these problems we are dealing with little mathematical theories, each with its own history, its own complex of problems and its own characteristic methods, all, however, closely connected with the history and methods of “great mathematics”.
The theory of Fibonacci numbers is just such a theory. Derived from the famous “rabbit problem”, going back nearly 750 years, Fibonacci numbers, even now, provide one of the most fascinating chapters of elementary mathematics. Problems connected with Fibonacci numbers occur in many popular books on mathematics, are discussed at
meetings of school mathematical societies, and feature in mathematical competitions.
The present book contains a set of problems which were the themes of several meetings of the schoolchildren*s mathematical club of Leningrad State University in the academic year 1949-50. In accordance with the wishes of those taking part, the questions discussed at these meetings were mostly number-theoretical, a theme which is
developed in greater detail here.
The investigation of three problems, that of perfect numbers, that of periodic decimals, and that of Pythagorean numbers has given rise to much of elementary number theory, and the author shows how each result gives rise to further results and conjectures. He treats not only results and theorems (”solved problems”) but also questions that are still open and conjectures (”unsolved problems”), making this a most exciting and unusual treatment. The author, a past editor of Mathematics of Computation, presents research done in the fifteen years between the first and second editions, with emphasis on results that were achieved with the aid of computers. The volume includes a substantial Bibliography.
While optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature.
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. The book contains definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ).
Mathematical Logic with Special Reference to the Natural Numbers
About ten years ago I conceived the idea of writing a book on the natural numbers because I thought that what had appeared up till then seemed to have reached a point where there was a certain amount of completeness - of course there never will be absolute completeness — and this is one of the attractions of the subject. Anyway it was not until I had retired that I had the time to get down to the task properly. The result is a book which begins with an account of formal languages including the two most basic, namely, the propositional calculus and the predicate calculus, and then goes on to arithmetic; beginning with a very simple arithmetic; finding this inadequate; extending it to overcome this inadequacy; finding the resulting system, though richer in modes of expression, still, but for a different reason, inadequate; extending this in turn to remedy this inadequacy; finding the resulting system has lost some of the ‘ nice’ qualities of its predecessor, but is again, for a new reason, inadequate; extending this and so on. Before I come to develop arithmetic formally, it is convenient to have a primitive notation for the natural numbers (mainly to avoid lengthy circumlocutions) from which the concept of order and the operations of addition and multiplication can easily be obtained. I use sequences of tally marks, this is sufficient for our purposes. The real difficulty with arithmetic, as with other things, enters with the universal quantifier, when we want to make statements about all natural numbers. This use of tally marks is mentioned in the text but in the main is left to the reader to fill in. There are several topics absent from the book which might have been
included, these are partly off the main line of development, partly appli-applications of the general theory developed, partly sidelines, etc. Among these topics are: recursive analysis, constructive ordinals, recursive
equivalence types, recursive probability theory, the word problem, algorithms, finite automata, A-conversion, combinations, productions, intuitionism, various forms of propositional calculus, many-valued logics, and so on. Of these the constructive ordinals are mentioned several [XV] times because now and again we come across a process which can be continued into the constructible transfinite, but we do not go into it further.
The matter developed in this book was developed over the years in a course of lectures delivered at Cambridge, except that very little was said about the contents of Chs. 10, 11, 12, so these three chapters have not come under the fire of criticism of young scholars, and I feel that in consequence that they are not of the same quality as the earlier chapters, particularly the account of cut elimination in Ch. 10. The remaining chapters have been fairly well thrashed out in lecture and I am very appreciative of the comments of my classes and of the elegant onstra- demonstrations they gave me from time to time. I hope that I have acknowledged them all. With regard to the language in which the book is written, this is meant to consist of instructions and descriptions and occasionally of pointing out that such and such a procedure would lead to an impossible situation. Later in the book, when treating with ultra products I have transgressed and used Zorn’s lemma, but a purist can tear that piece out of the book. Each chapter is followed by a short historical account of the matter treated in that chapter, it is this way that I make acknowledgement to those who first invented the matter, if I have made omissions then I apologize. After the historical account there follow a few examples. Many more examples can be found in books by Rogers A967), Shoenfeld A967) and Church A956). I must thank Professor R. Harrop and Dr N. Routledge for comments
on a former, now completely discarded draft which developed a much more complicated system. The present system owes its simplicity to the iterator symbol. I must also thank Dr G. T. Kneebone for reading the draft of Chs. 1-7 inclusive and providing valuable comments, and Drs T. J. Smiley and L. Drake for reading the draft of the remaining chapters and again providing valuable comments; also to the University Press for courtesy and consideration during the production of the book, and finally to my wife for help with the tedious business of making an index.
History of the Theory of Numbers, Volume III: Quadratic and Higher Forms (History of the Theory of Numbers)
Leonard Eugene Dickson, “History of the Theory of Numbers, Volume III: Quadratic and Higher Forms (History of the Theory of Numbers)”
Publisher: Dover Publications | Pages: 320 | ISBN: 0486442349 | DjVu | 3.64 MB
This 3rd volume in the series History of the Theory of Numbers presents material related to Quadratic and Higher Forms. Volume III is mainly concerned with general theories rather than with special problems and special theorems. The investigations deal with the most advanced parts of the theory of numbers. 1919 edition.
History of the Theory of Numbers, Volume ll: Diophantine Analysis (History of the Theory of Numbers)
Leonard Eugene Dickson, “History of the Theory of Numbers, Volume ll: Diophantine Analysis (History of the Theory of Numbers)”
Publisher: Dover Publications | Pages: 832 | ISBN: 0486442330 | DjVu | 12 MB
This 2nd volume in the series History of the Theory of Numbers presents material related to Diophantine Analysis. This series is the work of a distinguished mathematician who taught at the University of Chicago for 4 decades and is celebrated for his many contributions to number theory and group theory. 1919 edition.
History of the Theory of Numbers, Volume I: Divisibility and Primality (History of the Theory of Numbers)
Leonard Eugene Dickson, “History of the Theory of Numbers, Volume I: Divisibility and Primality (History of the Theory of Numbers)”
Publisher: Dover Publications | Pages: 512 | ISBN : 0486442322 | DjVu | 7.09 MB
This 1st volume in the series History of the Theory of Numbers presents the material related to the subjects of divisibility and primality. This series is the work of a distinguished mathematician who taught at the University of Chicago for 4 decades and is celebrated for his many contributions to number theory and group theory. 1919 edition.