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What is the best mathematics reference book or site you have ever seen?
Under the assumption that by reading a book, you intend to use the book as both a reference for mathematical principles, as well as a source for a diverse set of exercises, I have a few recommendations, with very distinct strengths. Algebra and Trigonometry. Sheldon Axler is my first recommendation. It is a very dense book, with a very diverse set of exercises. This book will be very good if you are self-motivated and comfortable struggling through novel exercises as part of the learning process. Axler provides only the most fundamental explanation of concepts and a very sparse collection of example calculations. While this increases the "mathematics per page" in a sense, the real source of this books density is the exercise sets. Many books (including my second and third recommendations) provide many example problems and build the exercise sets out of variations of those example problems. Axler's text is very different from these books. Axler provides a succinct collection of founding principles, demonstrates them and builds his exercise sets to be diverse rather than repetitive. For each problem type, he offers two adjacent exercises, one of which has a solution provided in an appendix while the other is left unsolved. Many students find this sort of book frustrating since problems cannot be readily solved by a template-style solution, in contrast to problem sets from other books. I believe this is a better book for this very reason. It demands that the exercises be addressed not with a practiced rote method, but rather with the students reflection and insight regarding the principle addressed in the lesson. Problems of the latter sort will help the student learn the material far more than any verbose introduction of a subject loaded with worked-out examples. If on the other hand, you would prefer to see lots of worked-out examples and copy the methods in those examples to solve the exercises in the book, there are two titles that are good which come to mind. If you want to focus on Algebra without Trigonometry, I recommend Intermediate Algebra (Hardcover) (Developmental Math). Julie Miller, Molly O'Neill, Nancy Hyde (MOH), because the development of the Algebraic concepts are carefully developed to provide a broad, consistent foundation for material the builds upon it. If you need to also include Trigonometry, I recommend Precalculus (5th Edition). Robert F. Blitzer. It is basically the same as Algebra and Trigonometry also by Blitzer, except there is one additional chapter at the beginning of the latter and two at the end of the former. Again, Blitzer is very careful to ensure that his development is both consistent and rigorous. Since his book includes the trigonometric material it hasn't the room to do as much explaining as MOH, and it is more expensive. I recommend these two titles over others of this ilk, because t start from basic principles of number systems without overlooking too many necessary formalities and without introducing glaring inaccuracies. Having selected books for curricula, I have been shocked by some of the material that can be published in the introductory chapters of a math book at this level. Other rigorous books at this level can assume a certain level of understanding which could require that the reader look for references elsewhere, which is undesirable for many reasons which I will omit here. How can you decide which type of book will be better for you? I would ask yourself whether you enjoy working through a problem on your own without guidance. If so, choose Axler or another book with a very diverse set of exercises. If not try MOH or Blitzer, t will provide guidance without misleading you. Good luck.
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What mathematics is--and how to learn it! An Introduction to Mathematical Thinking. A comprehensive guide to the fundamental concepts of mathematics, presented in an accessible manner, and illustrated with real-world examples. For example, in the first chapter of this book, Alan Turing, the greatest logician of his age, describes the inner workings of his own thinking in relation to mathematics. Mathematical Plots. Mathematician Daniel Steiner defines a mathematical plot with five key features: the plot is structured; there is a beginning, a middle, and an end; the plot involves two or more things happening simultaneously; there is some kind of internal logic; and there is logic or correspondence among the plot elements. Mathematician Daniel Steiner's book “Mathematical Plots” presents one way to think about and represent mathematical plots. Mathematical Logic. In addition to presenting many familiar figures of representation--the “square circle,” “cat's breakfast,” “lazy Susan,” and “Fourier.
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