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Advanced Math Intended Audience
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The intended audience for the book is people who have a serious interest in mathematics, but who do not necessarily intend to become mathematicians. If you are interested in science or engineering, you may want to extend your study of high school mathematics as you pursue your technical education. This will give a deeper understanding of the overall unity, logic and rigorousness of mathematics. This book is designed to help provide that understanding by considering material you have seen before from a different viewpoint.
This book is designed for self-study, as well as possible classroom use. As a result, this book provides detailed proofs of the theorems discussed. This is particularly important for self-study, since the reader may not have access to someone who can critique their understanding of what is needed for a rigorous proof. If the steps of the proof require a lot of algebraic manipulation, it is explicitly given. If the student is capable of providing the intermediate steps himself, he can easily pass over the text. But if the student has not seen extensive algebraic manipulation previously, he can study the proofs in this book. While rigorous proofs may not be necessary for the simplest math one studies first, they help considerably to confirm intuition about the properties of mathematics that arises in situations that are less obvious.
There are three kinds of people who might enjoy reading this book. This book should also be appropriate for home schoolers.
Initially, this book was intended as a book for the general reader. The author has run across many people who impressed him as very intelligent, but who claimed "I just don't understand mathematics." He got curious as to why this might be so.
His basic conclusion is that people may have been introduced to mathematics in a manner that could be supplemented by material that emphasized different aspects of the subject. Most people seem to have a good intuitive feeling for everyday mathematics. If you ask them how much it would cost to buy enough gas to drive to New York, they can probably figure that out. If you ask them how much it costs to buy groceries for their family for a month, they don't throw up their hands and say they don't have any idea. However, when someone claims they can't understand fractions, it might be worth while to consider the same material in a different light.
Often one major difficulty encountered when considering the more abstract ideas behind mathematics is that is most people don't read equations the same way that mathematicians do. An equations is a statement, saying that one thing is the same thing as another, as in y = x + 5. A mathematician seeing this will say to himself "y is equal to x plus 5," and is comfortable with this even though x and y are symbols for numbers that haven't been explicitly specified. A person who is uncomfortable with math may freeze up at such an equation, perhaps because of the symbolic nature of x and y, or perhaps because he is not used to reading the equation as shorthand for a normal sentence.
For more complicated equations, the difficulties are compounded. An experienced mathematician won't necessarily read the whole equation in detail. The mathematician is first likely to note that a complicated expression is telling them that the quantity on the left hand side of the equal sign is the same thing as a complicated combination of quantities defined by the right hand side of the equation. Only next does one look at the actual structure on the right. The general reader also needs to develop this skill.
This book attempts to help the less experienced reader in this situation by starting with arithmetic that is likely to already be familar. If the reader first looks at equations that express what one already knows to be true, it is hoped that one will become more comfortable with the the idea of equations in general. Furthermore, far more than in a book intended for an experienced audience, this book fills in the intermediate steps. This is because it is not assumed that the audience could do this themselves. To use a trivial example, if one claims x2 - 2·x + 1 = 0 implies x = 1, in this book the intermediate logic would be explicitly spelled out.
The mathematics described here is the kind of mathematics one is exposed to up through high school in the United States. It mainly concerns algebra, but also includes some geometry and trigonometry. For high school students interested in scientific or technical studies, this book collects many of the important points of the mathematics that they have learned before and gives them a unified foundation. Such a student will not find the discussions of natural numbers, integers and rational numbers difficult, but the discuusion in the earlier chapters of the book will help considerably when understanding the new features added with the real and complex numbers are discussed. This should also help when one undertakes to master calculus and other advanced subjects one needs for a technical career.
This book makes clear that topics such as sequences and infinite series are not subjects that require calculus to understand. Rather, they are a fundamental tool for describing real numbers. If these topics are not introduced until one studies calculus, one has the additional burden of learning about these fundamental aspects of real numbers at the same time he is learning about the details of differential and integral calculus.
The author thinks that this book would make an especially appropriate book for the last semester of a high school program. It is felt that the normal experience with high school algebra makes the subject matter of the entire book within the grasp of a well-prepared student. And seeing this material in high school will be a considerable help when the student starts a university-level program in a scientific or engineering field, where the student will find it a challenge to master all the additional mathematics he will need for his principal area of interest.
In his own career, the author has been surprised at the number of times that his own work has caused him to approach a familiar field from a different perspective. When this happens, it is often amazing how much help it is to reexamine the foundations to remind oneself what are its fundamental principles. One might not think that this would come up in such a fundamental field such as mathematics, but consider an obvious example.
The engineers who first developed electronic calculators needed to design electronic circuits that implemented the mathematical operations of addition and multiplication. This is about as fundamental as you could ask. That it wasn't obvious how to proceed follows from the fact that different approaches were taken. Even though a binary representation of the data was dictated by the properties of the electronics, early calculators were also based on binary-coded decimal as well as purely binary representations.
So it is hoped that this book will be helpful to experienced technical people desiring a review of a subject they have been using for years, but for which they desire a reveiw of the basic foundations.