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PDF Free Download | Number Theory A Historical Approach by John Watkins
Many years ago, I was sitting in my second-grade classroom when I made what I thought was a remarkable discovery: there is no largest number.
Whatever number I thought of, I realized I could just add one to it and get a larger number. This remains to this day one of my most vivid childhood memories. What I had “discovered” was the “dot, dot, dot” in the infinite collection of the numbers.
1, 2, 3, 4, 5, ... ,
which we know as the natural numbers. As simple as this collection of numbers may appear, humans have been studying these numbers for thousands of years,
learning their properties, uncovering their secrets, finding one marvelous thing after another about them, and still we have only barely begun to tap this remarkable and ever-flowing current of ideas. These are the numbers we intend to study.
This book is an introduction to the study of the natural numbers; it evolved from courses I have taught at Colorado College, ranging from a general math course designed for nonmajors to a far more rigorous sophomore-level course required of all math majors. I hope to preserve several fundamental features of these courses in this book:
• Number theory is beautiful. It is fun. That’s why people have done it for thousands of years and why people still do it today.
Number theory is so naturally appealing that it provides a perfect introduction—either for math majors or for nonmajors—to the idea of doing mathematics for its own sake and for the pleasure we derive from it.
• Although number theory will always remain a part of pure mathematics (as opposed to applied mathematics), it has also in modern times become a spectacular instance of what the physicist Eugene Wigner called the “unreasonable effectiveness of mathematics” in that there are now important real-world applications of number theory.
One of the most useful of these applications came along several centuries after the original concepts in number theory were developed and will be explored in the chapter on cryptography.
• Number theory is a subject with an extraordinarily long and rich history. Studying number theory with due attention to its history reminds us that this subject has always been an intensely human activity.
Many other mathematical subjects, calculus, for example, would have undoubtedly evolved much as they are today quite independent of the individual people involved in the actual development,
but number theory has had a wonderfully quirky evolution that depended heavily upon the particular interests of the people who developed the subject over the years.
• Reading mathematics is very different from, say, reading a novel. It requires enormous patience to read mathematics. You cannot expect to digest new, and often complex, mathematical ideas in a single reading.
It is frequently the case that multiple readings are needed. You will discover that individual sentences, paragraphs, and even whole chapters must be read carefully several times before the key ideas all fall into place.
• One of the primary goals of the book is to use the study of number theory as a context within which we learn to prove things. Proof plays a vital role in mathematics and is the way we bridge the gap between what our intuition tells us might be true and the certainty about what is true.
You will encounter several quite different styles of proof as you read (and should feel free to skip any that you find either too difficult or simply not very interesting).
In many cases, an informal argument or even a carefully examined example is sufficient to discover truth, but in other cases a far more rigorous and formal argument will be required to achieve certainty.
Another feature of our courses at Colorado College I hope to preserve in this book is the interactive nature of our classes. Learning mathematics requires active participation,
and this book should be read with paper and pencil in hand, and a good calculator or computer nearby, checking details and working things through as you go. Sometimes, in order to understand an idea, it is best to go through a few examples by hand.
Other times it is better to let a computer do the computations, and so an introduction to the computer software Sage has been provided at the back of the book. Sage is an extremely powerful aide to such computations and is a wonderful resource that can be used online or downloaded for free.