An Introduction to Number Theory is a great introduction to the field for anyone who loves numbers, is fascinated by math, and wants to go further into the relationships among these mysterious objects.
What Is Number Theory?
Called "the queen of mathematics" by the legendary mathematician Carl Friedrich Gauss, number theory is one of the oldest and largest branches of pure mathematics. Practitioners of number theory delve deep into the structure and nature of numbers, and explore the remarkable, often beautiful relationships among them.
In this course, you'll cover all the fundamentals of this exciting discipline and explore the many different types of numbers:
- Natural numbers
- Prime numbers
- Integers
- Negative and irrational numbers
- Algebraic numbers
- Imaginary numbers
- Transcendental numbers
But in An Introduction to Number Theory, you'll also develop a deeper understanding of their nature and the relationships among them, and gain insights into the many branches of number theory, including:
- Elementary number theory, an exploration of the fundamental patterns involving the natural numbers
- Analytic number theory, which uses the techniques of calculus to establish its results, and focuses on the prime numbers and their patterns
- Algebraic number theory, which relies on arithmetic—the operations of addition, subtraction, multiplication, division—to find solutions to equations
- Algebraic geometry, a combination of algebra and geometry which reveals the important connection between solutions to certain equations and points on certain curves.
An Intriguing Landscape of Calculation
Using brain-teasing problems and fascinating anecdotes, Professor Burger offers deep insights into the complex and beautiful patterns that structure the world of numbers. For example, you'll encounter:
- The sieve of Eratosthenes: A simple but brilliant method for "sifting out" all the prime numbers within any sequence of natural numbers
- The golden ratio: A mysterious and powerful number that recurs throughout the natural and scientific worlds, and is believed by some to hold the key to aesthetic beauty
- The method of modular arithmetic: An alternative way of counting that focuses on remainders rather than quotients to open up new possibilities for the manipulation of numbers
- The relationship between algebra and geometry: Exactly how does an algebraic formula correspond to a line or curve? What can we understand about these two mathematical entities?
Number Theory in Everyday Life
You'll quickly see that number theory—though complex and intellectually challenging—is no "ivory tower" endeavor. Throughout the course, Professor Burger answers intriguing questions about how the work of number theorists touches our daily lives:
- How do modern computers "talk" to each other through vast remote networks?
- What technologies lay behind crucial security procedures, such as the encryption used when you pay with a credit card on an e-commerce website?
- How do the puzzling numbers found on everyday consumer products—the UPC labels, or bar codes—allow you to move quickly and easily through the check-out stand at the local market?
- Why does the chromatic musical scale include 12 notes?
- What is the meaning of the mysterious bank routing numbers on checks? How do they provide a way to identify specific bank accounts?
Mathematical Proofs: The Agony and the Ecstasy
You'll also experience the exhilarating but often heartbreaking process of mathematical proof. In creating a proof, a mathematician is at once akin to the lawyer and the artist, constructing logical structures that deduce newly discovered truths from previously accepted premises.
While some mathematical problems readily yield proofs, others prove to be more intractable. In this course, you'll hear about the trials, tribulations, and triumphs of great minds as they attempted to solve some of the most vexing "open questions" that have tickled the curiosity of mathematicians for centuries.
Consider, for example, the quest of Andrew Wiles. Since childhood, he dedicated himself to finding a proof to one of the most notorious open questions in mathematics, Fermat's last theorem. Professor Burger follows Wiles's frustrating disappointments and miraculous breakthrough in proving this elusive theorem. It's a story that demonstrates the discipline, mental acuity, and remarkable creativity required to establish new areas of knowledge.
And you'll learn about some of the open questions that remain, including the Collatz conjecture, a tricky puzzle for which the great 20th-century Hungarian mathematician Paul Erdös offered to pay $500 to anyone who could provide a solution.
Are You Ready for a Challenge?
These are the kinds of mathematical puzzles that make An Introduction to Number Theory a treat for anyone who loves numbers. While the course requires confidence with basic math concepts, Professor Burger provides clear and effective guidance for students at all levels.
With each tantalizing problem, Professor Burger begins with an overview of the high-level concepts. Next, he provides a step-by-step explanation of the formulas and calculations that lay at the heart of each conundrum. Through clear explanations, entertaining anecdotes, and enlightening demonstrations, Professor Burger makes this intriguing field of study accessible for anyone who appreciates the fascinating nature of numbers.
And while the course is demanding, you'll soon find that An Introduction to Number Theory offers rewards that are equal to its tests. Through the study of number theory, you'll experience an enlightened perspective on the world around us. As Professor Burger says of number theory: "Great secrets and structures lie close by, to be uncovered only if we open our imagination to all possibilities."