In this opening lecture, we take our first steps into this ever-growing area of intellectual pursuit and see how it fits within the larger mathematical landscape.
The journey begins with the numbers we have always counted upon—the natural numbers 1, 2, 3, 4, and so forth.
Using an example involving billiard balls and equilateral triangles, Professor Burger demonstrates the fundamental mathematical concept of arithmetic progressions and introduces a famous collection of numbers: the triangular numbers.
Professor Burger introduces the concept of the geometric progression, a process by which a list of numbers is generated through repeated multiplication. Later, we consider various real-world examples of geometric progressions, from the 12-note musical scale to the take-home prize money of a game-show winner.
The famous Fibonacci numbers make their debut in this study of number patterns called recurrence sequences. Professor Burger explores the structure and patterns hidden within these sequences and derives one of the most controversial numbers in human history: the golden ratio.
Is it possible to find a formula that will produce any specific number within a recurrence sequence without generating all the numbers in the list? To tackle this challenge, Professor Burger reveals the famous Binet formula for the Fibonacci numbers.
The 2,000-year-old struggle to understand the prime numbers started in ancient Greece with important contributions by Euclid and Eratosthenes. Today, we can view primes as the atoms of the natural numbers—those that cannot be split into smaller pieces. Here, we'll take a first look at these numerical atoms.
As we look more closely at the prime numbers, we encounter the great 18th-century Swiss mathematician Leonhard Euler, who proffered a crucial formula about these enigmatic numbers that ultimately gave rise to modern analytic number theory.
Can we estimate how many primes there are up to a certain size? In this lecture, we tackle this question and explore one of the most famous unsolved problems in mathematics: the notorious Riemann hypothesis, an "open question" whose answer is worth $1 million in prize money.
How can clocks help us do calculations? In this lecture, we learn how cyclical patterns similar to those used in telling time open up a whole new world of calculation, one that we encounter every time we make an appointment, read a clock, or purchase an item using a scanned UPC bar code.
After examining the history of cryptography—code making—we combine ideas from the theory of prime numbers and modular arithmetic to develop an extremely important application: "public" key cryptography.
We continue our consideration of cryptography and examine how Fermat's 350-year-old theorem about primes applies to the modern technological world, as seen in modern banking and credit card encryption.
When most people think of mathematics, they think of equations that are to be "solved for x." Here we study a very broad class of equations known as Diophantine equations and an important technique for solving them. We also encounter one of the most widely recognized equations, x2 + y2 = z2, the cornerstone of the Pythagorean theorem.
One of the most famous and romantic stories in number theory is the legendary tale of Fermat's last theorem. Professor Burger explicates this most mysterious of proposed "theorems" and describes how the greatest mathematical minds of the 18th and 19th centuries failed again and again in their attempts to provide a proof.
This lecture returns to a fundamental mathematical fact—that every natural number greater than 1 can be factored uniquely into a product of prime numbers—and pauses to imagine a world of numbers that does not exhibit the property of unique factorization.
In this lecture, Professor Burger returns to Pythagoras and his landmark theorem to identify an important series of numbers: the Pythagorean triples. After recounting an ingenious proof of this theorem, Professor Burger explores the structure of triples.
The shapes studied in geometry—circles, ellipses, parabolas, and hyperbolas—can also be described by quadratic (second-degree) equations from algebra. The fact that we can study these objects both geometrically and algebraically forms the foundation for algebraic geometry.
Here we study a particularly graceful shape, the elliptic curve, and learn that it can be viewed as contour curves describing the surface of—of all things—a doughnut. This delicious insight leads to many important theorems and conjectures, and leads to the dramatic conclusion of the story of Fermat's last theorem.
Ancient mathematicians recognized only rational numbers, which can be expressed neatly as fractions. But the overwhelming majority of numbers are irrational. Here, we'll meet these new characters, including the most famous irrational numbers, p, e, and the mysterious g.
We move next to the exotic and enigmatic transcendental numbers, which were discovered only in 1844. We return briefly to a consideration of irrationality and the moment of inspiration that led to their discovery by mathematician Joseph Liouville. We even get a glimpse of Professor Burger's original contributions to the field.
In this lecture, Professor Burger explores a technique for generating a list of rational numbers that are extremely close to the given real number. This technique, called Diophantine approximation, has interesting consequences, including new insights into the motion of billiard balls and planets.
Real numbers are often expressed as endless decimals. Here we study an algorithm for writing real numbers as an intriguing repeated fraction-within-a-fraction expansion. Along the way, we encounter new insights about the hidden structure within the real numbers.
This lecture returns to the consideration of continued fractions and examines what happens when we truncate the continued fraction of a real number. The result involves two of our old friends—the Fibonacci numbers and the golden ratio—and finally explains why the musical scale consists of 12 notes.
In this final lecture, we take a step back to view the entire panorama of number theory and celebrate some of the synergistic moments when seemingly unrelated ideas came together to tell a unified story of number.