1. Two Ideas, Vast Implications
Calculus is a subject of enormous importance and historical impact. It provides a dynamic view of the world and is an invaluable tool for measuring change. Calculus is applicable in many situations, from the trajectory of a baseball to changes in the Dow Jones average or elephant populations. Yet, at its core, calculus is the study of two ideas about motion and change.
1. Two Ideas, Vast Implications (info)
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13. Achilles, Tortoises, Limits, and Continuity
The integral's strategy of adding up little pieces solves a variety of problems, such as finding the volume of a pyramid or the total pressure on the face of a dam.
13. Achilles, Tortoises, Limits, and Continuity (info)
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2. Stop Sign Crime—The First Idea of Calculus—The Derivative
The example of a car moving down a straight road is a simple and effective way to study motion. An everyday scenario that involves running a stop sign and the use of a camera illustrates the first fundamental idea of calculus: the derivative.
2. Stop Sign Crime—The First Idea of Calculus—The Derivative (info)
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14. Calculators and Approximations
The Fundamental Theorem links the integral and the derivative. It shortcuts the integral's infinite process of summing and replaces it by a single subtraction.
14. Calculators and Approximations (info)
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3. Another Car, Another Crime—The Second Idea of Calculus—The Integral
You are kidnapped and driven away in a car. You can't see out the window, but you are able to shoot a videotape of the speedometer. The process by which you can use information about speed to compute the exact location of the car at the end of one hour is the second idea of calculus: the integral.
3. Another Car, Another Crime—The Second Idea of Calculus—The Integral (info)
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15. The Best of All Possible Worlds—Optimization
Calculus is useful in many branches of mathematics. The 18th-century French scientist Georges Louis Leclerc Compte de Buffon used calculus and breadsticks to perform an experiment in probability. His experiment showed how random events can ultimately lead to an exact number.
15. The Best of All Possible Worlds—Optimization (info)
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4. The Fundamental Theorem of Calculus
The moving car scenario illustrates the Fundamental Theorem of Calculus. This states that the derivative and the integral are two sides of the same coin. The insight of calculus, the Fundamental Theorem creates a method for finding a value that would otherwise be hard or impossible to get, even with a computer.
4. The Fundamental Theorem of Calculus (info)
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16. Economics and Architecture
Zeno's Arrow Paradox concerns itself with the fact that an arrow traveling to a target must cover half the total distance, then half the remaining distance, etc. How does it ever get there? The concept of limit solves the problem.
16. Economics and Architecture (info)
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5. Visualizing the Derivative—Slopes
Change is so fundamental to our vision of the world that we view it as the driving force in our understanding of physics, biology, economics—virtually anything. Graphs are a way to visualize the derivative's ability to analyze and quantify change.
5. Visualizing the Derivative—Slopes (info)
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17. Galileo, Newton, and Baseball
The real numbers in toto constitute a smooth, seamless continuum. Viewing the world as continuous in time and space allows us to make mathematical models that are helpful and predictive.
17. Galileo, Newton, and Baseball (info)
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6. Derivatives the Easy Way—Symbol Pushing
The derivative lets us understand how a change in one variable affects a dependent quantity. We have studied this relationship with respect to time. But the derivative can be abstracted to many other dependencies, such as that of the area of a circle on the length of its radius, or supply or demand on price.
6. Derivatives the Easy Way—Symbol Pushing (info)
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18. Getting off the Line—Motion in Space
Zeno's Arrow Paradox shows us that an infinite addition problem (1/2 + 1/4 + 1/8 + . . .) can result in a single number: 1. Similarly, it is possible to approximate values such as π or the square root of 2 by adding up the first few hundred terms of infinite sum. Calculators use this method when we push the "sin" or square root keys.
18. Getting off the Line—Motion in Space (info)
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7. Abstracting the Derivative—Circles and Belts
One of the most useful ways to consider derivatives is to view them algebraically. We can find the derivative of a function expressed algebraically by using a mechanical process, bypassing the infinite process of taking derivatives at each point.
7. Abstracting the Derivative—Circles and Belts (info)
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19. Mountain Slopes and Tangent Planes
We have seen how to analyze change and dependency according to one varying quantity. But many processes and things in nature vary according to several features. The steepness of a mountain slope is one example. To describe these real-world situations, we must use planes instead of lines to capture the philosophy of the derivative.
19. Mountain Slopes and Tangent Planes (info)
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8. Circles, Pyramids, Cones, and Spheres
The description of moving objects is one of the most direct applications of calculus. Analyzing the trajectories and speeds of projectiles has an illustrious history. This includes Galileo's famous experiments in Pisa and Newton's theories that allow us to compute the path and speed of projectiles, from baseballs to planets.
8. Circles, Pyramids, Cones, and Spheres (info)
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20. Several Variables—Volumes Galore
After developing the ideas of calculus for cars moving in a straight line, we have gained enough expertise to apply the same reasoning to anything moving in space—from mosquitoes to planets.
20. Several Variables—Volumes Galore (info)
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9. Archimedes and the Tractrix
Optimization problems—for example, maximizing the area that can be enclosed by a certain amount of fencing—often bring students to tears. But they illustrate questions of enormous importance in the real world. The strategy for solving these problems involves an intriguing application of derivatives.
9. Archimedes and the Tractrix (info)
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21. The Fundamental Theorem Extended
Calculus plays a central role in describing much of physics. It is integral to the description of planetary motion, mechanics, fluid dynamics, waves, thermodynamics, electricity, optics, and more. It can describe the physics of sound, but can't explain why we enjoy Bach.
21. The Fundamental Theorem Extended (info)
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10. The Integral and the Fundamental Theorem
Formulas for areas and volumes can be deduced by dividing such objects as cones and spheres into thin pieces. Ancient examples of this method were precursors to the modern idea of the integral.
10. The Integral and the Fundamental Theorem (info)
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22. Fields of Arrows—Differential Equations
Many money matters are prime examples of rates of change. The difference between getting rich and going broke is often determined by our ability to predict future trends. The perspective and methods of calculus are helpful tools in attempts to decide such questions as what production levels of a good will maximize profit.
22. Fields of Arrows—Differential Equations (info)
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11. Abstracting the Integral—Pyramids and Dams
Archimedes devised an ingenious method that foreshadowed the idea of the integral in that it involved slicing a sphere into thin sections. Integrals provide effective techniques for computing volumes of solids and areas of surfaces. The image of an onion is useful in investigating how a solid ball can be viewed as layers of surfaces.
11. Abstracting the Integral—Pyramids and Dams (info)
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23. Owls, Rats, Waves, and Guitars
Whether looking at people or pachyderms, the models for predicting future populations all involve the rates of population change. Calculus is well suited to this task. However, the discrete version of the Verhulst Model is an example of chaotic behavior—an application for which calculus may not be appropriate.
23. Owls, Rats, Waves, and Guitars (info)
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12. Buffon’s Needle or π from Breadsticks
The integral involves breaking intervals of change into small pieces and then adding them up. We use Leibniz's notation for the integral because the long S shape reminds us that the definition of the integral involves sums.
12. Buffon’s Needle or π from Breadsticks (info)
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24. Calculus Everywhere
There are limits to the realms of applicability of calculus, but it would be difficult to exaggerate its importance and influence in our lives. When considered in all of its aspects, calculus truly has been—and will continue to be—one of the most effective and influential strategies for analyzing our world that has ever been devised.
24. Calculus Everywhere (info)
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1. Number Theory and Mathematical Research
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.
1. Number Theory and Mathematical Research (info)
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13. Fermat's Method of Ascent
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.
13. Fermat's Method of Ascent (info)
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2. Natural Numbers and Their Personalities
The journey begins with the numbers we have always counted upon—the natural numbers 1, 2, 3, 4, and so forth.
2. Natural Numbers and Their Personalities (info)
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14. Fermat's Last 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.
14. Fermat's Last Theorem (info)
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3. Triangular Numbers and Their Progressions
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.
3. Triangular Numbers and Their Progressions (info)
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15. Factorization and Algebraic Number Theory
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.
15. Factorization and Algebraic Number Theory (info)
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4. Geometric Progressions, Exponential Growth
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.
4. Geometric Progressions, Exponential Growth (info)
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16. Pythagorean Triples
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.
16. Pythagorean Triples (info)
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5. Recurrence Sequences
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.
5. Recurrence Sequences (info)
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17. An Introduction to Algebraic Geometry
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.
17. An Introduction to Algebraic Geometry (info)
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6. The Binet Formula and the Towers of Hanoi
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.
6. The Binet Formula and the Towers of Hanoi (info)
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18. The Complex Structure of Elliptic Curves
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.
18. The Complex Structure of Elliptic Curves (info)
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7. The Classical Theory of Prime 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.
7. The Classical Theory of Prime Numbers (info)
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19. The Abundance of Irrational Numbers
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.
19. The Abundance of Irrational Numbers (info)
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8. Euler's Product Formula and Divisibility
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.
8. Euler's Product Formula and Divisibility (info)
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20. Transcending the Algebraic Numbers
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.
20. Transcending the Algebraic Numbers (info)
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9. The Prime Number Theorem and Riemann
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.
9. The Prime Number Theorem and Riemann (info)
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21. Diophantine Approximation
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.
21. Diophantine Approximation (info)
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10. Division Algorithm and Modular Arithmetic
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.
10. Division Algorithm and Modular Arithmetic (info)
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22. Writing Real Numbers as Continued Fractions
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.
22. Writing Real Numbers as Continued Fractions (info)
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11. Cryptography and Fermat's Little Theorem
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.
11. Cryptography and Fermat's Little Theorem (info)
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23. Applications Involving Continued Fractions
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.
23. Applications Involving Continued Fractions (info)
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12. The RSA Encryption Scheme
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.
12. The RSA Encryption Scheme (info)
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24. A Journey's End and the Journey Ahead
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.
24. A Journey's End and the Journey Ahead (info)
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