TTC-Videolectures:A History of Numbers

by Nguyen Trung Hoa Sat 21 Aug 2010, 10:47 pm

Zero to Infinity: A History of Numbers
Course No. 1499 (24 lectures, 30 minutes/lecture)
Taught by Edward B. Burger
Williams College
Ph.D., The University of Texas at Austin

What is your definition of number?

The question is a challenging one because defining the abstract idea of number is extremely difficult. More than 2,500 years ago, the great number enthusiast Pythagoras described number as "the first principle, a thing which is undefined, incomprehensible, and having in itself all numbers." Even today, we still struggle with the notion of what numbers mean.

Numbers neither came to us fully formed in nature nor did they spring fully formed from the human mind. Like other ideas, they have evolved slowly throughout human history. Both practical and abstract, they are important in our everyday world but remain mysterious in our imaginations.

If numbers are precision personified, why does a precise, accurate, and satisfying definition of number still elude us?

The 24 lectures of Zero to Infinity: A History of Numbers explore this fascinating question and the equally fascinating history of numbers. Award-winning Professor of Mathematics Edward B. Burger's historical, global, and conceptual approach to numbers gives you not only a revealing tour of mathematical history but shows how and why numbers evolved, as well as the transforming implications of each advance for both mathematics and society.

The Fertile Ground of Mathematical Discovery

Zero to Infinity is, at heart, a human story of great minds applying themselves to major questions and an intellectual pilgrimage through the creative breakthroughs of pure mathematics.

"The study of number is a beautiful one that has captured humankind's imagination throughout the ages," Professor Burger says. "It continues to inspire us to explore its endless frontiers."

As Professor Burger notes throughout the course, the ancient construct of number is also the fertile ground of mathematical discovery. As our understanding of the world expanded and our capacity for abstract thinking grew, so too did our views of what number means. Slowly, numbers moved from useful tools for measuring quantities to their current state as abstract objects of independent interest.

Modern numbers now provide physicists with new insights into quantum physics and our not-so-foreign universe.

Three key themes that resound throughout the lectures:

* Despite what most people believe, numbers are truly difficult to define precisely.
* The notion of number is ever-evolving and is not a fixed idea.
* The history of numbers reflects the idea that what at first appears familiar and commonplace is in fact rare and exotic.

A thorough understanding of the concept of numbers shows just how we have expanded its complexity to compensate for more intricate mathematics; it also helps develop a greater understanding of how and why the larger field of mathematics works the way it does. In addition, the history of numbers illustrates the structure, beauty, and pattern that exists in nature?even before humankind began naming numbers.

Numbers in Life, Life in Numbers

Professor Burger structures this investigation into the concept of number with five main points of focus, each of which informs the other and allows our
understanding to grow and evolve in step with the numbers themselves.

* The Numbers within Our Lives: Early conceptual underpinnings of numbers were used to express different ideas throughout different cultures, all of which led to our current common notion of number.
* The Lives within Our Numbers: Born from our imagination, numbers eventually took on a life of their own within the larger structure of mathematics. This area of study is known as number theory, and the more it is explored, the more insight we gain into the nature of numbers.
* Transcendental Meditation?The pi and e Stories: Perhaps the two most important numbers in our universe, pi and e help us better understand nature and our universe. They are also the gateway into an exploration of transcendental numbers.
* Algebraic and Analytic Evolutions of Number: Two mathematical perspectives on how to create numbers, the algebraic view leads us to imaginary numbers, while the analytical view challenges our intuitive sense of what number should mean.
* Infinity?"Numbers" Beyond Numbers: The idea of infinity, just like the idea of numbers, can be understood and holds many fascinating features. Some of these features, paradoxically, require us to return to the earliest notions of number.

A mathematics course within a historical framework, Zero to Infinity offers moments in time in order to produce a mosaic of ideas, individuals, questions, and concepts that, when viewed from afar, allow you to see how numbers grew in our understanding and sparked our imagination.

There are many different types of numbers, each of which plays an important role within both mathematics and the larger world. Professor Burger takes care, while explaining the concepts behind these number types, to stress their own unique structure and beauty. Among the types of numbers you explore throughout the course are:

* real numbers: numbers that can be given by an infinite decimal representation (e.g., 34.5837 ... )
* natural numbers: also known as counting numbers, these are numbers used primarily for counting and ordering (e.g., 3)
* prime numbers: natural numbers greater than 1 that can be divided by only 1 and itself (e.g., 43)
* rational numbers: numbers that can be expressed as the ratio of two integers (e.g., ?)
* irrational numbers: numbers that cannot be expressed as simple fractions (e.g., v2)
* transcendental numbers: irrational numbers that are not algebraic (e.g., pi)

A Richer Perspective on Mathematics

Regardless of your level of mathematical expertise, you will find Professor Burger's approach to the world of numbers accessible and stimulating.
A number theorist, Professor Burger holds numerous teaching prizes and awards, including the Chauvenet Prize?the oldest and most prestigious prize awarded by the Mathematical Association of America?and the Nelson Bushnell Prize for Scholarship and Teaching. In 2006, Professor Burger was listed in the Reader's Digest annual "100 Best of America" special issue as "Best Math Teacher."

Zero to Infinity provides convincing evidence of Professor Burger's assertion that mathematicians are both artists and explorers. He presents you with an illuminating and lively atlas of the realms of number they have imagined and have discovered.

As you complete your epic journey from the birth of zero to the frontiers of infinity, you will gain a richer understanding of numbers, mathematics, and its practitioners, and develop a newfound lens through which to see the world in a whole new way.

Course Lecture Titles

01. The Ever-Evolving Notion of Number
02. The Dawn of Numbers
03. Speaking the Language of Numbers
04. The Dramatic Digits?The Power of Zero
05. The Magical and Spiritual Allure of Numbers
06. Nature's Numbers?Patterns without People
07. Numbers of Prime Importance
08. Challenging the Rationality of Numbers
09. Walk the (Number) Line
10. The Commonplace Chaos among Real Numbers
11. A Beautiful Dusting of Zeroes and Twos
12. An Intuitive Sojourn into Arithmetic
13. The Story of pi
14. The Story of Euler's e
15. Transcendental Numbers
16. An Algebraic Approach to Numbers
17. The Five Most Important Numbers
18. An Analytic Approach to Numbers
19. A New Breed of Numbers
20. The Notion of Transfinite Numbers
21. Collections Too Infinite to Count
22. In and Out?The Road to a Third Infinity
23. Infinity?What We Know and What We Don't
24. The Endless Frontier of Number

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