Zero to Infinity: A History of Numbers

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Rated 4 out of 5 by Diver Don from Find your inner nerd I am a retired scientist, who knows a fair bit of math, but little about number theory. I found the presentation delightfully nerdy, and quite interesting. If you want a deeper appreciation for the theory of numbers, dive in!

Date published: 2025-04-16

Rated 5 out of 5 by Tapia2025 from La historia es el mejor metodo El enfoque histórico es el más adecuado porque llena la aridez de un tema de "humanidad" que, muchas veces, es eliminada en las presentaciones académicas tradicionales.

Date published: 2025-01-26

Rated 5 out of 5 by Campanile_ from Great course on numbers and number theory This is a good course that gently introduces you to the historical development of what is commonly referred to as "number theory". This course can serve as a good prequal to Burger's over course Introduction to Number Theory. That second course is a little more "advanced" then this one is. Here in this course Burger opts to give a general, as well as historical overview on the usage of numbers in mathematics. Make sure to grab this course while you can. I'll be sad to see it go. Burger is excellent at showing us the joy of mathematics.

Date published: 2024-08-28

Rated 5 out of 5 by Canadian from Strength in Numbers! Growing up in the 1950s and 1960s, I enjoyed my math classes up to and including those at the university level, although the breadth of information covered did not turn out to be required during my later work experiences. In a somewhat nostalgic mood, I ordered Dr. Edward B. Burger’s Great Course, “Zero to Infinity: A History of Numbers,” and found it to be a delight. I consider myself representative of one category of prospective purchasers for this course; namely, those of us who already appreciate math for its own sake. Who else might really enjoy the present course? I suggest that these are other likely groups: * Young people considering possible career plans, hoping to get a sense of whether or not math is their “cup of tea.” * People of any age with a general interest in history, especially the history of human creative accomplishments. They may find satisfaction in Dr. Burger’s tracing of math creativity from the earliest archaeological evidence of humans keeping counting records, through to highly sophisticated and seemingly fantastical modern mathematical researches. * People who are simply curious about the goals, problems, and methods of committed mathematicians. This category of the curious might be similar, for example, to the many folks who like to keep informed about modern astronomy without necessarily having the desire to undertake astronomy projects themselves. * People who enjoy clever solutions to logic puzzles. Note: Dr. Burger manages to share such puzzle-solving aspects of math without heavy dependence on advance knowledge of complicated operations/calculations. Students of the course will need to do considerable thinking, but will not need to get out paper and pencil and “show their work.” * Those who have felt intimidated by math in the past, who might now have at least the chance to get over some of those feelings during Dr. Burger’s fun and fascinating lectures. Dr. Burger is a clear, excellent lecturer. I loved his course and give it my enthusiastic recommendation. While summing up what he had presented, the professor stated that, “far beyond numbers’ utility, numbers are objects of independent beauty, intrigue, and curiosity.” He certainly convinced me!

Date published: 2024-04-07

Rated 5 out of 5 by Hotel Whiskey from Thoroughly Satisfactory. Some time back, in 2019, my wife and I took Prof. Edward Burger’s “An Introduction to Number Theory,” Course #1495. That was excellent, but I wondered if this would be a rehash. I am pleased to report that this second course is a fine extension and expansion of its predecessor. It’s all new material. The important distinctions between algebraic, transcendental and normal numbers are very clearly explained. The most fascinating lectures, at least for us, were #18 (An Analytic Approach to Numbers) and #19 (A New Breed of Numbers) wherein the p-adic absolute value is introduced, and Chapter #24 (The Endless Frontier of Number) that provides a taste of Gaussian prime numbers. And best of all were the stories of the mathematician explorers who built this edifice. This course is a real winner. HWF & ISF, Mesa AZ.

Date published: 2024-03-11

Rated 5 out of 5 by Zaxx from Excellent Lecturer 3/4 through the title - excellent lecturer, good graphics. I'm thoroughly enjoying this title, and gaining additional understanding of subject matter.

Date published: 2022-08-29

Rated 5 out of 5 by Big Ed from Zero to infinity This is an amazing course, proof that even the simplest topic can lead to cosmic results if it is developed with strict attention to clarity.

Date published: 2022-07-12

Rated 1 out of 5 by HalTaylor from slow, plodding The first math used by human beings in the middle east was Base 60! This professor is woefully lacking in the the history of language in the middle east. He would do well to read up on Zecharia Sitchin before discussing the development of language in Sumeria.

Date published: 2022-01-07

01: The Ever-Evolving Notion of Number

While numbers are precision personified, the exact definition of "number" is elusive, because it's still evolving. The course will trace progress in understanding and using numbers while also exploring discoveries that have advanced our grasp of number. We will meet the great thinkers who made these discoveries - from old friends Pythagoras and Euclid to the more modern but equally brilliant Euler, Gauss, and Cantor.

32 min

02: The Dawn of Numbers

One of the earliest questions was "How many?" Humans have been answering this question for thousands of years - since Sumerian shepherds used pebbles to keep track of their sheep, Mesopotamian merchants kept their accounts on clay tablets, and Darius of Persia used a knotted cord as a calendar.

29 min

03: Speaking the Language of Numbers

As numbers became useful to count and record as well as calculate and predict, many societies, including the Sumerians, Egyptians, Mayans, and Chinese, invented sophisticated numeral systems; arithmetic developed. Negative numbers, Arabic numerals, multiplication, and division made number an area for abstract, imaginative study as well as for everyday use.

30 min

04: The Dramatic Digits—The Power of Zero

When calculation became more important, a crucial breakthrough was born. Unwieldy additive number systems, like Babylonian nails and dovetails, or Roman numerals, gave way to compact place-based systems. These systems, which include the modern base-10 system we use today, made modern mathematics possible.

32 min

05: The Magical and Spiritual Allure of Numbers

As numbers developed from tools into a branch of learning, they gained power and mystery. Mesopotamians used numbers to name their gods, for example, while Pythagoreans believed that numbers were divine gifts. Humans have invoked the power of numbers for millennia to ward off bad luck, to attract good luck, and to entertain the curious - uses that have drawn some to explore numbers' more serious and subtle properties.

31 min

06: Nature's Numbers—Patterns without People

Those who studied them found numbers captivating and soon realized that numerical structure, pattern, and beauty existed long before our ancestors named the numbers. In this lecture, our studies of pattern and structure in nature lead us to Fibonacci numbers and to connect them in turn to the golden ratio studied by the Pythagoreans centuries earlier.

29 min

07: Numbers of Prime Importance

Now we study prime numbers, the building blocks of all natural (counting) numbers larger than 1. This area of inquiry dates to ancient Greece, where, using one of the most elegant arguments in all of mathematics, Euclid proved that there are infinitely many primes. Some of the great questions about primes still remain unanswered; the study of primes is an active area of research known as analytic number theory.

32 min

08: Challenging the Rationality of Numbers

Babylonians and Egyptians used rational numbers, better known as fractions, perhaps as early as 2000 B.C. Pythagoreans believed rational and natural numbers made it possible to measure all possible lengths. When the Pythagoreans encountered lengths not measurable in this way, irrational numbers were born, and the world of number expanded.

30 min

09: Walk the (Number) Line

We have learned about natural numbers, integers, rational numbers, and irrationals. In this lecture, we'll encounter real numbers, an extended notion of number. We'll learn what distinguishes rational numbers within real numbers, and we'll also prove that the endless decimal 0.9999... exactly equals 1.

31 min

10: The Commonplace Chaos among Real Numbers

Rational and irrational numbers have a defining difference that leads us to an intuitive and correct conclusion, and to a new understanding about how common rationals and irrationals really are. Examining random base-10 real numbers introduces us to ";normal" numbers and shows that "almost all" real numbers are normal and "almost all" real numbers are, in fact, irrational.

30 min

11: A Beautiful Dusting of Zeroes and Twos

In base-3, real numbers reveal an even deeper and more amazing structure, and we can detect and visualize a famous, and famously vexing, collection of real numbers - the Cantor Set first described by German mathematician Georg Cantor in 1883.

32 min

12: An Intuitive Sojourn into Arithmetic

We begin with a historical overview of addition, subtraction, multiplication, division, and exponentiation, in the course of which we'll prove why a negative number times a negative number equals a positive number. We'll revisit Euclid's Five Common Notions (having learned in Lecture 11 that one of these notions is not always true), and we'll see what happens when we raise a number to a fractional or irrational power.

29 min

13: The Story of pi

Pi is one of the most famous numbers in history. The Babylonians had approximated it by 1800 B.C., and computers have calculated it to the trillions of digits, but we'll see that major questions about this amazing number remain unanswered.

30 min

14: The Story of Euler's"e"

Compared to pi, "e" is a newcomer, but it quickly became another important number in mathematics and science. Now known as Euler's number, it is fundamental to understanding growth. This lecture traces the evolution of "e".

30 min

15: Transcendental Numbers

Pi and "e" take us into the mysterious world of transcendental numbers, where we'll learn the difference between algebraic numbers, known since the Babylonians.

30 min

16: An Algebraic Approach to Numbers

This part of the course invites us to take two views of number, the algebraic and the analytical. The algebraic perspective takes us to imaginary numbers, while the analytical perspective challenges our sense of what number even means.

30 min

17: The Five Most Important Numbers

Looking at complex numbers geometrically shows a way to connect the five most important numbers in mathematics: 0, 1, p, "e", and "i", through the most beautiful equation in mathematics, Euler's identity.

31 min

18: An Analytic Approach to Numbers

We'll explore real numbers from another perspective: the analytical approach, which uses the distance between numbers to discover and fill in holes on a rational number line. This exploration leads to a new kind of absolute value based on prime numbers.

30 min

19: A New Breed of Numbers

Pythagoreans found irrational numbers not only counterintuitive but threatening to their world-view. In this lecture, we'll get acquainted with and use some numbers that we may find equally bizarre: p-adic numbers. We'll learn a new way of looking at number, and about a lens through which all triangles become isosceles.

31 min

20: The Notion of Transfinite Numbers

Although it seems that we've looked at all possible worlds of number, we soon find that these worlds open onto a universe of number. In this lecture, we'll learn not only how humans arrived at the notion of infinity but how to compare infinities.

29 min

21: Collections Too Infinite to Count

Now that we are comfortable thinking about the infinite, we'll look more closely at various collections of numbers, thereby discovering that infinity comes in at least two sizes.

30 min

22: In and Out—The Road to a Third Infinity

If infinity comes in two sizes, does it come in three? We'll use set theory to understand how it might. Then we'll apply this insight to infinite sets as well, a process that leads us to a third kind of infinity.

29 min

23: Infinity—What We Know and What We Don't

If there are several sizes of infinity, are there infinitely many sizes of it? Is there a largest infinity? And is there a size of infinity between the infinity of natural numbers and real numbers? We'll answer two of these questions and learn why the answer to the other is neither provable nor disprovable mathematically.

28 min

24: The Endless Frontier of Number

Now that we've traversed the universe of number, we can look back and understand how the idea of number has changed and evolved. In this lecture, we'll get a sense of how mathematicians expand the frontiers of number, and we'll look at a couple of questions occupying today's number theorists - the Riemann Hypothesis and prime factorization.

29 min

Overview Course No. 1499

Numbers surround us. They mark our days, light our nights, foretell our weather, and keep us on course. They drive commerce and sustain civilization. But what are they? Whether you struggled through algebra or you majored in mathematics, you will find Professor Edward B. Burger's approach accessible and stimulating. If you think math is just problems and formulas, prepare to be amazed.

In this course, we will paint a picture of an ever-evolving notion of "number"—coming to the realization that the notion of number is a difficult if not impossible idea to define precisely. Our course will have two main points of focus: the historical evolution of the representation of numbers for communication and manipulation, "the numbers in life," and the intrinsic structure of those numbers, “the life of numbers.” We will discover throughout the course that these two perspectives synergistically inform each other and allow our understanding to grow and evolve with the numbers themselves. While we will explore these two vantage points and their interconnectedness, we will also offer the recurring historical theme of number as a means to count, quantify, measure, and compare, and surprisingly, all of our intuitions about these basic notions will be challenged.

About

Edward B. Burger

For the truly wise individual, learning never ends.

ALMA MATER

University of Texas, Austin

INSTITUTION

Southwestern University

Dr. Edward B. Burger is President of Southwestern University in Georgetown, Texas. Previously, he was Francis Christopher Oakley Third Century Professor of Mathematics at Williams College. He graduated summa cum laude from Connecticut College, where he earned a BA with distinction in Mathematics. He earned his PhD in Mathematics from The University of Texas at Austin. Professor Burger is the recipient of many teaching awards and accolades. He was named by Baylor University as the 2010 recipient of the Robert Foster Cherry Award for Great Teaching for his proven record as an extraordinary teacher and distinguished scholar. Baylor University lauded Dr. Burger as truly one of our nation's most outstanding, passionate, and creative mathematics professors. His other teaching awards include the Nelson Bushnell Prize for Scholarship and Teaching from Williams College, the Distinguished Achievement Award for Educational Video Technology from the Association of Educational Publishers, and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America. In 2006 Reader's Digest honored him in its annual 100 Best of America special issue as Best Math Teacher. Professor Burger is the author of more than 40 scholarly papers and books, including Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, which has been translated into seven languages. He was honored with the Robert W. Hamilton Book Award for his coauthored work with Michael Starbird, The Heart of Mathematics: An invitation to effective thinking. He served for three years as mathematics advisor for the educational series NUMB3RS, produced by CBS, Paramount Studios, and Texas Instruments.