Zero to Infinity: A History of Numbers

Course No. 1499
Professor Edward B. Burger, Ph.D.
Southwestern University
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Course No. 1499
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Course Overview

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.

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24 lectures
 |  Average 30 minutes each
  • 1
    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. x
  • 2
    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. x
  • 3
    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. x
  • 4
    The Dramatic Digits—The Power of Zero
    When calculation became more important, zero—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. x
  • 5
    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. x
  • 6
    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. x
  • 7
    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. x
  • 8
    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. x
  • 9
    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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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, and the new—and teeming—realm of transcendentals. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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—and further still. In this lecture, we'll learn not only how humans arrived at the notion of infinity but how to compare infinities. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x

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  • 24 lectures on 4 DVDs
  • 128-page printed course guidebook

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Your professor

Edward B. Burger

About Your Professor

Edward B. Burger, Ph.D.
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 B.A. with distinction in Mathematics. He earned his Ph.D. in Mathematics from The University of Texas at Austin. Professor Burger is the recipient of many teaching...
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Zero to Infinity: A History of Numbers is rated 4.5 out of 5 by 52.
Rated 5 out of 5 by from Intriguing For someone that has feared math since college, it is remarkably illuminating!
Date published: 2020-08-01
Rated 5 out of 5 by from Highly recommended! An excellent and entertaining course! I have a master's degree in mathematics, although not a specialty in number theory, and this one was very informative. The professor is excellent - communicates extremely well. And even more important - it's fun!
Date published: 2020-07-26
Rated 4 out of 5 by from In depth presentation of theoretical subjects I got this course mostly for the initial lectures regarding the formation of numbers and the calculations derived by earlier civilizations. These were quite enlightening given the limited foundations of mathematics available in those times. Professor Burger, as a rule, attempts to build to the next topic by establishing a foundation for the topic through historical references to earlier mathematicians and the theorems each proposed. Unfortunately, as the lectures progressed, the material delved in areas more theoretical than practical. I.e. Discussions on infinity of different types of numbers might be a good topic for a PhD parties, but otherwise for myself it's a non interest. Professor Burger joy in his subject shows in his continued use of the word 'beautiful' as a description of the proofs. I enjoyed his method of presentation. Just not the last few lectures.
Date published: 2019-10-28
Rated 5 out of 5 by from A truly Illuminating Course I expected to receive an interesting course. What I received was an astonishing group of lectures, each one containing material which was totally new to me and totally unexpected. I feel that I have gained much knowledge and am really happy that I bought this course.
Date published: 2019-06-18
Rated 4 out of 5 by from Fun information Much interesting. Liked most quick courses has to treat a couple of points rather superficially but a great fun for those interesting in math. New historical information about some numerical facts. I enjoyed the whole course.
Date published: 2018-07-30
Rated 5 out of 5 by from Great combination of math and history! If you enjoy history and you enjoy math, this is a great combination of both. The professor knows both perspectives of this topic extremely well and makes it entertaining and accessible to his audience.
Date published: 2018-03-02
Rated 5 out of 5 by from Fascinating Journey I bought this course years ago and just got around to watching it completely; Dr Burger is a wonderful lecturer and his passion for his subject comes through clearly. I'm a teacher in my own field, and I recognize a great teacher when I encounter one--he makes you want to go further than the lectures do! I recommend the video version as there are on-screen examples which are helpful, and my only unhappiness is that TC apparently couldn't get streaming rights, so you can't watch this anywhere on whatever gadget you use--I love being able to watch my Great Courses on my tablet anywhere, it's like having a University in your briefcase.
Date published: 2017-07-21
Rated 4 out of 5 by from We enjoyed watching this program. The history is great!
Date published: 2017-07-05
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