All Seasons

Season 1

  • S01E01 The Ever-Evolving Notion of Number

    • January 1, 2007
    • The Great Courses

    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.

  • S01E02 The Dawn of Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E03 Speaking the Language of Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E04 The Dramatic Digits - The Power of Zero

    • January 1, 2007
    • The Great Courses

    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.

  • S01E05 The Magical and Spiritual Allure of Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E06 Nature's Numbers - Patterns without People

    • January 1, 2007
    • The Great Courses

    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.

  • S01E07 Numbers of Prime Importance

    • January 1, 2007
    • The Great Courses

    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.

  • S01E08 Challenging the Rationality of Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E09 Walk the (Number) Line

    • January 1, 2007
    • The Great Courses

    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.

  • S01E10 The Commonplace Chaos among Real Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E11 A Beautiful Dusting of Zeroes and Twos

    • January 1, 2007
    • The Great Courses

    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.

  • S01E12 An Intuitive Sojourn into Arithmetic

    • January 1, 2007
    • The Great Courses

    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.

  • S01E13 The Story of pi

    • January 1, 2007
    • The Great Courses

    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.

  • S01E14 The Story of Euler's e

    • January 1, 2007
    • The Great Courses

    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.

  • S01E15 Transcendental Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E16 An Algebraic Approach to Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E17 The Five Most Important Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E18 An Analytic Approach to Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E19 A Bew Breed of Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E20 The Notion of Transfinite Numbers

    • January 1, 2007
    • The Great Courses

    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.

  • S01E21 Collections Too Infinite to Count

    • January 1, 2007
    • The Great Courses

    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.

  • S01E22 In and Out - The Road to a Third Infinity

    • January 1, 2007
    • The Great Courses

    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.

  • S01E23 Infinity - What We Know and What We Don't

    • January 1, 2007
    • The Great Courses

    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.

  • S01E24 The Endless Frontier of Number

    • January 1, 2007
    • The Great Courses

    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.