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Zero to Infinity: A History of Numbers

This journey to infinity and beyond will be the final brush stroke on our painting of the endless frontier of the notion of number.

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The Ever-Evolving Notion of Number

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
The Dawn of Numbers

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
Speaking the Language of Numbers

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
The Dramatic Digits—The Power of Zero

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
The Magical and Spiritual Allure of Numbers

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
Nature's Numbers—Patterns without People

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
Numbers of Prime Importance

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
Challenging the Rationality of Numbers

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
Walk the (Number) Line

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
The Commonplace Chaos among Real Numbers

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
A Beautiful Dusting of Zeroes and Twos

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
An Intuitive Sojourn into Arithmetic

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
The Story of pi

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
The Story of Euler's

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
Transcendental Numbers

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
An Algebraic Approach to Numbers

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
The Five Most Important Numbers

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
An Analytic Approach to Numbers

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
A New Breed of Numbers

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
The Notion of Transfinite Numbers

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
Collections Too Infinite to Count

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
In and Out—The Road to a Third Infinity

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
Infinity—What We Know and What We Don't

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
The Endless Frontier of Number

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

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About

Edward B. Burger
Edward B. Burger

For the truly wise individual, learning never ends.

INSTITUTION

Southwestern University
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By This Professor

An Introduction to Number Theory
854
Zero to Infinity: A History of Numbers
854

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