Joy of Mathematics
(24
lectures,
30
minutes/lecture)

Course No.
1411

Taught by
Arthur T. Benjamin

Harvey Mudd College

Ph.D., Johns Hopkins University

Humans
have been having fun and games with mathematics for thousands of years.
Along the way, they've discovered the amazing utility of this field—in
science, engineering, finance, games of chance, and many other aspects
of life.

Join Professor Arthur T. Benjamin as he celebrates the sheer joy of
math in this course of 24 half-hour lectures. Nationally renowned for
feats of mental calculation performed before audiences at schools,
theaters, museums, conferences, and other venues, he shows you that
there are simple tricks that allow anyone to look like a math magician.

Professor Benjamin has another goal in this course: Throughout these
lectures, he shows how everything in mathematics is connected—how the
beautiful and often imposing edifice that has given us algebra,
geometry, trigonometry, calculus, probability, and so much else is
based on nothing more than fooling around with numbers.

Fun with Numbers

Here is an example:

Think of a number between 1
and 10. Triple it. Add 6. Then triple again. Now take your answer,
probably a two-digit number, and add the digits of your answer. If you
still have a two-digit number, add those digits again. You should now
be thinking of the magical number 9. The reason this works is based on
algebra and the fact that the digits of any multiple of 9 must sum to a
multiple of 9.

This is one of the many wonders of
modular arithmetic,
sometimes called clock arithmetic, where numbers wrap around in a
circle. A useful application of this field is casting out nines, a
simple and ancient technique for checking the answers to arithmetical
problems.

Modular arithmetic also provides a very handy method for mentally computing the day of the week for any date in history.

This connection between entertaining number tricks and the deeper
properties of mathematics reflects Dr. Benjamin's specialty, which is
combinatorics,
the branch of mathematics that deals with the subtleties of counting.
Some examples: How many different six-symbol license plates are
possible? And for the book collector, how many ways are there of
arranging 10 books on a shelf? (Would you believe more than 3 million?)
These simple questions introduce concepts such as the
factorial function.

Drawing on his dual fascination with combinatorics and games, Dr.
Benjamin used his analytical skill to win first place in the American
Backgammon Tour in 1997.

Have You Forgotten Math? Worry Not!

Professor Benjamin gives his presentation on the number 9 in—where
else?—Lecture 9. Other lectures are devoted to pi, the imaginary number
i, the transcendental number
e,
and infinity. These numbers are gateways to intriguing realms of
mathematics, which you explore under Dr. Benjamin's enthusiastic
guidance.

He also introduces you to prime numbers, Fibonacci numbers, and
infinite series. And you investigate the powerful techniques for
manipulating numbers using algebra, geometry, trigonometry, calculus,
and probability in lectures that may hark back to subjects you studied
in high school and college. You will find Dr. Benjamin's introduction
to these fields both a useful refresher and a bird's-eye view of the
most important areas of mathematics. Intriguingly, he approaches these
topics from the novel perspective of combinatorics and mathematical
games, providing a fun entry into subjects that are often taught in a
lackluster way.

In the next-to-last lecture, you look at the application of
probability to games. And finally, you splurge on feats of mathematical
magic: For instance, did you know that you don't have to be a genius to
calculate cube roots in your head?

Throughout the course, Dr. Benjamin assumes you have no more than a
distant memory of high school math. He believes that it is his job to
fan those embers into a burning interest in the subject he loves so
much—and in which he takes such exquisite joy.

A Math Course Designed for You

This course is especially well suited for:

- Anyone attracted by Dr. Benjamin's promise of a joyful attitude to an often-imposing subject
- Anyone for whom high school and college math courses are a distant
memory who would like to revisit these subjects to explore topics they
skipped the first time
- Anyone now taking math who would like a big-picture perspective on
the major areas of the field from a playful, joyous point of view
- Budding math mavens, who love numbers and the magic that can be done with them.

Be prepared to encounter strange equations, novel ways of thinking,
and symbols and computational methods that may be new to you. But also
prepare to sharpen your wits in ways you never thought possible. Math
is a challenging subject, but it pays immense rewards. Few people
understand everything the first time through an unfamiliar domain of
math. "But that's OK," says Dr. Benjamin. "You can rewind me and have
me explain it all over again! All of this material bears repeating, and
I hope you get to enjoy it many times over."

Patterns, Patterns Everywhere

One of Dr. Benjamin's greatest loves is the Fibonacci sequence,
which shows up in many spheres of mathematics, as well as in nature,
art, computer science, and poetry. The distinctive meter of a limerick
encodes Fibonacci numbers, and Dr. Benjamin has even composed his own
limerick to show how the sequence begins:

I think Fibonacci is fun;

We start with a 1 and a 1.

Then 2, 3, 5, 8,

But don't stop there, mate!

The fun has just barely begun.

The series continues on: 13, 21, 34, 55, 89, … with each successive
Fibonacci number being the sum of the previous two. This simple pattern
is named for a 12th-century mathematician who described a problem
involving imaginary rabbits that never die. Starting with a pair of
baby rabbits, the animals take a month to mature, then mate and produce
a male and a female; these mature after a month and mate, along with
their parents. The total number of pairs after each month follows the
Fibonacci sequence.

In our own day, Fibonacci numbers appear as a critical plot element in
The Da Vinci Code,
notably under the guise of the golden ratio, an ideal proportion
favored by artists and architects that is intimately connected to the
Fibonacci sequence. However, Dr. Benjamin cautions that the quest for
instances of the golden ratio in nature can get out of hand.

A New Way to Experience Beauty

Anyone who has ever witnessed a feat of mathematical prowess and
chalked it up to unfathomable intellect will be interested to learn
that there is often a simple shortcut at work. For example, the first
24 digits of pi, the famous ratio of the circumference of a circle to
its diameter, can be memorized with the help of a silly sentence
starting “My turtle Pancho …” Four more sentences take you to 100
digits, making you look like a prodigy indeed!

Similarly, if someone asked you to add up all the numbers from 1 to
100, you might take out a sheet of paper and start to work, little
realizing that this difficult-looking problem can be done in your head
in seconds by a method devised by the famous mathematician Carl
Friedrich Gauss when he was a boy. The same goes for squaring and
multiplying multidigit numbers in your head.

Mathematics "helps you think precisely, decisively, and creatively,
and helps you look at the world from multiple perspectives," says Dr.
Benjamin. "Naturally, it comes in handy when you're shopping around for
the best bargain or trying to understand the statistics you read in the
newspaper.

"But I hope that you come away from this course with a new way to
experience beauty—in the form of a surprising pattern or an elegant
logical argument. Many people find joy in fine music, poetry, and other
works of art—and mathematics offers joys that I hope you, too, will
learn to experience. If Elizabeth Barrett Browning had been a
mathematician, she might have said, ‘How do I count thee? Let me love
the ways!'"

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