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Math Tricks and the Joy of Math I The Great Courses

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Welcome to the Joy of Math. I'm Arthur Benjamin, Professor of Mathematics at Harvey Mudd College. Many people, when they hear the "joy of math," think that sounds like a contradiction in terms. I mean, for many people, "math" is a four-letter word—something to be afraid of, not something to be in love with. Yet, in these lectures, I hope to show you why mathematics is indeed something to love. There are many, many reasons to love mathematics. In fact, I call these the ABCs of loving mathematics. You can love mathematics for its applications, for its beauty and structure, and for its certainty; and I'll say more about these.

For instance, what are some of the applications of mathematics? Mathematics is, after all, the language of science. The laws of nature are written in the language of mathematics, particularly in calculus and differential equations. We'll learn about calculus in these lectures. Calculus is how things change and grow over time, modeling everything from the motion of pendulums to galaxies. In more down-to-earth subjects, mathematics can be used to model how your money grows. We'll talk about the mathematics of compound interest and how it connects to the mysterious number e.

When I was a kid, I loved mathematics primarily for its consistency. I loved the fact that you could take a math problem, do it lots and lots of different ways, and if you were careful, you'd always get the same answer. Let's multiply two numbers that are close to 100. Look at the numbers 104 and 109. I'll write next to them how far each of those numbers are away from 100, so 104 is 4 away and 109 is 9 away from 100. All right, now next step, all we do is we take 104 + 9 or 109 + 4, take your pick, you'll always get the same answer. What do you get? Let's see, 104 + 9 is 113, good. Next, multiply those two one-digit numbers together: 4× 9 is 36. Write that down. Believe it or not, you've now written down the answer. The answer to that multiplication problem is 11336—11,336.

When I was a teenager, I loved to play games. Boy, I was addicted to backgammon, chess, poker—and I still am, as an adult, I confess. I saw that many of these games used math in some way. So, what I found was that by understanding math, especially areas of math like probability, or my favorite area of math, combinatorics—which is a fancy way of saying clever ways of counting things—I found that I could be a better game player and actually win more. And so, I'm going to share some of those secrets with you in this course, as well. We'll use math to analyze games like poker, roulette, and craps. We'll learn about the long-term effects of playing some of these games.

You'll also be exposed in this class to math ideas from high-school level mathematics to college level mathematics, all the way to unsolved problems in mathematics. You'll learn what is 5 factorial and why should we care. We'll learn about many of the fundamental theorems in mathematics. You'll learn the fundamental theorem of arithmetic, the fundamental theorem of algebra, and even the fundamental theorem of calculus.

In addition to being a professor of mathematics, I'm also a professional magician. I've performed magic most of my life, and I find that math and magic have an awful lot in common. For instance, in both cases, when you're looking at math and magic, you're solving a problem or a puzzle based on limited information. Part of the fun of doing a math problem or figuring out a magic trick is trying to understand how it's done. What's really going on here? When I teach my courses at Harvey Mudd College and my courses for you, I always try to present the material almost as a magician would, with elements of humor, surprise, and elegance.

Let me conclude by saying that if you don't know the difference between a logarithm and an algorithm, a radian from a radius, or a polygon from a polynomial, then please join me, as together we explore the joy of math.

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