# Arthur T. Benjamin

### Arthur T. Benjamin >

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The Joy of Math - The Big Picture

Episode 1 of The Joy of Mathematics

Episode 1 of The Joy of Mathematics

Professor Benjamin introduces the ABCs of math appreciation: The field can be loved for its applications, its beauty and structure, and its certainty. Most of all, mathematics is a source of endless delight through creative play with numbers.

The Joy of Numbers

Episode 2 of The Joy of Mathematics

Episode 2 of The Joy of Mathematics

How do you add all the numbers from 1 to 100--instantly? What makes a square number square and a triangular number triangular? Why do the rules of arithmetic really work, and how do you calculate in bases other than 10?

The Joy of Primes

Episode 3 of The Joy of Mathematics

Episode 3 of The Joy of Mathematics

A number is prime if it is evenly divisible by only itself and one: for example, 2, 3, 5, 7, 11. Professor Benjamin proves that there are an infinite number of primes and shows how they are the building blocks…

The Joy of Algebra

Episode 6 of The Joy of Mathematics

Episode 6 of The Joy of Mathematics

Arguably the most important area of mathematics, algebra introduces the powerful idea of using an abstract variable to represent an unknown quantity. This lecture demonstrates algebra's golden rule: Do unto one side of an equation as you do unto the…

The Joy of Differential Calculus

Episode 18 of The Joy of Mathematics

Episode 18 of The Joy of Mathematics

Calculus is the mathematics of change, and answers questions such as: How fast is a function growing? This lecture introduces the concepts of limits and derivatives, which allow the slope of a curve to be measured at any point.

The Joy of Counting

Episode 4 of The Joy of Mathematics

Episode 4 of The Joy of Mathematics

Combinatorics is the study of counting questions such as: How many outfits are possible if you own 8 shirts, 5 pairs of pants, and 10 ties? A trickier question: How many ways are there to arrange 10 books on a…

The Joy of Fibonacci Numbers

Episode 5 of The Joy of Mathematics

Episode 5 of The Joy of Mathematics

The Fibonacci numbers follow the simple pattern 1, 1, 2, 3, 5, 8, etc., in which each number is the sum of the two preceding numbers. Fibonacci numbers have many beautiful and unexpected properties, and show up in nature, art,…

The Joy of the Number e

Episode 15 of The Joy of Mathematics

Episode 15 of The Joy of Mathematics

Another indispensable number to learn is e = 2.71828 ... Defined as the base of the natural logarithm, e plays a central role in calculus, and it arises naturally in many spheres of mathematics, including calculations of compound interest.

The Joy of Geometry

Episode 11 of The Joy of Mathematics

Episode 11 of The Joy of Mathematics

Geometry is based on a handful of definitions and axioms involving points, lines, and angles. These lead to important conclusions about the properties of polygons. This lecture uses geometric reasoning to derive the Pythagorean theorem and other interesting results.

The Joy of 9

Episode 9 of The Joy of Mathematics

Episode 9 of The Joy of Mathematics

Adding the digits of a multiple of 9 always gives a multiple of 9. For example: 9 x 4 = 36, and 3 + 6 = 9. In modular arithmetic, this property allows checking answers by "casting out nines." A…

The Joy of Higher Algebra

Episode 7 of The Joy of Mathematics

Episode 7 of The Joy of Mathematics

This lecture shows how to solve quadratic (second-degree) equations from the technique of completing the square and the quadratic formula. The quadratic formula reveals the connection between Fibonacci numbers and the golden ratio.

The Joy of Integral Calculus

Episode 20 of The Joy of Mathematics

Episode 20 of The Joy of Mathematics

Geometry and trigonometry are used to determine the areas of simple figures such as triangles and circles. But how are more complex shapes measured? Calculus comes to the rescue with a technique called integration, which adds the simple areas of…

The Joy of Trigonometry

Episode 13 of The Joy of Mathematics

Episode 13 of The Joy of Mathematics

Trigonometry deals with the sides and angles of triangles. This lecture defines sine, cosine, and tangent, along with their reciprocals, the cosecant, secant, and cotangent. Extending these definitions to the unit circle allows a handy measure of angle: the radian.

The Joy of Approximating with Calculus

Episode 19 of The Joy of Mathematics

Episode 19 of The Joy of Mathematics

Exploiting the idea of the derivative, we can approximate just about any function using simple polynomials. This lecture also shows why a formula sometimes known as "God's equation" (involving e, i, p, 1, and 0) is true, and how to…

The Joy of Proofs

Episode 10 of The Joy of Mathematics

Episode 10 of The Joy of Mathematics

Professor Benjamin begins his discussion of mathematical proofs with intuitive cases like "even plus even is even" and "odd times odd is odd." He builds to more complex proofs by existence and induction, and ends with a checkerboard challenge.

The Joy of Mathematical Games

Episode 23 of The Joy of Mathematics

Episode 23 of The Joy of Mathematics

This lecture applies the law of total probability and other concepts from the course to predict the long-term losses to be expected from playing games such as roulette and craps and understand what is known as the "Gambler's Ruin Problem."

The Joy of Pascal's Triangle

Episode 21 of The Joy of Mathematics

Episode 21 of The Joy of Mathematics

A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of beautiful number patterns. It even provides an answer to the song "The Twelve Days of Christmas": Exactly how many gifts did my true love give to…

The Joy of Mathematical Magic

Episode 24 of The Joy of Mathematics

Episode 24 of The Joy of Mathematics

Closing the course with a magician's flair, Professor Benjamin shows a trick for producing anyone's phone number, how to create a magic square based on your birthday, how to play "mathematical survivor," a technique for computing cube roots in your…

The Joy of Algebra Made Visual

Episode 8 of The Joy of Mathematics

Episode 8 of The Joy of Mathematics

Algebra can be used to solve geometrical problems, such as finding where two lines cross. The technique is useful in real-life problems, for example, in choosing a telephone plan. Graphs help us better understand everything from lines to equations with…

The Joy of the Imaginary Number i

Episode 14 of The Joy of Mathematics

Episode 14 of The Joy of Mathematics

Could the apparently nonsensical number the square root of -1 be of any use? Very much so, as this lecture shows. Such imaginary and complex numbers play an indispensable role in physics and other fields, and are easier to understand…