The Joy of Mathematics

Professor Benjamin introduces the ABCs of math appreciation; the field's applications, its beauty and structure and its certainty; mathematics is a source of endless delight through creative play with numbers.

How to add all the numbers from 1 to 100 instantly; what makes a square number square and a triangular number triangular; why the rules of arithmetic really work; how one calculates in bases other than 10.

A number is prime if it is evenly divisible by only itself and the number one; Professor Benjamin proves that there are an infinite number of primes and how primes are the building blocks of the number system.

Combinatorics is the study of counting questions, such as how many outfits are possible if one owns eight shirts, five pairs of pants and 10 ties; combinatorics can also be used to analyze numbering systems and games of chance.

The Fibonacci numbers follow a simple pattern in which each number is the sum of the two preceding numbers; Fibonacci numbers have many beautiful and unexpected properties; Fibonacci numbers in nature, art and poetry.

Arguably the most important area of mathematics, algebra introduces the idea of using an abstract variable to represent an unknown quantity; discovering algebra's golden rule, which is to do unto one side of an equation as one does unto the other.

How to solve quadratic equations using the technique of completing the square and the quadratic formula; the quadratic formula reveals the connection between Fibonacci numbers and the golden ratio.

Algebra can be used to solve geometrical problems, such as finding where two lines cross; the technique is useful in real-life situations; graphs help people better understand everything from lines to equations with negative or fractional exponents.

Adding the digits of a multiple of nine always gives a multiple of nine; in modular arithmetic, this property allows checking answers by casting out nines; a related trick is mentally computing the day of the week for any date in history.

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; he ends with a checkerboard challenge.

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, using geometric reasoning to derive the Pythagorean theorem and other results.

Pi is the ratio of the circumference of a circle to its diameter; it starts at 3.14 and continues in an infinite non-repeating sequence; Professor Benjamin shows how to learn the first hundred digits of this celebrated number, making it look easy.

Trigonometry deals with the sides and angles of triangles; defining sine, cosine, and tangent, along with their reciprocals; extending these definitions to the unit circle allows a handy measure of angle, which is the radian.

Whether the apparently nonsensical number of the square root of -1 could be of any use; such imaginary and complex numbers play an indispensable role in physics and other fields; they are easier to understand than they appear.

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

The meaning of infinity; whether some infinite sets are more infinite than others; the possibility of an infinite number of levels of infinity; exploring some of the strange ideas associated with mathematical infinity.

Starting with the analysis of the proposition 0.999999999... = 1; what it means to add up an infinite series of numbers; some infinite series converge on a definite value, while others grow arbitrarily large.

Calculus is the mathematics of change; it answers questions such as how fast a function is growing, discovering the concepts of limits and derivatives, which allow the slope of a curve to be measured at any point.

By exploiting the idea of the derivative, one can approximate just about any function using simple polynomials; why a formula sometimes known as God's equation is true; how to calculate square roots in one's head.

Geometry and trigonometry are used to determine the areas of simple figures, and how more complex shapes are measured; calculus comes to the rescue with a technique called integration, which adds the simple areas of many tiny quantities.

A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of number patterns; it provides an answer to the song "The Twelve Days of Christmas" and exactly how many gifts it refers to.

Mathematics draws detailed inferences about random events, covering major concepts in probability, such as the law of large numbers, the central limit theorem and how to measure variance.

The law of total probability and other concepts can be applied to predict the long-term losses to be expected from playing games such as roulette and craps, understanding what is known as the gambler's ruin.

Professor Benjamin shows a trick for producing anyone's phone number; how to create a magic square based on one's birthday; how to play mathematical survivor; a technique for computing cube roots; an interesting card trick.