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# The Joy of Pascal's Triangle Episode 21 of The Joy of Mathematics

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## 24 episodes in this series

Episode 1 The Joy of Math - The Big Picture
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…
Episode 2 The Joy of Numbers
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,…
Episode 3 The Joy of Primes
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…
Episode 4 The Joy of Counting
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…
Episode 5 The Joy of Fibonacci Numbers
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…
Episode 6 The Joy of Algebra
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…
Episode 7 The Joy of Higher Algebra
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…
Episode 8 The Joy of Algebra Made Visual
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…
Episode 9 The Joy of 9
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,…
Episode 10 The Joy of Proofs
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…
Episode 11 The Joy of Geometry
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…
Episode 12 The Joy of Pi
Pi is the ratio of the circumference of a circle to its diameter. It starts 3.14 and continues in an infinite nonrepeating sequence. Professor Benjamin shows how to learn the…
Episode 13 The Joy of Trigonometry
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…
Episode 14 The Joy of the Imaginary Number i
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…
Episode 15 The Joy of the Number e
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…
Episode 16 The Joy of Infinity
What is the meaning of infinity? Are some infinite sets "more" infinite than others? Could there possibly be an infinite number of levels of infinity? This lecture explores some of…
Episode 17 The Joy of Infinite Series
Starting with the analysis of the proposition 0.999999999 ... = 1, this lecture explores what it means to add up an infinite series of numbers. Some infinite series converge on…
Episode 18 The Joy of Differential Calculus
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…
Episode 19 The Joy of Approximating with Calculus
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,…
Episode 20 The Joy of Integral Calculus
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…
Episode 21 The Joy of Pascal's Triangle
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":…
Episode 22 The Joy of Probability
Mathematics can draw detailed inferences about random events. This lecture covers major concepts in probability, such as the law of large numbers, the central limit theorem, and how to measure…
Episode 23 The Joy of Mathematical Games
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…
Episode 24 The Joy of Mathematical Magic
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…