Ponder another surprising appearance of geometry--the mathematics of numbers and number theory. Look into the properties of square and triangular numbers, and use geometry to do some fancy arithmetic without a calculator.

Running Time

33 mins

Year

2014

Kanopy ID

1338455

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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, and how do you calculate in bases other than 10?

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 beautiful and unexpected properties, and show up in nature, art, and poetry.

Visualizing the Fibonacci Numbers

Learn how a rabbit-breeding question in the 13th century led to the celebrated Fibonacci numbers. Investigate the properties of this sequence by focusing on the single picture that explains it all. Then hear the world premiere of Professor Tanton's amazing Fibonacci theorem!

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 to derive the Pythagorean theorem and other interesting results.

Complex Numbers in Geometry

In lecture 6, you saw how 17th-century mathematician Rene Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane. Examine how using complex numbers can help solve several tricky…

The Equation of a Circle

In your study of lines, you used the combination of geometry and algebra to determine all kinds of interesting properties and characteristics. Now, you'll do the same for circles, including deriving the algebraic equation for a circle.

Visualizing Mathematical Infinities

Ponder a question posed by mathematician Georg Cantor: what makes two sets the same size? Start by matching the infinite counting numbers with other infinite sets, proving they're the same size. Then discover an infinite set that's infinitely larger than the counting numbers. In fact, find an infinite number of…

Divide and Conquer

Turn now to the last fundamental operation of arithmetic: division. Explore a variety of shortcuts for dividing by one- and two-digit numbers; learn how to convert fractions such as 1/7 and 3/16 into decimals; and discover methods for determining when a large number is divisible by numbers such as 3,…

Pushing the Picture of Fractions

Delve into irrational numbers--those that can't be expressed as the ratio of two whole numbers (i.e., as fractions) and therefore don't repeat. But how can we be sure they don't repeat? Prove that a famous irrational number, the square root of two, can't possibly be a fraction.

Advanced Multiplication

Professor Benjamin shows you how to do enormous multiplication problems in your head, such as squaring three-digit and four-digit numbers; cubing two-digit numbers, and multiplying two-digit and three-digit numbers. While you may not frequently encounter these large problems, knowing how to mentally solve them cements your knowledge of basic mental…

The Joy of the Imaginary NumberĀ 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 in physics and other fields, and are easier to understand than they appear.

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 the golden ratio.

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