Episode 35 of 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 geometry problems.

Running Time

33 mins

Year

2014

Kanopy ID

1338457

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Distance, Midpoints, and Folding Ties

Learn how watching a fly on his ceiling inspired the mathematician Rene Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more.

The Geometry of Figurate Numbers

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.

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.

Proofs and Proof Writing

The beauty of geometry is that each result logically builds on the others. Mathematicians demonstrate this chain of deduction using proofs. Learn this step-by-step process of logic and see how to construct your own proofs.

The Pythagorean Theorem

We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and how it's a critical foundation for the rest of geometry.

BeginningsāJargon and Undefined Terms

Lay the basic building blocks of geometry by examining what we mean by the terms point, line, angle, plane, straight, and flat. Then learn the postulates or axioms for how those building blocks interact. Finally, work through your first proof--the vertical angle theorem.

The Power of Mathematical Visualization

World-renowned math educator Dr. James Tanton shows you how to think visually in mathematics, solving problems in arithmetic, algebra, geometry, probability, and other fields with the help of imaginative graphics that he designed. Also featured are his fun do-it-yourself projects using poker chips, marbles, paper, and other props, designed to…

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.

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…

Memorizing Numbers

Think that memorizing long numbers sounds impossible? Think again. Investigate a fun: and effective: way to memorize numbers using a phonetic code in which every digit is given a consonant sound. Then practice your knowledge by trying to memorize the first 24 digits of pi, all of your credit card…

Visualizing Negative Numbers

Negative numbers are often confusing, especially negative parenthetical expressions in algebra problems. Discover a simple visual model that makes it easy to keep track of what's negative and what's not, allowing you to tackle long strings of negatives and positives--with parentheses galore.

The Visuals of Graphs

Inspired by a question about the Fibonacci numbers, probe the power of graphs. First, experiment with scatter plots. Then see how plotting data is like graphing functions in algebra. Use graphs to prove the fixed-point theorem and answer the Fibonacci question that opened the lecture.

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