# MathPages

Free Calculus The Calculus Here is a free online calculus course. This is essentially an ordinary text, but you can read it online. There are lots of exercises and examples. This text is somewhat unusual for two reasons. The text is rigorous. Both points are no doubt controversial, but conceptually the approach gives a kind of clean synergy which generates important examples and unifies calculus to a great extent. Unfortunately, there are no doubt uncorrected typographical errors and logical errors. This text © 1993, 2001 Copyright William V. Table of Contents

The Top 10 Coolest Websites: Best of 2011 | All My Faves | Blog Posted on Monday, January 2nd, 2012 by Danny Davies Well folks, happy new year. In a blaze of fireworks and e-greetings, 2012 has finally arrived. And as we stride over the finishing line of a tumultuous year for the internet, All My Faves – your one stop easy guide to the Web – has decided to look past Facebook and Twitter to find 2011′s most unique and interesting sites. We selected the 10 coolest sites in categories like music, travel and education; and then handed the job of picking the absolute best in each category down to you, allowing you to vote on your favorites. 1. Decide.com – I’m sure you’ve heard the phrase “The Age of Austerity” being bandied around by media commentators this year. In other words, Decide.com uses clever prediction algorithms to warn you if the price of any given electrical product is about to drop, increase, stay the same, or if a new model is about to be released. 2. 3. 4. What Movie Should I Watch Tonight? WMSIWT is a really, really, simple site. 5. 6.

Prehistoric Calculus: Discovering Pi Pi is mysterious. Sure, you “know” it’s about 3.14159 because you read it in some book. But what if you had no textbooks, no computers, and no calculus (egads!) — just your brain and a piece of paper. Could you find pi? Archimedes found pi to 99.9% accuracy 2000 years ago — without decimal points or even the number zero! How do we find pi? Pi is the circumference of a circle with diameter 1. Say pi = 3 and call it a day.Draw a circle with a steady hand, wrap it with string, and measure with your finest ruler.Use door #3 What’s behind door #3? How did Archimedes do it? Archimedes didn’t know the circumference of a circle. We don’t know a circle’s circumference, but for kicks let’s draw it between two squares: Neat — it’s like a racetrack with inner and outer edges. And since squares are, well, square, we find their perimeters easily: Outside square (easy): side = 1, therefore perimeter = 4Inside square (not so easy): The diagonal is 1 (top-to-bottom). Squares drool, octagons rule Cool!

Take It to the Limit In middle school my friends and I enjoyed chewing on the classic conundrums. What happens when an irresistible force meets an immovable object? Easy — they both explode. Philosophy’s trivial when you’re 13. But one puzzle bothered us: if you keep moving halfway to the wall, will you ever get there? Questions like this have always caused headaches. But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite. In the coming weeks we’ll delve into the great ideas at the heart of calculus. If you’re a careful thinker, you might be worried about something already. Imagine using a photocopier to reduce an image of a circle by, say, 50 percent. Of course, this doesn’t tell us how big pi is. Before turning to Archimedes’s brilliant solution, we should mention one other place where pi appears in connection with circles. Here A is the area, π is the Greek letter pi, and r is the radius of the circle, defined as half the diameter. Yes, it is. More in This Series

Integral Table Khan Academy Banach-Tarski Paradox -- Math Fun Facts Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox. So why can't you do this in real life, say, with a block of gold? If matter were infinitely divisible (which it is not) then it might be possible. But the pieces involved are so "jagged" and exotic that they do not have a well-defined notion of volume, or measure, associated to them. In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"! Presentation Suggestions: Students will find this Fun Fact hard to believe. The Math Behind the Fact: First of all, if we didn't restrict ourselves to rigid motions, this paradox would be more believable.

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