Math Gems. MathPages. Banach-Tarski Paradox. 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"!

Khan Academy. Integral Table. 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. Calculus Mega Cheat Sheet. 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! 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. MathPages.