Hello everyone!

Much as it pains me to say this, I need to take a break.  My life is getting busy, and I don't have the energy to do this too.  I promise I'm not going away forever.  And I'll still be around in the comments, if anyone has questions or an interesting topic to discuss.  But I need the next few months to sort things out.  Thank you for sticking with me thus far, and I hope you'll join me when updates resume.

Mostly Mental will return November 9, 2014.

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Hello everyone!

For those of you who missed it, I posted a proof of the Pythagorean Theorem every day last week.  For easy reference, here are the links:

Square Rearrangement - Trapezoid Area - Sliding Parallelograms

Euclid - Power of a Point - Similar Triangles - Scaled Copies

All these proofs, and many more (103, as I write this) can be found here.  There are all sorts of spiffy ideas in there, including dissections, tessellations, and even a proof based on Newtonian mechanics.  The language is a bit dense, so I'm happy to clarify any of those proofs in the comments.

Continue reading Pythagoras Week: Conclusion →

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Hello, everyone!

Before I launch into the content, I should probably introduce myself.  I'm Kyle.  I'm currently a college senior studying math and computer science.  In particular, I like areas such as combinatorics (fancy counting), graph theory (the study of networks), and algorithms (getting useful information out of lots of data), but my mathematical interests are pretty broad.  I'm also a huge fan of puzzles, in pretty much every form.  Jigsaws, crosswords, sudoku, chess, etc.  I like things that make me stop and think.

So why am I making this blog?  I've heard a lot of people say they hate math.  The thing is, what they think of as "math" is calculation.  It's taking a few numbers, following a set of rules, and arriving at another number.  I suppose that could be considered useful, but it's computation, not math.  We've invented computers to do that for us.  Math is something else entirely.  It's useful, it's logical, and, if you ask me, it's beautiful.  It's a way of systematically reasoning about the world.  It's about proving that assumptions lead to conclusions without any room for doubt.

That's a pretty vague statement, so let me clarify with an example.

You may have heard that the whole numbers (1, 2, ) go on forever.  But how do we know that?  Well, let's say they don't.  Then there has to be a largest whole number.  It has to be pretty big, as it's larger than every other number you can think of, so let's call it   What happens if we add one to it?  We really don't know much about but we do know that adding one to any number makes it bigger, so   But that can't be right; we've got a whole number bigger than when we assumed that was the largest.  That means one of our assumptions must have been wrong.  The only assumption we made along the way was that there is a biggest whole number, so that must not be true.  Thus, the whole numbers must go on forever.

This is a pretty simple example, but we can derive much more interesting results in a lot of fun ways.  I hope you'll follow along to see some of the power and beauty of mathematics.

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