(no troll)

  1/2 + 1/4 + 1/8 + 1/16 + ... = 1

is like a binary version of

  9/10 + 9/100 + 9/1000 + ... = 0.999... = 1

Good observation. This is more apparent if you switch to decimal notation, (For want of a better term. I'm sure there's a name for it, I just don't know what to call the decimal point when you're talking about binary. Surely "decimal" point is wrong.)

In binary:

0.1 + 0.01 + 0.001 + 0.0001 + ... = 0.1111... = 1

In decimal:

0.9 + 0.09 + 0.009 = ... = 0.999... = 1

I discovered that 5th one while doodling in high school. It blew my mind at the time (and it can be extended to dimensions other than 2).

If you like this, you'll love the book Proofs without Words: Exercises in Visual Thinking: http://www.amazon.com/Proofs-without-Words-Exercises-Classro...

Wow. Thanks!

I got the idea for this post from something I first saw on reddit/r/math earlier this week, but I also so it here on HN (you might recognize the first proof), so I thought I'd submit this here also. Hope you like it.

I also saw it here...and I need to learn how to proof read.

Oh my god. The Pythagorean one just blew my mind. I am so embarrassed.

IIRC that proof is from India around 900 C.E.

I'm not sure if you'll check back on your post, but if you do I think you might really enjoy http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/byrne.ht... (it's an illustrated version of Euclid's "The Elements").

What made the 1/4 + 1/16 + 1/256... = 1/3 that was posted earlier this week so good was that it did not use numeric labeling, but purely visual content to make its point. If you could figure out a way to graphically represent what is now labeled, it'd be a complete success.

I'm pretty sure "1/4 + 1/16 + 1/256... = 1/3" was in the title of the post.

true enough, buts its quite different if it is wedded to the image.

Awesome post. I think I upvoted every comment too.