I just saw an interesting article on /r/math detailing how Polynesians had invented binary notation 600 years ago. I love when these fundamental concepts are rediscovered over and over. I've been reading a lot of Iain Banks' science fiction lately, and one great passage of his says something along the lines of "one of the few things that all sufficiently developed species can agree upon is that integers should be represented in base 2." It wouldn't even require so much conversion: a week becomes 8 days, a month 32 days, a "thousand" becomes 1024, and so on. Somehow I don't think this is going to get much popular support...
I read a great article in Quanta about the recent flurry of progress on the twin prime conjecture. I'm not a number theorist, and the article did a good job of relating the big ideas at a high level for a wide mathematical audience. I absolutely love that Polymath8, and now Polymath8b, have had so much to contribute. It'll be interesting to see how far the current approach can bring down the bound.
Doing and teaching mathematics.