1 + 2 + 3 + 4 + ... = -1/12
Now I'm the first to admit that this result is at least counter-intuitive and definitely requires some pretty sophisticated mathematical thinking. If you'd like a pretty good explanation and have a hankering for some math this morning, you can check out this blog post that includes the original video.
Infinity is a pretty interesting concept, one I love to use to challenge my students to think a little differently, to think outside the box. I offer a couple of examples here that may help you understand why infinity is such an interesting concept.
Zeno's Paradox: The Archer's Arrow
An archer picks up an arrow, places it against the bowstring,
draws the string back and lets the arrow fly towards the target. It looks like it will be a bulls-eye, but of course we won't know for sure until it hits the target.
Before the arrow reaches the target, it must first travel half of the distance to the target. From there the arrow travels half of the remaining distance to the target. Quickly, the arrow travels half of the distance which remains after that, then half of the distance that is still between it and the target. In fact, before it can get to the target, the arrow must always first go half of the distance that remains between it and the target. Does the arrow ever hit the target?
Consider the counting numbers (called Natural Numbers). They are infinite. The first is 1, the second is 2, the fifth is 5, the 100th is 100. They go on forever, as whatever number you pick, I can add one to it and keep going. Because I can count them, this kind of infinity is called countably infinite.
Now consider the numbers 0 and 1 How many number lie between them? Well, there's all the fractions, and I hope you can see that there's an infinite number of those. I'll mention a few, 1/100, 2/100, 3/100.... 1/1000, 2/1000, 3/1000... Can you see how there a huge number of these? There's also a whole lot of other numbers in there called irrational numbers. They are decimals that neither repeat nor end. Here's one: .1234567891011.... Here's another one... .101001000100001000001.... In fact, there's an infinite number of numbers between any pair of numbers and an infinite number of pairs of numbers between 0 and 1. In fact, there's an infinite infinities between 0 and 1, in this case called uncountably infinite. I can't even imagine where to start counting here. Hence it's name, uncountably infinite.
Point 9 repeating, .9999.... equals 1.
There's a simple little proof that shows that .9999.... = 1 as shown in the diagram. The lovely little video explains it and a lot more. I know, I know... this whole discussion is also counter-intuitive. As a student said in my class a few years ago, "I get it, but I just don't buy it." I just love infinity!