[Twenty-eighth] Guest post!

So, as you may gather, this post isn't by Hong Wei. This post is by her "irritating" [sic] brother, because, well, she uses the same password for (almost) everything, and I happen to know what the password is. So, if you know her password for anything, you know it for everything.

So, anyway, we can have some real content now. Real intellect. Not like all the nonsense you read below or in the previous posts (ahem auroar). This one will be real stuff. Or maybe not. You have the option of scrolling down and be flooded with randomness or spam as you decide, or continue reading this stuff.

I could write about plenty of things in my mind, but most of them wouldn't really fit in here, so I'll just pick one at random. Mathematics. In my school, maths is the most popular subject, but, I figure people in other schools don't have such a great liking for maths. But, regardless, I don't care.

You see, maths is awesome. What you learn in school is maths, but it's not the real stuff. Not the real hardcore stuff. (Er, not hardcore as in porn, y'know?) The real stuff lies beyond that. There are plenty of interesting mathematics around, that is undeniable. Things like divergent series. I'm not sure whether divergent series are covered much in the A-level syllabus, but I'm pretty sure you encountered them while learning geometric progressions. I'm not sure whether you learned that yet, but unimportant to current discussion.

Divergent series are basically infinite sums that, well, don't end up at a single value. A classic example is 1+2+4+8+16+…. The terms keep doubling, and we may as well say that the sum is infinity. But what if, we be a little lax, and start manipulating it a bit? Be warned, such manipulations can cause it to have any value. But trudging on anyway, we can find that, since this is a geometric progression with term ratio 2, the value can be taken to be 1/(1-2), which is −1 !

You might be thinking that this is all bullshit. How can a sum of ever-increasing positive numbers become negative? Well, note that it was ever-increasing. Sometimes, we can allow ourselves to think like Euler: the number line actually wraps; once you go beyond infinity you come back at negative infinity. And somehow, this causes it to end up at −1. This sounds a bit wacky, I know, but it sorta makes sense. xkcd agrees. (Actually, not really.)

We can even have 1-3+5-7+9-11+…, which I can show equals 0. Ah, you might counter that for not only does it alternate, it keeps getting bigger! Well, I don't care. It just is.

Of course, please do not use these methods in your A-level examinations. You will fail if you do. I am not responsible if you fail. Do not blame me. Blame Hong Wei, because she was the one who let me type out this post.