a=bAnd then divide both sides by b. Voilà! 2=1. So you see, numbers are actually highly deceptive unless you know how to spot missteps - and not just the accidental kind that randomly crop up here and there, but the conceptual kind of missteps that prevent even the most scrupulous of mathemancers from conjuring a correct answer.
Now multiply both sides by a.
Now subtract b2 from both sides.
Now factor out the quantity a minus b.
You usually factor things out in order to divide by them, so do that.
Now swap out that a for a b, since their equivalence was our starting point.
I was reading Lucifer's Hammer the other day, which is about how the world ended like 35 years ago, and an astronaut thinks to himself that humans burn brighter than the Sun. Yeah, just like they sing in that song.
Well, not them. But they're cooler.
So there's no way biology burns hotter than motherfucking nuclear fusion, right? Right. But if you do the math wrong in just the right subtle way, you can get fooled. Rick Delanty reflects that there's just so much more Sun (or was it Johnny Baker? I don't know, I can't find the quote any more, and it's not important anyway), but he's still wrong because that isn't what bridges the gap between the Sun burning my pale flesh from a hundred million miles off and me not zapping it back with my laser eyes. Because if I burned brighter than the Sun, I'd have those. Tru fax.
Never one to do work when there's a chance someone beat me to it, I turned to Google. Answer: Phil Plait beat me to it two and a half years ago. First he compares surface temperatures (Sun wins), but Lucifer's Hammer puts it in terms of cubic centimeter against cubic centimeter - which Phil obligingly tackles next. Shit. Do I still have an article? Why, yes! Phil arrives at the final figures: Sun, 2.8 ergs/sec/cc; Human, 170,000 ergs/sec/cc. This says that humans are brighter than stars, and he also blames it (partially) on "so damn much Sun." The other part he blames is that the Sun gets hotter the farther in you go, from about 6,000 Kelvins on the surface to 15,000,000 Kelvins at the core. And since luminosity functions off of temperature to the fourth power, that factor of about two and a half thousand gets blown up quite a bit more (well more than enough to eclipse humanity's lead factor of 60,000).
So what's going on here? I mean, since the inside of the Sun is hotter, shouldn't it also be brighter? And if so, how come that brightness gets, what, trapped inside or something? Well, yeah - that's actually more or less exactly it. See, using the Sun's surface brightness to compare who's the badassinest burner only makes sense if you don't know how the Sun works. If you could slice the Sun open and look inside, you'd get something that looks a lot like this:
Found here (click the Playskool-lookin' one), but credited to NASA.
Now, that brightest chunk in the center is where the magic happens. This is the only area of the Sun that actually produces energy, i.e. "makes it bright." Every other part of the Sun, which is to say most of it, simply reacts with this energy. The core is where you get that hot, hot fusing of one proton into two protons, and that energy then spends millions of years in the radiative zone. Like it's in line or something. (And now can you guess why stars, as a rule, get bigger over time?) But eventually, that built-up energy passes into the convective zone, where it swirls around like a bunch of those chimney-lookin' coal starters. And then it finally, finally enters the photosphere, the very outermost layer, where it piddles out. Relatively speaking. And that piddly leakage is enough to power an entire planet at a hundred million miles off - as long as it's rotating, or it'll scorch - in every direction and all at once.
Our Sun won't go supernova, it's not massive enough. But it will likely enter a red giant phase which would inflate it enough to engulf Mars (who's brighter now?). Anyway, the point is: proving with math that humans are brighter than the Sun, by using the Sun's surface as its measure, is as meaningless as proving that 2=1 by surreptitiously dividing by zero. Because the real action happens at the core of the Sun, and everything between there and the hundreds of thousands of miles to the surface is just hydrogen getting in the way. This means that when we're talking about what makes the Sun bright, it's what's inside that counts. I guess we and stars aren't really that different after all? Well, the apple doesn't fall very far, as they say.