Calling all math people:

urahonky

I took a quiz a week ago and couldn't figure out how to do it at all. My fault. But now that I'm looking at his answer I have no idea how he got there. And I'd like to know.

http://img178.imageshack.us/img178/7089/noidea.jpg

This is the image. I understand the -b+-sqrt(blahblah) stuff, but towards the end....


He has 2(10^121 + sqrt(10^242 - 4)) and that somehow turns into: 4 * 10^121

My question is: HOW? He says: Since 4 is small compared to 10^242, but that is just for the = sign right?

Doc

He's just ignoring the 4 in sqrt(10^242 - 4). Since sqrt(10^242) = 10^121, sqrt(10^242 - 4) is going to be ridiculously close to the same number.

2(10^121 + 10^121) = 4 * 10^121

urahonky

Good God... I feel like an idiot. I stared at that for so, so long and couldn't figure out how he did that. Ugh. Thanks Doc... Seriously I owe you.

GoodOmens

Is there more to the solution? I ask because the picture you posted doesn't seem to answer the question (though the answer is pretty obvious now).

urahonky

Oh yeah there's more. The other end of the root wasn't hard to understand since he didn't have to multiply both sides by the reciprocal. Also thanks for the help Doc, one of those questions were on the exam I took yesterday.

Jealous Deva

That is honestly a bullshit way to solve that problem.

Clearly he doesn't want to ignore the 4 in the second step, since that would simplify to the answer 0. But it's just as valid an approximation to suggest in the second step that 10^242-4 = 10^242 and simplify to 0, as it is to multiply top and bottom by the entire equation. Either way you are getting an approximation. There's no justification for going the extra step just to get a nonzero value.

rabidrabbits

I have some issues with the *solution* to this question, but first I'm going to address the "extra step" problem. The reason for going that extra step is that the root is very definitively not 0. Let's try this out:

f(0) = 1. This is not a solution, nor is it close to a solution.
f(1.0*10^(-121)) = 1.0*10^(-242). Not exactly 0, I know, but it is still remarkably close. In fact, to get any closer, we'd need to include roughly 242 more significant figures on our roots.

So why not go a third step, and get those additional 242 significant figures? Or a fourth, to get 242 more? Once we get going, why even stop?

The answer is simple. 0 is not the root, and has 0 sig figs. It may be close, but it tells us nothing about the nature of the root (order of magnitude, positive or negative, etc). 1.0 * 10^-121 is not the root either, but it has 2 sig figs (we really only need 1, but a 2nd is nice) and tells us the order of magnitude of the root as well as the fact it is positive. With any further approximations, we'd get 0.9999... with 241 or so 9's and then maybe a .6 or something, which is waaaay too much to write down on a page when it is only 10^(-240)% different than our 1st order approximation.

I hope that clears up the reason for approximating.



Now my issue with our *solution*. We don't even need to do any approximation in the first place. If you add the two roots of a quadratic, you get -b/a. You can even solve for them (and leave them in the complicated form with an integer adding a square root), then when you add them together the square roots cancel each other out.

urahonky

Absolutely makes sense now guys, thanks. The class is called Numerical Methods for Computation Science, and this example was supposed to show rounding errors and such. I hope that makes more sense on why that is the solution.