Math Is Fun Forum

Hi;

You probably thought that finding the square root will simplify the expression, but that is only true once you assume that x is positive (which it is).

Hi;

Won't there be infinitely many solutions to this? Perhaps it should be restricted to integer coefficients or something. Nevertheless, I am unfamiliar with the methods of answering this question.

Hi Solvitur, welcome back (:

The question of whether the idea of an infinite universe makes sense is discussed by all sorts of thoughtful yet unqualified folk xD It puzzles a lot of people that, for instance, one thing could be twice as large as another, while both are equally insignificant parts of an infinitely large whole. Or, similarly, that one thing is made up of infinitesimal bits, while another seems to be made up of twice as many. It is my equally unqualified understanding that in physics, 10^-100 metres is nonsense; nothing related to length occurs at that scale to distinguish it. If my statement that the infinite divisibility hypothesis is unverifiable is valid, then it is a significant objection, because the scientific method takes experimentally unverifiable claims to be worthless.

I used to think, and some philosophers do think, that the idea of an essentially random event is logically incoherent. Your first question is rhetorical unless this can be demonstrated, and I am quite sure it cannot. As to your second question ... I suppose you have given me a very concise explanation xD The quantum theory has led us to disregard our intuitions about causality, and I think rightly so.
Incidentally, Einstein himself claimed that the universe is essentially determined despite quantum theory (which he regarded as provisional); he also admitted that this belief was entirely based on faith.

PS: Regarding your last question, you may find the OP here http://www.mathisfunforum.com/viewtopic.php?id=22752 to be relevant. Basically, there is almost nothing about the world we can be absolutely sure of. If you are doubtful enough, you can virtually always find a way that reality might not at all correspond to what you observe or think.
The scientific method is so successful because it comprises the set of principles (read: assumptions) that best identify consistency in what is observed. That, I think, is the basis of gaining knowledge about the world.

Not really. Although I can see why you'd link it, since the Gettier problem is about how difficult it is to specify the "justified" criterion of knowledge. But it's not the same because most people do not want to define knowledge as certainty; most people are fallibilist. So we aren't specifically discussing the general definition of knowledge at all; unless you want to make knowledge and absolute certainty synonymous.

I think that is the simplest interpretation, yes (:
There are an infinite number of pages in this book. Every day has a page that represents it. Assume they are in order. It appears, at first, that he will not get to have a birthday very often, since for every birthday page, there are 364.2425 non-birthday pages (if he isn't born on leap day; if he is, then there are 1,505 and 15/97 non-birthday pages for each birthday page!). But nevertheless, he can always flip to the next birthday page and tear that out for tomorrow, so effectively he has more birthdays.

I honestly think that the role of the book and the gods is unimportant. The principle is that he can choose the order of an infinite set, such that only the least common elements appear, and that is somewhat counterintuitive.

I understood that I'm not sure if you can say that 0/0 could be anything, honestly, unless you create a framework for it. But in elementary arithmetic, although no answer is satisfactory, I think some answers make more sense than others.

In particular, if you graph a few functions, you will get answers for what functions approach when various things get arbitrarily close to 0, and they will not be 3.9425. I stated in the thread "What is 0^0?" what these limits are.
For example, you mentioned that the answer to 1/0 could be infinity. But if you graph 1/x, you will find that it could be infinity ... or it could be minus infinity, depending on which direction you approach x=0 from. The same holds true for any nonzero real number in the numerator.

As for 0/0, if you graph x/x, it is always 1. If you graph 0/x, it is always 0 (although x has to be positive). So, the answer to 0/0 could be either 1 or 0, or plus or minus infinity, in a sense. But it doesn't work unless there is one consistent answer.

Hello,
I am not totally sure of the interpretation, but I have assumed that you are after a single a-value that gives two x-values where one is double the other.

Hi;
Oh, you just average them (:

Hi!

My answer agrees with yours bobbym, even though it is a strange result for this kind of problem P:

Hello!

Hello;

It's actually quite enlightening to graph the various functions and see why it's so ambiguous what happens at 0.

For 0^0, if you graph:
x^x you get 1,
x^0 you get 1, and
0^x you get 0.

For 0/0, if you graph:
x/x you get 1, and
0/x you get 0.

For ordinary division by 0, if you graph y/x you get plus or minus infinity. Plus the implication that if, for instance 1/0 = 2/0, then 1 = 2