John

You could do (1+1) mod 2, but that's just 0. 1+1 in binary is, in fact, 10. And yes, (10+1) mod 10 = 1.

Wikipedia is fairly accurate on most topics. Also, the specific part of the article I linked is a mathematical proof that switching is the rational decision. It's also followed, a bit further down, by other mathematical proofs. Below that, there is a list of variants of the problem that lead to different results, including the 50/50 you intuitively expect. Edit: Here is another article detailing programs written to test the idea: http://en.wikipedia.org/wiki/Empirical_solution_of_the_Monty_Hall_problem This is, of course, another Wikipedia article, but in this case you can easily run the code yourself to verify the results. In addition, http://leeps.ucsc.edu/misc/page/monty-hall-puzzle/ also provides references to studies done confirming the result.

Again, the Wiki: http://en.wikipedia.org/wiki/Monty_Hall_problem#Bayes.27_theorem

When the contestant chooses a door, the probability that the car is behind that door is 1/3. This means there's a probability of 2/3 that the car is behind one of the other two doors. One of the two others is opened to reveal a goat. The probability that the car is behind the contestant's chosen door is still 1/3; therefore, the probability that the car is behind the unopened door the contestant didn't initially choose is 2/3, and the contestant should switch. To put it another way, consider three doors. There are only three possibilities here (considering the goats to be interchangeable): car, goat, goat; goat, car, goat; and goat, goat, car. Let the contestant choose any door, and then let a second door be opened to reveal a goat. In one of the three possible orderings, switching will cause the contestant to go from car to goat, losing. With either of the other two orderings, switching will cause the contestant to go from goat to car, winning. Therefore, the probability that switching will result in a win is 2/3. There are several explanations of the solution in the Wikipedia article.

I think at first glance it almost looks like an example of the gambler's fallacy, but most mathematically literate people probably accept it at this point.

Euclid probably wasn't aware of the curvature of spacetime. There are others geometries we can use to deal with such things, though Euclidean geometry is still fairly useful, of course. You might want to read through the Wikipedia article on "axiom."

To answer the second specifically (and using [math]h = \Delta x[/math] to help avoid careless errors), set it up as the following: [math]\lim_{h \to 0}\frac{\frac{3}{(x + h)^2} - \frac{3}{x^2}}{h}[/math] and simplify. The first can be handled in a similar fashion.

LaTeX is integrated with the forums. The way it works is you use [math] and [/math] around the LaTeX, like so: [math]\sigma_f=\sqrt{\left(\frac{\partial}{\partial x_1}f\sigma_{x_1}\right)^2+\left(\frac{\partial}{\partial x_2}f\sigma_{x_2}\right)^2+\ldots+\left(\frac{\partial}{\partial x_n}f\sigma_{x_n}\right)^2}[/math] Clicking on that will show the code used and a link to the SFN LaTeX tutorial.

I would imagine the graph would look completely full. Given the line between two points, no matter how close a third point is to that line, a fourth point can be placed even closer without being on the line.

What you might do first is express the xor in terms of and and or. Your problem is saying that either the left side is true or the right side is true, but not both, so [math]\neg c \oplus (\neg b \vee c)[/math] is the same as [math](\neg c \wedge \neg (\neg b \vee c)) \vee (\neg (\neg c) \wedge (\neg b \vee c))[/math]. You can then use DeMorgan's law on the [math]\neg (\neg b \vee c)[/math] part and go from there. Does that help at all? As for verifying your result, you can construct a truth table (if you've learned how to do so) and see whether the values for your result match the values for the original statement. In fact, constructing a truth table for the original statement will make it clear what the simplified statement will be.

I think most of us who haven't been on a constant upward path throughout our lives struggle with confidence issues in this situation. I myself am pretty much convinced I'm fairly intelligent, and even I keep wondering if I've made a mistake deciding to pursue mathematics. At least for me, I find myself thinking about the giants---Newton, Euler, Gauss, Pauling, and so many others---and I wonder why I even bother, since the odds are I'll never be on their level in their respective fields. However, at the end of the day, science owes its current state to far more than just the relatively few names that have made the history books. If you're the next Einstein, then great, but if not, you can still make valuable contributions to your field. Intelligence is required, of course, but at the risk of sounding like I'm catering to political correctness, I think for the bulk of people who would even consider going into science, it's more a matter of interest and dedication. Then, of course, there's the feel-good notion that you should at least give it a shot, because if you do you may fail, but if you don't you'll never know whether you would have succeeded. I know the risk of failure is hard to swallow (at least for me, being well past the age most people attend and even graduate college, and therefore worried about wasting more time), but avoiding all risk is impossible, and attempts to do so leave life a bit dull. Try a few classes out, and see what seems to work best for you, keeping in mind that lower-level courses aren't necessarily the best indicators of what later classes will be like. Don't be caught up quite so much in choosing the "right" field. There are numerous examples of scientists who completed their undergraduate work in field X, then went to graduate school and started careers in field Y. It may not be the most direct or efficient route, but these examples at least indicate that if you do get to the end and realize you've made the "wrong" decision, it's not as if you've ruined your chances of ever being happy. Good luck, in any case. I'm afraid I don't consider myself to be in a position to really comment on the general topic at hand.

I think you misread that. In the first limit, x is decreasing to negative infinity, not increasing to infinity. I'm not certain of how to solve it myself (I really need to brush up, apparently), but going through the first several negative integers it does seem to be going to -1 after being non-real on (-2,0).

Well, thank you. This is an example of the social factors I mentioned, which are an unfortunate result of the human element in science. I have a perhaps naive expectation that the better theory would eventually win out, though. The recent neutrino results at CERN, if confirmed, will open up new avenues for theorists to explore. While it's probable (and understandable) that new theories presented by established scientists in the field will be taken seriously more easily than those presented by laymen (like myself, though I'm not likely to try amateur theoretical physics any time soon), if a random amateur does come up with something amazing, I imagine it will gain traction in time. Sometimes it takes decades or even centuries, but the truth has historically found a way to gain acceptance, despite people's efforts to stop it.

There are a variety of ways in which species can evolve to suit a particular environment. In the case of human beings, intelligence was apparently advantageous, and was selected for over other traits that might have been beneficial in other ways. I think I read that humans are superb long-distance runners, for what it's worth. But, ultimately, our intelligence has allowed our species to adapt and thrive. Of course, evolution continues to act on all species, and human intelligence may eventually be selected against for whatever reason.

Perhaps, but if you disagree with mainstream physical theories, formulate better ones that match experimental results at least as accurately as the current ones. Physicists aren't out to worship Einstein. Their goal is to understand how the universe works. Relativity and quantum mechanics are the best descriptions we've come up with so far, but research to improve upon them continues. If it becomes apparent that relativity and quantum mechanics are wrong, then there will be skepticism/resistance perhaps for a while, but eventually they'll be discarded in favor of whatever more accurate theory is developed. This is how science works--there are social factors to consider, of course, but the field as a whole adapts to new discoveries.

It should be noted that human beings evolved to interact with the universe on scales at which the more startling results of relativity and quantum mechanics aren't immediately obvious (which, of course, is why they're startling). Therefore, it's understandable that said results aren't intuitive and don't necessarily "make sense" until one reaches a sufficient level of education in physics to see how they're derived. I myself am willing to accept them despite my own lack of said education, based on the facts that a) experimental results agree with the theories to a high degree of accuracy, and b) the scientific consensus is therefore that, while said theories may not be complete, they are at least accurate enough to be useful, and will likely play some part in whatever final theory is eventually found. Of course, it is possible that some result, or set of results, will be found that will completely destroy both theories. However, given how accurate they are, the odds that they're totally wrong seem fairly slim. This is why the scientific community is attached to them. It's not some religious adherence to dogma, devoid of evidence or logical basis. It's confidence borne of mountains of evidence supporting both theories. As for the form a final theory might take, who knows? While there is an idealistic notion of some simple equation that neatly ties together everything we observe, it's possible that the universe doesn't ultimately fit into something to tidy. Calculations based on the final theory might still require loads of processing power to perform, and simplified approximations may be necessary. There may end up being several competing theories that all explain the universe equally well, and then we'll be in a pickle trying to decide which one truly reflects reality. And then maybe the next Ed Witten (or the current one, if the theories are developed soon enough) can announce that he's found some wonderful symmetry that ties them all together. We'll see.

John figured Cyclops had experienced neural adaptation leading to the motion aftereffect, and, failing to formulate a hilarious reponse, replied with links to Wikipedia.

Well, as I mentioned, there is overlap. I made the distinction into three (or four) separate categories more to illustrate that even if psychology is viewed simply as the study of some abstract and subjective notion of the mind (which it isn't, of course, but if it is viewed that way), it's still a field of science. I suppose I could have worded it better, and perhaps even my own view of the field is rather elementary. In any case, thanks for the clarification.

Precise definitions of what constitutes a "science" may vary over time or with who you ask, but I tend to go with the categorization of various disciplines into three broad fields (note that there is some overlap in some subdisciplines): the formal sciences (e.g. mathematics, theoretical computer science), the natural sciences (e.g. physics, chemistry, biology---sometimes further broken down into the physical and life sciences), and the social sciences (e.g. psychology, sociology). Each category has its own methods of arriving at conclusions about the world, despite limitations inherent in each. In general, psychologists, like other scientists, design and perform controlled experiments to test hypotheses and develop theories. While certain aspects of psychology may be open to interpretation, this in no way diminishes its status as a science as valid as any other. Indeed, certain aspects of any science may be open to interpretation. I wonder if perhaps some people recoil from the notion that something as personal as the human mind can be systematically studied, and try to diminish the legitimacy of psychological research as a result. In any case, tl;dr "yes, psychology is a science."

Let [math]x[/math] represent the number of apples and [math]y[/math] represent the number of oranges. Then, if there are twice as many apples as oranges, then the number of apples ([math]x[/math]) will be two times the number of oranges ([math]y[/math]). Expressed as an equation, this means [math]x = 2y[/math].

Really an excellent little video. Thank you for sharing.

Oh, really?

Well, that's anticlimactic. The methods described so far will provide you with the answer, but it's good that you got your teacher's input. Have fun with the rest of your geometry class, and feel free to ask any other questions that might come up as you progress.

I'm really not much of a forum person.