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gib65

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About gib65

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    Protist
  • Birthday 12/29/1976

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    http://www.shahspace.com/art

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  • Location
    Calgary
  • Interests
    writing, drawing, philosophizing...
  • College Major/Degree
    B.Sc. computer science & B.A. psychology
  • Favorite Area of Science
    quantum/relativity/modern... the deep stuff
  • Occupation
    web/software developer

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  1. So would you say that the sequence (0.9, 0.99, 0.999, ... , 1) is not a sequence since 1 can't be mapped?
  2. Well, the point is, it has a last without a second last.
  3. Hello, I'm in a online debate with someone about some deep mathematical concepts. My opponent was trying to convince me that you can have a sequence of numbers for which there is a first element, a last element, but no second element and no second last element (where the sequence contains more than 2 elements). I thought that was absurd until he gave me an example: all the real numbers between 0 and 1. It definitely has a first member (0) and it definitely has a last member (1), but after 0 there is no "next" real number. Likewise, there is no real number that comes just before 1. Yet there are obviously real numbers between 0 and 1. That stumped me until I figured that couldn't possible count as a sequence because sequences must consist of well-define discrete elements and real numbers aren't well-defined or discrete. I thought that was a terrible way of putting it, so I looked up the definition of sequences online and the key word I found was "enumerable". The members of a sequence must be enumerable. And I don't believe the reals are enumerable. But then he came up with this other example: take the sum \(\sum_{i=1}^{n}\frac{9}{10^i}\). If you define each member of the sequence as the value of this sum for every value for n > 0 and order them by each incremental value of n, then you will have the sequence (0.9, 0.99, 0.999, ...). And if you allow n = \(\infty\), then we know this sum equals 1. Therefore, 1 is the last value in the sequence. Therefore, the sequence starts with 0.9 and ends with 1. Furthermore, each member is well-defined and discrete. We know each member by the sum \(\sum_{n}{i^1}\frac{9}{10^i}\) and the value of n. Yet, it has no second last member. Is this a legitimate example of a sequence?
  4. Hello, I understand that scienceforums.net is not about giving medical advice, but I have some questions about how drug tolerance works and I'm wondering if anyone knows about a good source that will answer my questions. Here's some example questions: 1) Can I expect that the time it takes to get over tolerance is relatively equal to the time it takes to become tolerant? So let's say it takes 4 days to become tolerant to a drug at a specific dose. Would it then take 4 days to undo the tolerance if I abstain from the drug for those 4 days? 2) Would tolerance buildup occur at the same rate if I consumed a certain dosage of a drug once per day vs. consuming half that dose twice a day. For example, suppose I consistently had 2 cups of coffee every morning, one immediately after the other. Compare that to having 1 cup every morning and another cup in the afternoon. Would I become tolerant to caffeine at the same rate in both cases? Or would it happen quicker in one case or the other? 3) Can I avoid tolerance buildup by taking a drug at a low enough dose each day? 4) Does tolerance buildup begin to occur after just one consumption, or do I have to consume a drug several times in a row before tolerance begins to occur? If so, how many times in a row? If anyone can answer these questions, great! But if it is inappropriate to answer them hear, I would appreciate being directed to another place where I can get my questions answered. Thank you.
  5. What about the imaginary number, the square root of -1? <-- That's said to be imaginary, therefore not one of the reals. Does that make it a hyperreal? Do the hyperreals include imaginary numbers?
  6. Here's the youtube video where Michael Stevens talks about cardinals and ordinals: Note that at 7:55, he explains how only cardinals refer to amounts. Note the statement: "Omega plus one isn't bigger than omega, it just comes after omega." Another question I have is: infinitesimals--are they divisible? In other words, is an infinitesimal defined as the "smallest possible number" (in terms of magnitude, not how far below 0 it is)? Or is it more of a set of numbers that are infinitely smaller than any real number? I would think its a set. Just as for any infinitely large hyperreal number R, you can have R + 1, R + 2, etc., and R - 1, R - 2, etc., I would think for any infinitely small hyperreal number e, you can have e/2, e/3, etc. or 2e, 3e, etc. That is, e doesn't represent a limit to how small numbers can get, it just represent an infinite amount of division you would have to do on a real number to get to it. That means that no matter how many times you multiply e, you will still only have an infinitely small hyperreal number. What happens if you multiply the infinitely small hyperreal number e by the infinitely large hyperreal number R? Do you get a real number?
  7. Yes, the definitions and explanations on the web are very complex. They just confuse me. I'm a better learner if I just ask someone and they explain it in plane English. Ah, but you're essentially saying the 0 of the hyperreals is just the 0 of the reals, correct? But is it correct to think of 2 x R as twice the distance from 0 as R is from 0? If so, and if 2 x R is just another hyperreal number, then is it fair to say that there can be infinite distances between hyperreal numbers (assuming both hyperreal numbers are greater than any real number). Do you mean to say even the cardinals can be extended passed infinity? As in, there are cardinal numbers (representing quantities) greater than infinity?
  8. Hello, I've been getting into the concept of hyperreal numbers lately, and I've got tons of questions. What I understand about the hyperreals is that they are numbers larger than any real number or smaller than any real number. I'm sure you can imagine how counterintuitive this sounds to someone like me who's new to the concept. It's like talking about numbers greater than infinity. I always thought that was impossible. So it shouldn't be surprising that someone like me would have a ton of questions. I'll start with a couple. 1) Assume that R is a hyperreal number greater than any real number. What does 2 x R equal? It's clear what 2 x n means where n is a real number because there is a 0 value for reference--i.e. 2 x n is a number twice the distance from 0 as n is from 0. But do the hyperreals have their own 0 point? How could they if they are greater than any real number (I realize some hyperreals are smaller than any real number, but for this question I'm only focused on the infinitely large hyperreals)? If 2 x R means twice the distance from 0 the real number as R is from 0 the real number, the you get a number another infinite distance away--sort of like a hyper-hyperreal number. <-- Does that make sense? Do the infinitely large hyperreals have their own infinity beyond which are numbers that are hyperreal even to the hyperreals? 2) I remember watching a vsauce episode on youtube where Michael Stevens explained the difference between cardinals and ordinals, which as I understand it is the difference between numbers that represent quantities and numbers that represent orders. He explained that while there is no cardinal number greater than infinity, you could talk about ordinal numbers greater than infinity. He didn't explicitly link ordinals to hyperreals but it seemed like the same idea. He stressed that since ordinals don't stand for quantities, you cannot use ordinals to speak of "how much" something is, but simply whether they come "before" or "after" another number. Is this true of hyperreals as well? If so, this would seem to imply that there is no 0 point on the hyperreal number line as that would mean you could quantify any hyperreal number R (the ones greater than infinity). It's quantity would just be how many whole hyperreal numbers it is away from "hyper-zero" (just as we say the number 5 represents the quantity of whole numbers it is away from 0). But if there is no such "hyper-zero" number, then there isn't a reference point relative to which we can say "how much" a hyperreal number (greater than infinity) represents (except that it's greater than any real number). We could still quantify the difference between any two (greater than infinity) hyperreal numbers. So we could say R+3 is 3 greater than R, but without knowing how much R really is, we don't really know how much R+3 is either. So I guess the question is: should hyperreal numbers greater than infinity be thought of as ordinals only--they represent orders of number, not quantities--or is there a way of talking about their quantities as well? I'll stop there for now. Thanks for any forthcoming responses.
  9. Why isn't the universe just a big soup of chemicals? Why is it that when we find water, or dirt, or air, we find it with more water, dirt, or air. In other words, why not just a single water molecule by itself? Why do water molecules tend to stick around with other water molecules? When we look around the world, we don't see one uniform substance making up everything--we see rocks, trees, sky, clouds, rivers, animals, and so on. In other words, different substances clumping together and staying separate from other substances. Take the sky for example. It is not only composed of oxygen, nitrogen, and carbon dioxide, but water molecules. Water molecules tend to clump together as clouds, separating itself from the rest. Of course, the rest of the molecules--the oxygen, nitrogen, and carbon dioxide--seem to stay evenly mixed (I think), but in general, there seems to be this tendency of molecules and atoms of one kind to stick together with other molecules and atoms of the same kind. The consequence is that, at the macroscopic level, we see objects made of specific substances separately from other objects made of different substances rather than a uniform soup of chemicals permeating everything. Why do molecules and atoms do this?
  10. Thanks very much Nevim, those are good links
  11. Hello, I remember reading an article a long time ago about the tendency of people to disagree with depressed people. So if a person suffers from depression, people are more likely to disagree with that person's statements and opinions. It doesn't seem to matter what those statements or opinions are (positive or negative, offensive or flattering), and it doesn't seem to matter whether the depressed person makes their depression evident or acts as if they are happy. I can't find that article. I can't seem to find any research on any studies that would support the above. No doubt, my google search skills aren't as refined as they could be. I'm wondering if anyone can corroborate with the above or link me to some research that supports the above (or perhaps disproves it). Thanks.
  12. They say that to get over a fear, one has to expose themselves to that fear over and over until the fear goes away (assuming of course they experience no adverse effects). For example, I have a fear of public speaking. I'm trying to get over it by going to public speaking sessions. For example, toastmasters. I've been going for the past several months and I'm not experiencing the effects I was expecting. I still get very nervous speaking in front of crowds and it shows. What I'm wondering is, is there any research to show that to get over a fear of public speaking (or any phobia), one has to expose themselves to that fear at a sufficient frequency? I mean, to take a ridiculously extreme example, I don't think one would ever get over a fear by exposing one's self to it once a year. But do it once a day, then maybe. I go to Toastmasters once a week and I'm wondering if that's not frequent enough. I'm wondering if it should be more like twice or three times a week. Has there been any research to show that the frequency with which one is exposed to a certain fear makes a difference? In particular, is there a frequency below which it has no effect whatsoever?
  13. Thank you Strange, What is the earliest that scientists can "see" the state of the early universe? I mean, I know scientists can look at the early universe by observing the CMBR in deep space, but this happened much after the first picosecond. Can they actually "look" that far back, to the first picosecond, or is it all based on mathematical models at that point?
  14. I am told that scientists don’t know what happened within the first picosecond of the universe. If this is true, I have a question: How do they know it was a picosecond? I guess the assumption is that if you extrapolate the current expansion of the universe back to its origin, you don’t have to assume anything unusual in the first picosecond. But what if scientists could somehow see into the first picosecond? Is it possible that what they find is that it was way longer than a picosecond? In other words, the universe originally started expanding very slowly, and for the longest time didn’t grow much bigger than its original size, and then for some reason went through the explosive expansion scientists are familiar with?
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