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Possible Maths FAQ

  1. Sep 12, 2004 #1

    matt grime

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    Is there support for a sticky thread answering commonly asked questions? I am specifically thinking of the 0/0, infinity/infinity, what is infinity, 0.999..=/=1 questions that come up far too often.

    I am not suggesting this thread becomes it, nor that I should write it. If anyone wants to add a suggestion for a *general* topic please feel free, but please don't post disputing the fact that 0.999...=1 is true.

    I'd be happy to write one, but don't think I'm the best person to do so.


    Ideally it should simply explain why these are mathematical truths, not offer opinions on the "real world implication" of what it means to, say, divide infinity by infinity. I mean, the fact that 0.999 ..=1 cannot be in dispute, and the answer should simply explain what the mathematical ideas behind this are; it should not attempt to rebuff the crankish views on these subjects since they are often not mathematical. It should instead address people asking genuine mathematically based questions about why this is true, or that isn't a valid operation. I'd suggest that the final thread, if there were one, should be locked.

    Any pros cons I've not mentioned and so on?
     
  2. jcsd
  3. Sep 12, 2004 #2
    Surely wanna help on this, matt

    I suggest we also talk about some-integration rules and elementary vector-calculus since lot's of students are experiencing difficulties on these matters...

    regards
    marlon
     
  4. Sep 12, 2004 #3

    jcsd

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    I was thinking that a sticky on basic vector analysis might be useful in the homework section as over 50% of the questions seem to be about this topic.

    A sticky on 0.999.. =/= 1 and other common misconceptions in maths would be a good idea as looking into my crystal ball I see the topic will come up again.
     
  5. Sep 12, 2004 #4

    matt grime

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    I don't think that counts as general maths, and would be better in the proper forum for that topic.
    I can't say I bother reading most of the calc threads, what kinds of things crop up there frequently?
     
  6. Sep 12, 2004 #5

    Gokul43201

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    Here's a hypothetical sequence of events :

    (CON) Folks are not likely to read through the sticky before posting their query.

    (PRO) They can simply be directed to the sticky once it is seen that they've posted a question that is addressed by the FAQ.

    (CON) More often than not, they will be dissatisfied with the explanation given in the FAQ, and will pursue the matter further in their thread. And as often tends to happen, these threads will attract more cranks, all wanting to chip in their tuppence. Back to square 1 !

    (PRO) A mentor need not feel pangs of guilt when locking down a thread that has been addressed by the FAQ.


    Despite the cons, I think the idea is worth a shot. It is a small one-time investment, that may produce huge long-term returns.
     
  7. Sep 12, 2004 #6

    matt grime

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    I was worried about the possible effects it might have on spurring people to post their own (nonmathematical) theories on the subject. But if it is expressly written to explain the maths behind it, with the intention of helping those who want to learn the facts, then there can be no harm... right? Besides there will always be those who don't want to listen to the maths.
     
  8. Sep 12, 2004 #7

    arildno

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    I am quite certain that this will occur to some extent.
    However, by having at their disposal a thread which goes through these examples in a satisfactory way, mentors/admins can more easily (sooner) choose to lock/delete threads on these topics as being irrelevant/not worth answering again.

    On the whole, I think this is a good idea.
     
  9. Sep 12, 2004 #8

    Hurkyl

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    IMHO It's certainly a good idea. I've been considering it myself for quite a while, now, but I've never found the time to sit down and draft just what I'd want it to say.
     
  10. Sep 12, 2004 #9

    matt grime

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    Well, let's start with the 0.999..=1 answer, since it's about time for that to come up in the rotation.

    Why 0.999..=1

    This is a fairly common topic, and the explanation is simple, but disguises some quite deep mathematics.
    Rather than go too deeply into the mathematics straight away, let's start with a simple direct proof:

    let 0.9[n] mean 0.9...9 with n nines.

    0.9[n]<0.99....<=1

    right? So

    0<=1-0.999....<1-0.9[n]

    for all choices of integer n. Now, 1-0.9[n]=10**(-n) (my six key is broken so I can't do powers), but we can pick n as big as we want and make that number as small as we care. Now, it follows from the definition of the real numbers that 1-0.999.. must thus be zero and therefore they are equal.

    Of course, this may be the first time you've met the phrase "real numbers", and that might be part of the problem. No explanation would be complete without talking about them, since even if you know the name you might not know what it means.
    At school you start using whole numbers, and fractions, and then you just start using infinitely long strings of decimal expansions without explaining what they are properly. This is necessary to make life easier but creates exactly the problem about not being sure why 0.99..=1.
    We're all happy with fractions, and we're all happy that 1/2 = 2/4. Let's examine a little more what that means: 1/2 and 2/4 are just symbols, representations, of fractions. You're happy to say they're equal, and happy to manipulate them according to the rules you're given at school.
    We also soon learn that fractions "aren't enough", that there are quantities that we want to describe that can't be written as fractions, such as pi and the square root of 2. At this point we start to think of these quantities as decimals, but really they aren't. Decimals are representations of these other quantities, which we call the REAL numbers (formal definition to follow), and just as fractions have may have different representations, so may real numbers.

    <insert more explanation here>

    back to me talking to the other posters here:

    i'm very much unhappy with that, by the way, but I can't think of a way to start it. I mean, we all know that it's true simply because of the definition of the real numbers, but the chances are that the person who has the question has never heard the phrase "real number" and doesn't know anything about analysis. so, what should be done to improve the introduction, apart from a wholesale rewrite (i am not at all protective of this attempt, which was written off the cuff).

    is it the right way to be explaining it, or should it just go for the full on: give the definition of a complete metric space.

    if someone wants another view, here's a Field's Medal winner's view on a similar thing.

    www.dpmms.cam.ac.uk/~wtg10/decimals.html
     
  11. Sep 12, 2004 #10

    Hurkyl

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    Hrm, I've always considered starting with the definition of an ordered field to emphasize the algebraic properties we like in our number systems. Since even the crackpots presumably have some proficiency in arithmetic and elementary algebra, they will probably be willing to accept this part.

    The problem with this approach is then one has to figure out how to motivate the completeness axiom. :frown:


    But this isn't necessarily an advantage over your approach, since presumably the reader will have some proficiency with terminating decimals as well. You can avoid having to introduce the Archmedian axiom (or some form of completeness) if you talk about the "number" of digits in a decimal, which I think would need to be addressed in the FAQ anyways.


    I've always preferred simply defining 0.999... = 1, but I don't think that would be satisfying to most people.
     
  12. Sep 12, 2004 #11

    matt grime

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    How about this method of doing it?

    When a mathematician sees something like 0.999...=1, they are perfectly happy that it is true, perhaps even self evident, yet to a lot of people it seems just plain wrong. The difference is that a mathematician views this as a statement about "real numbers" and our representation of them as decimals. In order to understand why it is true it is necessrary to understand what the real numbers are. Different mathematicians have different ways of thinking about this, though they are all equivalent. For instance, if you were an algebraist you might think in terms of the real numbers as being an example of an "ordered field" with some other properties. An analyst might think in terms of sequences and "completeness".

    Then something about where to look to get more information about these things, and some brief indication of a proof perhaps.
     
  13. Sep 12, 2004 #12

    arildno

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    What I've noticed is quite typical of crackpots is how they assume that properties pertaining to finite quantities (in some sense) can unproblematically be transferred onto infinite quantities.
    Hence, to make it clear through simple examples that these intiutively reasonable ideas simply don't hold, might possibly be of some advantage.
     
  14. Sep 12, 2004 #13

    Hurkyl

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    Another thing that I've been unsuccessfully trying to put into words, and might not be useful at all in the FAQ, is how mathematics isn't exploring some mysterious "reality" about which we might be mistaken; it rigorously defines the reality that it explores.
     
  15. Sep 12, 2004 #14

    arildno

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    Agreed; IMO perhaps the most important insight to draw from this, is that unless a mathematical concept is rigourously defined, it is basically meaningless (without "reality").
    Even if we need some basic concepts that cannot easily be derived from others (concepts in basic set theory, I would think), we should at the very least have clear relations between such concepts.
     
  16. Sep 12, 2004 #15
    I agree. I'd defenitely find it useful and I can see myself looking to it as a great reference.
     
  17. Sep 12, 2004 #16

    Tide

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    [tex]1 = \frac {1}{3} + \frac {2}{3} = 0.333 \cdot \cdot \cdot + 0.666 \cdot \cdot \cdot = 0.999 \cdot \cdot \cdot[/tex]
     
  18. Sep 12, 2004 #17

    HallsofIvy

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    That's assuming, of course, that the person you are responding to accepts that
    1/3= 0.3333..., that 2/3= 0.6666..., and that you can add infinite decimals like that!
     
  19. Sep 12, 2004 #18

    Tide

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    Is there any other way to add decimals? There's nothing to "carry." :smile:
     
  20. Sep 12, 2004 #19

    mathwonk

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    another approach is simply to define what an infinite decimal means. i.e. an infinite decimal is by definition the smallest real number not smaller than any of its finite decimal approximations. Hence .99999..... = 1, the smallest number not smaller than any of the finite numbers .999....9, of any length.

    But it seems simpler just to say that 1/3 = .3333....., so 1 = 3/3 =.99999.....
     
  21. Sep 12, 2004 #20
    Oh no this is turning into yet another one of those dreaded threads.

    I think there should be an intuitive version and a rigorous version establishing the notion that 0.9...=1. The intuitive version was posted by mathwonk and the rigorous verison from Mr. Grime. And lastly, we can challenge anyone who thinks 0.9...!=1 to express the difference of the two numbers in decimal form which differs from 0! (Um.. I don't mean factorial)

    We should make a list here in this thread of all the FAQs and then we can start a fully moderated thread where they get answered.

    Some FAQs:
    Is 0 a natural number?

    What is 0^0?

    What is 0! and why? (BTW why can't one figure out (-1)! using the same pattern?)

    What's the deal with 0/0, "undefined", and "indeterminate"?

    What about 1/0?

    Why isn't oo/oo defined and why isn't oo-oo=0?

    I don't know these are all I could think of quickly.

    I suggest, again, that we collect some questions here and then someone can be either commissioned to answer them (like Mr. Grime or Hurkyl) and/or a fully moderated thread is started and/or Hurkyl collects PM'ed answers and posts the appropriate ones as a sticky.

    Cheers
     
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