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anonymous

  • 5 years ago

Gotta another annoying question. Last one for the night

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  1. anonymous
    • 5 years ago
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  2. anonymous
    • 5 years ago
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    This delves into the issue of convergence. Here's something to think about. Does \[ 0.\bar{9} = 1 \] ?

  3. anonymous
    • 5 years ago
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    Where I'm using the convention that the bar overtop means "repeating"

  4. anonymous
    • 5 years ago
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    umm nearly but i wldnt say it equals 1

  5. anonymous
    • 5 years ago
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    ya i know that

  6. anonymous
    • 5 years ago
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    Okay. By how much does it differ from one?

  7. anonymous
    • 5 years ago
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    By which I mean, if it's not equal, than how close is it?

  8. anonymous
    • 5 years ago
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    .0000000000000000000000000001

  9. anonymous
    • 5 years ago
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    it is an infinite number

  10. anonymous
    • 5 years ago
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    No, it must be closer than that, those nines go on forever.

  11. anonymous
    • 5 years ago
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    If I ask you "What's the difference between 0.9 repeating and 1", could you give me any real number?

  12. anonymous
    • 5 years ago
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    i dont think so but i may be wrong but if I had to give a number i wld say 0

  13. anonymous
    • 5 years ago
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    That would be reasonable. But if the difference between two numbers is zero, what does that mean?

  14. anonymous
    • 5 years ago
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    They are one in the same

  15. anonymous
    • 5 years ago
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    no difference lol

  16. anonymous
    • 5 years ago
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    Indeed they are. Mathematically, 1 and 0.9 repeating represent exactly the same number. A bit non-intuitive at first, I'd imagine.

  17. anonymous
    • 5 years ago
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    yes

  18. anonymous
    • 5 years ago
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    But that raises the question, what makes two quantities equal? Debating that 2 = 2 and 8 = 8 seems to be kind of silly, and I wouldn't argue with that. But you must understand that the quantity 2 and the shape that you draw on a piece of paper when you'd like to represent the quantity are not the same thing.

  19. anonymous
    • 5 years ago
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    ok

  20. anonymous
    • 5 years ago
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    I could just as easily express the quantity "two" as \[ \sum_{n=0}^\infty \frac{1}{2^n} \]

  21. anonymous
    • 5 years ago
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    huh didnt understand that part

  22. anonymous
    • 5 years ago
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    how cld 2 equal that?

  23. anonymous
    • 5 years ago
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    If you haven't studied infinite series then you probably wouldn't. But it does. I promise :)

  24. anonymous
    • 5 years ago
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    lol ok trust u on that one

  25. anonymous
    • 5 years ago
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    So this finally leads us to the issue of convergence. When you say that \[\lim_{x \rightarrow \infty} f(x) = y\] What does that mean? In non-rigorous terms, it means that you can make f(x) as close as you want to y by making x larger.

  26. anonymous
    • 5 years ago
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    ok in other words as when u use reimann sums u make many rectangles?

  27. anonymous
    • 5 years ago
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    Example: I argue that \[\lim_{x \rightarrow \infty} \space \frac{1}{x} = 0\]. My friend disagrees. His argument is that the equality is not correct.

  28. anonymous
    • 5 years ago
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    lol but like math books wld say it =0

  29. anonymous
    • 5 years ago
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    So, I would say to my friend what I said to you earlier: Alright, so if the two sides of the equals sign are different, by how much do they differ? And the key to this is that he could not provide me with a real number by which the two sides would differ that I could not disprove simply by making x larger.

  30. anonymous
    • 5 years ago
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    lol so i guess I shld take ur side of the arguement

  31. anonymous
    • 5 years ago
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    If he said, "oh, well, they differ by one billionth" then I would say "as soon as x becomes larger than a billion, that's not true". I could repeat that argument forever.

  32. anonymous
    • 5 years ago
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    Awesome explanation Jemurray3 :)

  33. anonymous
    • 5 years ago
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    jemurray is awesome

  34. anonymous
    • 5 years ago
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    He probably cant sleep cus he is too busy thinking abt math

  35. anonymous
    • 5 years ago
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    Oh no someone is spying on me again. It happens every night. Do u see that guest?

  36. anonymous
    • 5 years ago
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    Thank you guys, you are too kind :) But does that help in your understanding of convergence? That particular integral is defined to be \[\lim_{a \rightarrow \infty}\space \int_{1}^a \space \frac{1}{x^2} dx\] That integral is just \[\lim_{a \rightarrow \infty} \space 1-\frac{1}{a} \] Which we can make as close as we'd like to 1 by making a larger and larger and larger. That's the real idea behind a limit, and the convergence of improper integrals.

  37. anonymous
    • 5 years ago
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    And you're right about the not sleeping sometimes, my nerdiness is almost depressing ;)

  38. anonymous
    • 5 years ago
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    lol I got it thanks. That was very clear

  39. anonymous
    • 5 years ago
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    U always come save me

  40. anonymous
    • 5 years ago
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    I'm glad, it's a tricky and non-intuitive idea. It seems weird, but then you study it and it seems fine, but then you take a closer look and it gets weird again.... it takes awhile to really get comfortable with it, but I find this particular part of mathematics to be a little gem that is too often skipped over.

  41. anonymous
    • 5 years ago
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    LOL I guess my prof also thinks it is a gem that shldnt be skipped ove thanks gnite

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