anonymous
  • anonymous
Gotta another annoying question. Last one for the night
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
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anonymous
  • anonymous
This delves into the issue of convergence. Here's something to think about. Does \[ 0.\bar{9} = 1 \] ?
anonymous
  • anonymous
Where I'm using the convention that the bar overtop means "repeating"

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anonymous
  • anonymous
umm nearly but i wldnt say it equals 1
anonymous
  • anonymous
ya i know that
anonymous
  • anonymous
Okay. By how much does it differ from one?
anonymous
  • anonymous
By which I mean, if it's not equal, than how close is it?
anonymous
  • anonymous
.0000000000000000000000000001
anonymous
  • anonymous
it is an infinite number
anonymous
  • anonymous
No, it must be closer than that, those nines go on forever.
anonymous
  • anonymous
If I ask you "What's the difference between 0.9 repeating and 1", could you give me any real number?
anonymous
  • anonymous
i dont think so but i may be wrong but if I had to give a number i wld say 0
anonymous
  • anonymous
That would be reasonable. But if the difference between two numbers is zero, what does that mean?
anonymous
  • anonymous
They are one in the same
anonymous
  • anonymous
no difference lol
anonymous
  • anonymous
Indeed they are. Mathematically, 1 and 0.9 repeating represent exactly the same number. A bit non-intuitive at first, I'd imagine.
anonymous
  • anonymous
yes
anonymous
  • anonymous
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.
anonymous
  • anonymous
ok
anonymous
  • anonymous
I could just as easily express the quantity "two" as \[ \sum_{n=0}^\infty \frac{1}{2^n} \]
anonymous
  • anonymous
huh didnt understand that part
anonymous
  • anonymous
how cld 2 equal that?
anonymous
  • anonymous
If you haven't studied infinite series then you probably wouldn't. But it does. I promise :)
anonymous
  • anonymous
lol ok trust u on that one
anonymous
  • anonymous
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.
anonymous
  • anonymous
ok in other words as when u use reimann sums u make many rectangles?
anonymous
  • anonymous
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.
anonymous
  • anonymous
lol but like math books wld say it =0
anonymous
  • anonymous
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.
anonymous
  • anonymous
lol so i guess I shld take ur side of the arguement
anonymous
  • anonymous
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.
anonymous
  • anonymous
Awesome explanation Jemurray3 :)
anonymous
  • anonymous
jemurray is awesome
anonymous
  • anonymous
He probably cant sleep cus he is too busy thinking abt math
anonymous
  • anonymous
Oh no someone is spying on me again. It happens every night. Do u see that guest?
anonymous
  • anonymous
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.
anonymous
  • anonymous
And you're right about the not sleeping sometimes, my nerdiness is almost depressing ;)
anonymous
  • anonymous
lol I got it thanks. That was very clear
anonymous
  • anonymous
U always come save me
anonymous
  • anonymous
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.
anonymous
  • anonymous
LOL I guess my prof also thinks it is a gem that shldnt be skipped ove thanks gnite

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