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Its often stated in mathematics that 0.99999... (repeating until infinity) is equal to one. The proof that is commonly used is that we associate 1/3 with 0.3333..., and since 3*(1/3) = 1, it is clear that 0.9999... = 1.
But this proof seems inadequate because we are using what we are trying to prove in the proof itself...that is, it should be just as unacceptable that 1/3 can be written as an infinite 'series' of threes as that a one can be written as an infinite series of nines.
So to rephrase: Is 1/3 = 0.3333... just supposed to be taken as an axiom of mathematics, or is there actually a good reason to believe that an infinite series of threes can actually equal a rational number? I can certainly understand it in terms of limits, perhaps, where as we add additional threes to the decimal expansion it approaches the value of 1/3. But as far as I know there is no such thing as infinity as a real number in math, so the expression 0.333... doesn't really make sense, and so I have a hard time equating it with a real (rational) number.
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On January 06 2014 09:42 radscorpion9 wrote:
Its often stated in mathematics that 0.99999... (repeating until infinity) is equal to one. The proof that is commonly used is that we associate 1/3 with 0.3333..., and since 3*(1/3) = 1, it is clear that 0.9999... = 1.
But this proof seems inadequate because we are using what we are trying to prove in the proof itself...that is, it should be just as unacceptable that 1/3 can be written as an infinite 'series' of threes as that a one can be written as an infinite series of nines.
So to rephrase: Is 1/3 = 0.3333... just supposed to be taken as an axiom of mathematics, or is there actually a good reason to believe that an infinite series of threes can actually equal a rational number? I can certainly understand it in terms of limits, perhaps, where as we add additional threes to the decimal expansion it approaches the value of 1/3. But as far as I know there is no such thing as infinity as a real number in math, so the expression 0.333... doesn't really make sense, and so I have a hard time equating it with a real (rational) number.
Limits don't approach things in math; they equal things. The "approaching" talk (more commonly used for what the argument of a function does in a limit) is a metaphor that only results in confusion.
In particular, the limit of the sequence partial sums (3/10, 3/10+3/100,3/10+3/100+3/1000,...) equals 1/3. 0.3333... is defined as the limit of this sequence of partial sums, so it too equals 1/3.
Of course, if you're going to take this detour through sequence limits, you might as well show that .999... equals one directly using such limits rather than going through 1/3.
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There is a good reason. The reason is the convergence theorem from geometric series, which has been proven to be true. Look the proof up if you don't believe me in this part.
An infinite decimal number is a geometric series, namely
0.333...= 3/10 + 3/100 + 3/1000 +... = Sumi(1-inf) 3*(1/10)^i
And this series is
= 3/(1-1/10) - 3
= 30/9 - 3
= 1/3
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Why is this thread so full of math? Does this imply that math is stupid?
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On January 06 2014 11:48 rebdomine wrote:
Why is this thread so full of math? Does this imply that math is stupid?
Yes
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On January 06 2014 11:48 Ettick wrote:Show nested quote +On January 06 2014 11:48 rebdomine wrote:
Why is this thread so full of math? Does this imply that math is stupid? Yes
Day9 is a math major, are you calling Day9 stupid?
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On January 06 2014 11:54 [UoN]Sentinel wrote:Show nested quote +On January 06 2014 11:48 Ettick wrote:On January 06 2014 11:48 rebdomine wrote:
Why is this thread so full of math? Does this imply that math is stupid? Yes Day9 is a math major, are you calling Day9 stupid?
No, smart people can study stupid things and do all the time.
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I'm gonna order some pizza. Should I get it from Little Caesars, Pizza Hut, Pizza 73, Panago, Boston Pizza or Papa Johns?
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On January 06 2014 12:19 Orcasgt24 wrote:
I'm gonna order some pizza. Should I get it from Little Caesars, Pizza Hut, Pizza 73, Panago, Boston Pizza or Papa Johns?
Burger King.
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Pizza Hut if you don't mind getting your hands oily. I prefer them when I have napkins ready
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On January 06 2014 12:19 Orcasgt24 wrote:
I'm gonna order some pizza. Should I get it from Little Caesars, Pizza Hut, Pizza 73, Panago, Boston Pizza or Papa Johns?
Domino's
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Papa John's
They dipped their toe in eSports
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On January 06 2014 10:07 Simberto wrote:
There is a good reason. The reason is the convergence theorem from geometric series, which has been proven to be true. Look the proof up if you don't believe me in this part.
An infinite decimal number is a geometric series, namely
0.333...= 3/10 + 3/100 + 3/1000 +... = Sumi(1-inf) 3*(1/10)^i
And this series is
= 3/(1-1/10) - 3
= 30/9 - 3
= 1/3
Also this:
n = .9999999
10n = 9.999999999
10n - n = 9
9n = 9
n = 1
Unrelated, I'm at work with a ton of downtime. Any work-friendly websites that can be entertaining to fill the time?
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On January 04 2014 05:28 Najda wrote:Show nested quote +On January 03 2014 18:54 IAmWithStupid wrote:So I was watching Day[9]'s story time about graham's number. + Show Spoiler +I also know that π is irrational number, so somewhere in it there is every combination of digits, such as my birthday, yours, bank account of Bill Gates and its pin-code, etc. So my stupid question is: does the order happens in π after graham's number of written digits or somewhere in π the graham's number is written? (My opinion is the second option. Am I right?) Since π continues on for infinity, it therefore will contain graham's number within it at some point.
I don't have the mathematical proof for this, but statistically speaking, isn't that false? Flip a coin an infinite amount of times, and you could still get tails every time (going on to infinity could just mean an infinite amount of tails). Therefore, pi being infinite shouldn't be a sufficient reason for it containing every possible combination.
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On January 07 2014 08:59 Dark_Chill wrote:Show nested quote +On January 04 2014 05:28 Najda wrote:On January 03 2014 18:54 IAmWithStupid wrote:So I was watching Day[9]'s story time about graham's number. + Show Spoiler +I also know that π is irrational number, so somewhere in it there is every combination of digits, such as my birthday, yours, bank account of Bill Gates and its pin-code, etc. So my stupid question is: does the order happens in π after graham's number of written digits or somewhere in π the graham's number is written? (My opinion is the second option. Am I right?) Since π continues on for infinity, it therefore will contain graham's number within it at some point. I don't have the mathematical proof for this, but statistically speaking, isn't that false? Flip a coin an infinite amount of times, and you could still get tails every time (going on to infinity could just mean an infinite amount of tails). Therefore, pi being infinite shouldn't be a sufficient reason for it containing every possible combination.
I was wrong in saying that, but not for the reason you listed. If the digits of Pi were true random then it would indeed contain all possible combinations of digits, including Graham's number, but I was incorrect in assuming that Pi had that property. It apparently has yet to be proven, but the general thought is that it likely does have that attribute.
In response to your coin problem, I could be wrong in saying this, but I believe if you flipped a coin an infinite amount of times, somewhere within that series would be an infinite number of tails, as it would have every single possible outcome contained within the series.
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On January 07 2014 09:06 Najda wrote:
In response to your coin problem, I could be wrong in saying this, but I believe if you flipped a coin an infinite amount of times, somewhere within that series would be an infinite number of tails, as it would have every single possible outcome contained within the series.
Infinity. Filling math with bullshit since the beginning of time. Possibly longer.
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What is the value of infinity?
How stupid is stupid?
Is R1CH > Liquid'Nazgul?
Why does math exist?
Why does my school block educational sites with the label 'gaming' while leaving this site unblocked even though there is a picture of a gamer on the frontpage almost every day?
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Infinity doesn't have a value, it's just a concept.
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Is New York City a foreign country?
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On January 07 2014 22:39 Taekwon wrote:
Is New York City a foreign country?
basically.
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EPT
