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Operations over infinite decimals numbers

  1. Oct 17, 2012 #1
    Hello, i would like to ask the following question that has been troubling me.

    lets say i have 1/3 = 0.33333333333(3)

    it may seem clear that 1/3 - 1/3 = 0, but when operating over the decimals this doesnt seem clear. How can i perform an operation over infinite digits and consider it as finite? If i say 1/3 - 1/3 = 0, then i am saying that i was able to subtract all the the infinite digits, or not?

    The same for PI, if i say PI - PI = 0, am i not saying that i was able to subtract all the digits of pi?

    Maybe i am missing something, but this has been troubling me =( it would also lead to a contradiction to say that PI - PI != 0 since that would mean that they are not the same quantity, something i just asserted when i wrote them, yet i cant perform their subtraction to check that it is 0 in fact... =S

    Thanks in advance
  2. jcsd
  3. Oct 17, 2012 #2


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    Hello DarkFalz! :smile:
    You can't.

    You'll get an infinite number of digits in your result.

    Of course, if they're all 0, that's pretty easy to interpret! o:)
  4. Oct 17, 2012 #3


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    Change bases and it won't be infinite anymore, sometimes..
  5. Oct 17, 2012 #4
    Well, but can 1/3 exist in reality? can i truly have 1/3 of a cake? Or its just that i can have 1/3, but i cant measure it to confirm it? It has been confusing me!

    Also, if it is true that 0.333333... - 0.3333333 is 0 because 3-3 = 0 in each position, then is it true that 1/3 + 1/3 + 1/3 != 1 because

    0.99999... which is not 1 ?

    (i applied the same rule that makes 0.3333.. - 0.3333.. be 0)
  6. Oct 17, 2012 #5
    0.999... does equal 1, see www.physicsforums.com/showthread.php?t=507001 [Broken]
    Last edited by a moderator: May 6, 2017
  7. Oct 17, 2012 #6
    Uh-Oh. Prediction: this thread will be closed in 24 hours.

    Summary of converstaion:

    Person1: .99... isn't 1 because it never actually "gets" to 1.

    Person2: What do you mean "gets to". .9 repeating is a number, it isn't getting anywhere.

    Person1: Yeah, but how can it be?

    Person2: Here's a proof using series <insert proof>

    Person1: I don't know about series, you're wrong.

    Person3: <Some intuitive argument>

    Person4: <Another intuitive argument>

    Person5: <Some vague discussion about something that doesn't really relate to anything>

    Person1: No, y'all are all wrong because you never actually "get" to 1.

    Lather, rinse repeat.
  8. Oct 17, 2012 #7


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    This argument works for the Standard Reals:

    For any e>0 , 1-0.999999... <e ; just take enough 9's.

    This implies d( 1, 0.9999...)=0 (to rigorize, e.g., take a limit)

    In a metric space , d(x,y)=0 iff x=y. Then 1=0.999.....
  9. Oct 17, 2012 #8


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    That's nothing compared to what the Wikipedia guys go through.

    But we are getting sidetracked.

    DarkFalz, numbers don't exist physically. They are mental constructs which have agreed upon behaviour, and from time to time, allow us to represent quantities from nature.

    Secondly, you have a misconception where the operations on the reals are done one digit at a time. This probably originates from long multiplication you learned at primary school. We teach this algorithm because kids don't want memorise their 278-times-tables. With infinite decimal expansions, we don't calculate term by term, the calculation occurs across all the digits at the same time.
  10. Oct 18, 2012 #9
    The only thing that is still confusing me, is this step and subsequent rule:

    i remove a 9 and they remain the same? what the hell? infinity is very strange, is this truly intuitive to everyone else?
  11. Oct 18, 2012 #10


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    It tends to be a shock when you first see it. Imagine what the mathematicians at the time thought when they discovered stuff like this.

    Also read the article on Hilbert's Hotel.
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