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Reasons why infinity hasn't been implemented into modern math

  1. Jul 5, 2008 #1
    I don't really understand why it hasn't been numerically added into modern math. I mean, we make all kinds of properties for zero, but we can't make properties for infinity? It gets messy from time to time, but we could define everything strictly so that it still works algebraically.

    Example: (quoted from http://mathforum.org/dr.math/faq/faq.divideby0.html)

    5/0 = [tex]\infty[/tex]
    5 = 0*[tex]\infty[/tex]
    Multiplicative property of 0.
    5=0
    WRONG!

    If we defined [tex]\infty[/tex] numerically:
    5/0 = 1/0
    5 = 0/0
    We then, could define 0/0 as truly undefined, or all real numbers, or some other name. I don't think we need to classify something as undefined, or anything for that matter, unless it is 0/0, 0^0, [tex]\infty^\infty[/tex] etc.

    As for infinity, it should be implemented carefully into our modern math.

    Now it's time for you to post "Reasons why infinity hasn't been implemented into modern math."
     
  2. jcsd
  3. Jul 5, 2008 #2
  4. Jul 6, 2008 #3

    CRGreathouse

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    Don't forget the projective reals!
     
  5. Jul 6, 2008 #4

    matt grime

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    Or indeed the projective anything, the compactifications of spaces, the Riemann sphere, the theory of poles and singularities going back hundred+ years, Laurent series, Mobius transformations,....
     
  6. Jul 6, 2008 #5
    I'm talking numerically, 1/0. You won't ever see its notation in a function etc...
     
  7. Jul 6, 2008 #6

    Hurkyl

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    Unless you're working with extended real numbers, cardinal numbers, projective reals, projective anything, compactified spaces, the Riemann sphere, meromorphic functions, Laurent series, Möbius transformations....

    You don't see an infinite integer simply because the ring of integers doesn't contain such a thing. The ring of integers doesn't contain 1/2 either.
     
  8. Jul 6, 2008 #7
    You won't usually see it in a function over the real numbers, in the same sense you won't see 1/2 in functions over the integers. Infinity is a special class of number, just like rational numbers, irrational numbers, transcendental numbers, imaginary numbers, complex numbers, etc. If you're working with functions that have a domain and range over the real numbers you won't see infinity (OR imaginary numbers or complex numbers.)
     
  9. Jul 6, 2008 #8

    CRGreathouse

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    Actually, I'm pretty tired of the word "infinity". A good tenth of the posts on these math forums could be avoided if posters instead named the sort of infinite number they were thinking about. The one-point compactification of the reals? Aleph-2? Epsilon-naught? The IEEE +Infinity?
     
  10. Jul 6, 2008 #9
    location.reimannsphere(santa) ???
     
  11. Jul 6, 2008 #10
    My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.
     
  12. Jul 7, 2008 #11

    matt grime

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    It doesn't 'define infinity as undefined' (which is a contradiction in terms). It merely, and correctly, states that you can't cancel off zeroes in multiplicative expressions.
     
  13. Jul 7, 2008 #12

    arildno

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    So, according to you, low-level algebra is the heart of modern maths?
     
  14. Jul 7, 2008 #13

    Hurkyl

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    Well, epkid08, your post has been answered: mathematics does 'implement infinity'. Allow me to suggest that, before you continue ranting, you spend some time studying mathematics so as to gain an understanding both of how mathematics works and how the notion of infinite is treated and used in mathematics.

    Incidentally, your comment on zero is unfounded. It is so incredibly useful that natural languages have evolved to give it special treatment -- the difference between 'zero' and 'not zero' is quite deeply ingrained in grammar. (Apparently so deeply that some people have difficulty comprehending how they could be united into a notion of 'quantity') Grammar can also split the latter case into 'one' and 'more than one', but not in such an essential way. (There is some limited direct linguistic support for larger quantities, but going beyond one usually entails using ordinal or cardinal numbers)
     
    Last edited: Jul 7, 2008
  15. Jul 7, 2008 #14

    arildno

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    I hardly ever use 2.17, but I am really glad it has already been defined! :smile:
     
  16. Jul 7, 2008 #15

    D H

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    As an aside, one of my jobs is writing software standards. One such is "All magic numbers shall be named and referenced by that name. For example 'circumference=2*3.14159*radius' violates this rule and is wrong to boot. Pi is a magic number. Whether two is a magic number is debatable. A good starting point is that the only non-magical numbers are zero and one. You can use an unnamed small integer if the usage is well-commented." I never mentioned that zero and one, being the root of almost all mathematics, are actually the most magical numbers of all.
     
  17. Jul 7, 2008 #16

    D H

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    No. It does define infinity in terms of limits. It does not define operations with infinity as undefined. It simply does not define such operations, period, and for good reason. They are a superfluous and confusing distraction in the topic at hand, which is getting students to grasp the main concepts of elementary algebra.

    That's going a bit too far. Computer science can be viewed as an offshoot of applied logic / applied mathematics. Several computer programming languages explicitly define "undefined behaviors". Defining what is undefined is not inherently a contradiction in terms. It is just a warning about the potential for nasal demons.
     
  18. Jul 7, 2008 #17
    My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

    We call it undefined for the same reason we tell grade-schoolers they can't sqaure root negative numbers -- it'd confuse the hell out of them and it's off-topic. When their minds are ripe, and the contexts of our discussion is proper, these concepts are introduced. Crawl before you can walk.

    Oh, and, surprise, some infinities are bigger than others and we can divide by zero. Hehehe.
     
  19. Jul 7, 2008 #18

    matt grime

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    Declaring something to be undefined is not the same as *defining* it to be undefined. It might be a useful bastardisation of the language, but it is technically cobblers.... "coming up at ten: what is a pin head and how many angels can we make dance on it."
     
  20. Jul 7, 2008 #19
    To help explain what Mr. Grime has said, contemplate the following:

    "My talent is not having any talents." as opposed to "I have no talents." One makes a declaration, wheras the other implies a sentiment (and in such a way that in contradicts itself.)
     
  21. Jul 7, 2008 #20

    D H

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    Well OK then. Is it permissible to say that some operations in mathematics are declared to be undefined, or one must simply say that some operations in mathematics are not defined?

    To the OP: This is true even for the extended real number line. For example, the values of [itex]0/0[/itex] and [itex]\infty/\infty[/itex] are not defined.
     
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