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New maths debate to top 0.99...=1

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  • [Deleted User][Deleted User] Posts: 4,188
    edited September 2014
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    I had a dream my life would be
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  • KarlolinKarlolin Member Posts: 3,940 ✭✭✭✭✭
    Yes.
    Change is constant.
  • GouchnoxGouchnox Member, Friendly, Cool, Conversationalist Posts: 6,475 ✭✭✭✭✭✭
    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.
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  • ShylightShylight Moderator, Friendly, Helpful, Flagger Posts: 6,436 Mod
    edited September 2014
    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    lim(x→0)[1/x]=∞
    lim(x→∞)[1/x]=0
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  • [Deleted User][Deleted User] Posts: 4,188
    edited September 2014
    The user and all related content has been deleted.
    I had a dream my life would be
    So different from this hell I'm living
    So different now, from what it seemed
    Now life has killed the dream I dreamed
  • Deathranger999Deathranger999 Member Posts: 1,692 ✭✭✭
    Of course zero is a number. It's even (I'll explain if anybody actually needs me to) and it follows almost all rules of mathematics except for division (which you can actually account for if you think about infinity too).
    "Monotheism is progress of a sort because they're getting nearer the true figure all the time." -C. Hitchens
    "After all, to the well-organized mind, death is but the next great adventure." -A. Dumbledore
    "If you want to know what a man's like, take a good look at how he treats his inferiors, not his equals." -S. Black
    “Confidence is ignorance. If you're feeling cocky, it's because there's something you don't know.” -Foaly the Centaur
  • LauraLaura Member, Internet Detective, Friendly, Idle Game Master, Conversationalist Posts: 3,253 ✭✭✭✭✭
    edited September 2014
    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    theoretically it does
  • [Deleted User][Deleted User] Posts: 4,188
    The user and all related content has been deleted.
    I had a dream my life would be
    So different from this hell I'm living
    So different now, from what it seemed
    Now life has killed the dream I dreamed
  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    edited October 2014
    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.

    But zero is a number. It is a special number. It is neither odd, nor even; neither positive, nor negative. Even though it is equal to an integer squared, it isn't a perfect square (a trait that it shares with 1). It isn't prime. It is a placeholder digit, but only when used as a digit (ex. 101 means 1 group of a hundred, placeholder/no groups of ten, and 1 group of one).

    ∞ isn't actually a number, though. It is a placeholder that means "Really gigantic number so large that when you add 1 to it, it is so close to itself that there is no appreciable difference."
    Shylight said:

    (x→0)[1/x]=∞
    lim(x→∞)[1/x]=0

    This is absolutely correct! The only time that you can "divide by zero" is when you are playing around with limits. (Integrals and Derivatives are limits, just special kinds).


    Playing with math is FUN!
    Post edited by DarthCookie on
  • GouchnoxGouchnox Member, Friendly, Cool, Conversationalist Posts: 6,475 ✭✭✭✭✭✭
    If .999...=1, there is no doubt that 1/∞=0.
    If you are still doubting...
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  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    edited October 2014
    Gouchnox said:

    If .999...=1, there is no doubt that 1/∞=0.
    If you are still doubting...

    The reasons why .999...=1 (a point that I am not arguing against; it is a true statement) are different than the reasons why 1/∞≠0.

    Starting at 1:34, you can have a number that is infinitely close to 1 but still less than 1 (but it isn't .999... or any real number, but infinity isn't a real number either). Likewise, you can have a number that is infinitely close to 0, but still greater than 0. That is what you get when you divide any number by infinity. It gets really fancy when you start getting into indeterminate forms like 0/0, 00, 1, 0*∞, ∞/∞, or my favorite, ∞-∞.
  • PerfectionPerfection Member, Friendly, Conversationalist Posts: 2,745 ✭✭✭
    edited October 2014

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0


    But zero is a number. It is a special number. It is neither odd, nor even; neither positive, nor negative. Even though it is equal to an integer squared, it isn't a perfect square (a trait that it shares with 1). It isn't prime. It is a placeholder digit, but only when used as a digit (ex. 101 means 1 group of a hundred, placeholder/no groups of ten, and 1 group of one).

    Actually zero is even, when divided by 2 it has no remainder.
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  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    edited October 2014

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0
    If you disallow ∞/∞, you have to disallow any division by infinity. You can't just say that it doesn't doesn't work in this one very specific case (the reason why indeterminate forms are a thing is because they are where two rules conflict, e.g. n^0=1 && ∞^n=∞ =>∞^0=?). The only thing that I am really assuming in my example is that ∞=∞, but I could prove that given a base of 1/∞=0.
    1/∞=0=1/∞:
    1/∞=1/∞: (cross-multiply)
    ∞=∞;
    Let me put it another way:
    Let x=∞: If I use a variable, I can do algebra with it.
    1/x=0 && 2*0=0: therefore=>
    2*1/x=0=1/x:
    2*1/x=1/x:
    2*1/x*x=1/x*x: =>
    2=1;

    The only way that this doesn't work is if x≠x which would mean that 1/x≠1/x => 1/x≠0. Luckily, ∞≠∞, because ∞ isn't a real number; it is a placeholder for super big number.

    But zero is a number. It is a special number. It is neither odd, nor even; neither positive, nor negative. Even though it is equal to an integer squared, it isn't a perfect square (a trait that it shares with 1). It isn't prime. It is a placeholder digit, but only when used as a digit (ex. 101 means 1 group of a hundred, placeholder/no groups of ten, and 1 group of one).

    Actually zero is even, when divided by 2 it has no remainder.
    You are right on this one. I should have looked up the definition of parity before relying on something I learned in math class 8 years ago.

    EDIT: <sup> tags don't work.
  • PerfectionPerfection Member, Friendly, Conversationalist Posts: 2,745 ✭✭✭
    edited October 2014

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0
    If you disallow ∞/∞, you have to disallow any division by infinity. You can't just say that it doesn't doesn't work in this one very specific case
    Actually I can! You're making an aesthetic judgement not a logical one. One might find (as I do) this system (the extended real number system) to be ugly or inelegant but it's not inconsistent.

    (the reason why indeterminate forms are a thing is because they are where two rules conflict, e.g. n^0=1 && ∞^n=∞ =>∞^0=?).

    There is a conflict here, the rules n/∞=0 and n*∞=∞. This is why ∞/∞ is disallowed.



    The only thing that I am really assuming in my example is that ∞=∞, but I could prove that given a base of 1/∞=0.
    1/∞=0=1/∞:
    1/∞=1/∞: (cross-multiply)
    ∞=∞;
    Let me put it another way:
    Let x=∞: If I use a variable, I can do algebra with it.
    1/x=0 && 2*0=0: therefore=>
    2*1/x=0=1/x:
    2*1/x=1/x:
    2*1/x*x=1/x*x: =>
    2=1;

    The only way that this doesn't work is if x≠x which would mean that 1/x≠1/x => 1/x≠0.

    I can parody your argument with this one...

    Let x=0: If I use a variable, I can do algebra with it.
    1*x=0 && 2*0=0: therefore=>
    2*1*x=0=1*x:
    2*1*x=1*x:
    2*1*x/x=1*x/x: =>
    2=1;

    much like you can't divide zero by zero you can't multiply 1/∞ by ∞

    Luckily, ∞≠∞, because ∞ isn't a real number; it is a placeholder for super big number.

    ∞ certainly isn't a real number, but that doesn't mean it's not really a number (just like i isn't a real number but is still a number). ∞ is also not merely a placeholder for a super big number; ∞ is a representation of an infinite or limitless quantity,

    There are many systems that deal with infinitely large numbers, some of them (like the extended real number system) use ∞ as a number. Really, whether ∞ is a number depends on the number system you choose to use.

    Typically though, using ∞ as a number is an abuse of notation, but if one is careful it can be a rather useful one.
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  • Deathranger999Deathranger999 Member Posts: 1,692 ✭✭✭
    I think just as there are different sizes of infinity, there are also different sizes of zero, per se. Practically, they all have no value. But technically, their value is infinitely small.
    "Monotheism is progress of a sort because they're getting nearer the true figure all the time." -C. Hitchens
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  • Frayu1600Frayu1600 Member Posts: 4,177 ✭✭✭✭✭
    I am kinda... confused.
    What the freak is all this?
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  • ShylightShylight Moderator, Friendly, Helpful, Flagger Posts: 6,436 Mod
    Frayu1600 said:

    I am kinda... confused.
    What the freak is all this?

    Madness. Pure madness. And math.
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  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    edited November 2014
    Shylight said:

    Frayu1600 said:

    I am kinda... confused.
    What the freak is all this?

    Madness. Pure madness. And math.
    I found me a new sig quote.

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0
    If you disallow ∞/∞, you have to disallow any division by infinity. You can't just say that it doesn't doesn't work in this one very specific case
    Actually I can! You're making an aesthetic judgement not a logical one. One might find (as I do) this system (the extended real number system) to be ugly or inelegant but it's not inconsistent.
    I realize that I am not applying math entirely correctly; that was partially the point. I wasn't trying to show the number system as being inconsistent, just the statement 1/∞=0. I agree that n/∞ approximates 0, but once you start using the "=" things change.

    (the reason why indeterminate forms are a thing is because they are where two rules conflict, e.g. n^0=1 && ∞^n=∞ =>∞^0=?).

    There is a conflict here, the rules n/∞=0 and n*∞=∞. This is why ∞/∞ is disallowed.



    The only thing that I am really assuming in my example is that ∞=∞, but I could prove that given a base of 1/∞=0.
    1/∞=0=1/∞:
    1/∞=1/∞: (cross-multiply)
    ∞=∞;
    Let me put it another way:
    Let x=∞: If I use a variable, I can do algebra with it.
    1/x=0 && 2*0=0: therefore=>
    2*1/x=0=1/x:
    2*1/x=1/x:
    2*1/x*x=1/x*x: =>
    2=1;

    The only way that this doesn't work is if x≠x which would mean that 1/x≠1/x => 1/x≠0.

    I can parody your argument with this one...

    Let x=0: If I use a variable, I can do algebra with it.
    1*x=0 && 2*0=0: therefore=>
    2*1*x=0=1*x:
    2*1*x=1*x:
    2*1*x/x=1*x/x: =>
    2=1;

    much like you can't divide zero by zero you can't multiply 1/∞ by ∞
    That's the thing, n/∞=0 isn't a rule. lim[x->±∞](n/x)=0 is a rule. ∞/∞ is disallowed because ∞ doesn't always equal ∞. Indeterminate forms are forms that involve playing with limits where just replacing the parts of an equation with their limits doesn't tell you what the whole thing equals.

    For instance, lim[x->∞](x^2/x); lim[x->∞](x/x); and lim[x->∞](x/x^2) go to ∞, 1, and 0 respectively, even though lim[x->∞](x)=∞, ∞/∞ is, in this case, 1. I am allowed to simplify out the ∞ because I have already proved that it equals itself, therefore I am using the "lim[x->∞](x/x)" definition of ∞/∞.

    You can't divide anything by zero--in fact, [Newton/Leibniz] invented calculus just to get around this problem--but if you are playing with variables, you can simplify them out of the equation, regardless of what the variable equals. (<--If you disagree with this, you disagree with Calculus)

    <blockquote class="UserQuote">

    Luckily, ∞≠∞, because ∞ isn't a real number; it is a placeholder for super big number.

    ∞ certainly isn't a real number, but that doesn't mean it's not really a number (just like i isn't a real number but is still a number). ∞ is also not merely a placeholder for a super big number; ∞ is a representation of an infinite or limitless quantity,

    There are many systems that deal with infinitely large numbers, some of them (like the extended real number system) use ∞ as a number. Really, whether ∞ is a number depends on the number system you choose to use.

    Typically though, using ∞ as a number is an abuse of notation, but if one is careful it can be a rather useful one.


    I understand that ∞ is really a number, but not in a typical sense. I agree that generally using ∞ as a number is an abuse of notation; I was trying to show this by abusing the hell out of it. My point was that--outside of limits--you can't treat ∞ as a number/variable and expect to have any sort of consistency in your number system; a point I was trying to illustrate by treating it as a number in 1/∞=0.
    As for your first point, I agree that it depends on which number system you are using, so I will not debate this point further, it is a mere difference of opinion.

    (I swear this argument would be so much easier if it were in person and I were allowed to talk using my hands and a chalkboard. You would see that I agree with you on every point except n/∞=0. You can only do such abrupt things with ∞ if you are evaluating limits. This is because when you are evaluating limits, you aren't using "=" as "is equal to", you are using it as "approaches". Limits were invented because you can't actually allow ∞ to be used as a number.
    lim[x->∞](n/x)=n/∞=0
    but n/∞≠0)
    Post edited by DarthCookie on
  • [Deleted User][Deleted User] Posts: 4,188
    edited October 2014
    The user and all related content has been deleted.
    I had a dream my life would be
    So different from this hell I'm living
    So different now, from what it seemed
    Now life has killed the dream I dreamed
  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    Zyzzyzus said:

    Like Perfection said, it depends on the number system. You can't even do n/∞ in the real number system since it doesn't contain ∞. You can treat ∞ as a limit in calculus, but unless you extend to another system, n/∞ is undefined.

    Edit:
    You may as well try to find 1/pony :p

    In the extended complex plane, n/∞ = 0 for n ≠ ∞

    Yes and no. The extended complex plane is just the complex plane with a point attached at ∞. This singular point at complex infinity turns the complex coordinate plane into a sphere. (Imagine taking a square piece of latex and stretching it over the surface of a sphere, except all of the edges meet up at a single point.)
    "For all points in the complex plane, the chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending infinity to 0 and all other points z to 1/z." Wolfram MathWorld

    While this allows you to play with ∞ as though it were a real number, it isn't. The sphere works because z/0≣ for , and z/≣ 0 where is complex infinity, is the Extended Complex plane, and ≣ is "defined as". So, yes, within the extended complex plane you are allowed to divide by ∞ by definition, but you are also allowed to divide by 0. I would consider that cheating if it weren't for the fact that the entirety of math rests on the backs of turtles.

    Fun Fact: Bertrand Russell once wrote a 360-plus-page proof that 1+1=2.
    In other news, here are some definitions of/including infinity as stolen from Wolfram MathWorld. (I have spent the last 2 hours on this site like it was YouTube or Wikipedia.)
    "Infinity is an unbounded quantity that is greater than every real number."
    "The Cantor diagonal method is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence (i.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers)."
    "However, even though the formal statement 1/0=∞ is permitted in C-*, note that this does not mean that 1=0·∞. Zero does not have a multiplicative inverse under any circumstances."
    "Surreal Numbers"
  • [Deleted User][Deleted User] Posts: 4,188
    edited October 2014
    The user and all related content has been deleted.
    I had a dream my life would be
    So different from this hell I'm living
    So different now, from what it seemed
    Now life has killed the dream I dreamed
  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    Zyzzyzus said:

    Yeah I know and I took a complex analysis class. :p

    The fact that you brought it up proves that. Most of that explanation was for anyone else who tries to wade through this thread. :p
    Zyzzyzus said:


    Another fun thing is lines being circles with an infinite radius.

    The Riemann mapping theorem is also great.
    It shows any simply connected region that is an open subset of C but not the whole of C can be mapped onto any other such region by a bijective and holomorphic function.

    Most of the functions though will be impossible to express algebraically. But they still exist.

    Edit:
    Plus Liouville's theorem which states that any bounded entire function is constant.

    *looks some of this stuff up*
    *Spends an hour browsing Wolfram MathWorld*
    *Realizes what time it is*

    You, sir, have stolen a good deal of my free time, and I would like it back! :)
  • PerfectionPerfection Member, Friendly, Conversationalist Posts: 2,745 ✭✭✭
    edited October 2014

    Shylight said:

    Frayu1600 said:

    I am kinda... confused.
    What the freak is all this?

    Madness. Pure madness. And math.
    I found me a new sig quote.

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0
    If you disallow ∞/∞, you have to disallow any division by infinity. You can't just say that it doesn't doesn't work in this one very specific case
    Actually I can! You're making an aesthetic judgement not a logical one. One might find (as I do) this system (the extended real number system) to be ugly or inelegant but it's not inconsistent.
    I realize that I am not applying math entirely correctly; that was partially the point. I wasn't trying to show the number system as being inconsistent, just the statement 1/∞=0. I agree that n/∞ approximates 0, but once you start using the "=" things change.

    (the reason why indeterminate forms are a thing is because they are where two rules conflict, e.g. n^0=1 && ∞^n=∞ =>∞^0=?).

    There is a conflict here, the rules n/∞=0 and n*∞=∞. This is why ∞/∞ is disallowed.



    The only thing that I am really assuming in my example is that ∞=∞, but I could prove that given a base of 1/∞=0.
    1/∞=0=1/∞:
    1/∞=1/∞: (cross-multiply)
    ∞=∞;
    Let me put it another way:
    Let x=∞: If I use a variable, I can do algebra with it.
    1/x=0 && 2*0=0: therefore=>
    2*1/x=0=1/x:
    2*1/x=1/x:
    2*1/x*x=1/x*x: =>
    2=1;

    The only way that this doesn't work is if x≠x which would mean that 1/x≠1/x => 1/x≠0.

    I can parody your argument with this one...

    Let x=0: If I use a variable, I can do algebra with it.
    1*x=0 && 2*0=0: therefore=>
    2*1*x=0=1*x:
    2*1*x=1*x:
    2*1*x/x=1*x/x: =>
    2=1;

    much like you can't divide zero by zero you can't multiply 1/∞ by ∞
    That's the thing, n/∞=0 isn't a rule.
    Typically it isn't! My point is dependent on us defining n/∞=0 (as the extended real number system does). A consistent system can actually by put together with that taken to be true.

    Helpful link:

    http://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations


    I understand that ∞ is really a number, but not in a typical sense.

    My point is more nuanced. ∞ being a number depends on usage. When you use it in lim([->∞](1/x), it's not a number. The normal usage of ∞ as taught in calculus class is such that ∞ isn't a number. But if you use it in extended reals, you can make it one.

    I agree that generally using ∞ as a number is an abuse of notation; I was trying to show this by abusing the hell out of it. My point was that--outside of limits--you can't treat ∞ as a number/variable and expect to have any sort of consistency in your number system; a point I was trying to illustrate by treating it as a number in 1/∞=0.
    As for your first point, I agree that it depends on which number system you are using, so I will not debate this point further, it is a mere difference of opinion.

    (I swear this argument would be so much easier if it were in person and I were allowed to talk using my hands and a chalkboard. You would see that I agree with you on every point except n/∞=0. You can only do such abrupt things with ∞ if you are evaluating limits. This is because when you are evaluating limits, you aren't using "=" as "is equal to", you are using it as "approaches". Limits were invented because you can't actually allow ∞ to be used as a number.
    lim[x->∞](n/x)=n/∞=0
    but n/∞≠0)

    I generally get where you're at, and I agree there is a large amount of agreement here. My point is not that 1/∞=0 is a general rule for mathematics. Under standard calculus you're completely correct, this kind of thing only occurs when evaluating limits.

    My point is there is such a formulation (extended real number system) where you can use ∞ as a number and 1/∞=0 is a correct statement, and that system is not inconsistent.

    Edit: Quote pruning....
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  • [Deleted User][Deleted User] Posts: 4,188
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  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod

    Gouchnox said:

    1/∞=0
    But
    1/0≠∞

    Sure, it's a number... but a weird one.

    Technically, 1/∞≠0. It is so close that it makes no appreciable difference, except that if it were true, then:
    (proof language is hard to type outside of math programs, so I'm cheating)
    1/∞=0 && 2*0=0: therefore=>
    2*1/∞=0=1/∞: (simple substitution of equivalent terms)
    2*1/∞=1/∞: (now to multiply each side by infinity)
    2*1/∞*∞=1/∞*∞: (cancel out like terms) =>
    2=1;
    Math stops working, because once I prove 1=2, I can use that to prove that (2-1)=(1-1)=>1=0=2. Continuing on in this vein, I can prove that ANY NUMBER=ANY OTHER NUMBER.
    That's only problematic if you allow ∞/∞ = 1, if you disallow this operation, I believe you can be consistent in saying 1/∞ = 0
    If you disallow ∞/∞, you have to disallow any division by infinity. You can't just say that it doesn't doesn't work in this one very specific case
    Actually I can! You're making an aesthetic judgement not a logical one. One might find (as I do) this system (the extended real number system) to be ugly or inelegant but it's not inconsistent.
    I realize that I am not applying math entirely correctly; that was partially the point. I wasn't trying to show the number system as being inconsistent, just the statement 1/∞=0. I agree that n/∞ approximates 0, but once you start using the "=" things change.

    (the reason why indeterminate forms are a thing is because they are where two rules conflict, e.g. n^0=1 && ∞^n=∞ =>∞^0=?).

    There is a conflict here, the rules n/∞=0 and n*∞=∞. This is why ∞/∞ is disallowed.



    The only thing that I am really assuming in my example is that ∞=∞, but I could prove that given a base of 1/∞=0.
    1/∞=0=1/∞:
    1/∞=1/∞: (cross-multiply)
    ∞=∞;
    Let me put it another way:
    Let x=∞: If I use a variable, I can do algebra with it.
    1/x=0 && 2*0=0: therefore=>
    2*1/x=0=1/x:
    2*1/x=1/x:
    2*1/x*x=1/x*x: =>
    2=1;

    The only way that this doesn't work is if x≠x which would mean that 1/x≠1/x => 1/x≠0.

    I can parody your argument with this one...

    Let x=0: If I use a variable, I can do algebra with it.
    1*x=0 && 2*0=0: therefore=>
    2*1*x=0=1*x:
    2*1*x=1*x:
    2*1*x/x=1*x/x: =>
    2=1;

    much like you can't divide zero by zero you can't multiply 1/∞ by ∞
    That's the thing, n/∞=0 isn't a rule.
    Typically it isn't! My point is dependent on us defining n/∞=0 (as the extended real number system does). A consistent system can actually by put together with that taken to be true.

    Helpful link:

    http://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations


    I understand that ∞ is really a number, but not in a typical sense.

    My point is more nuanced. ∞ being a number depends on usage. When you use it in lim([->∞](1/x), it's not a number. The normal usage of ∞ as taught in calculus class is such that ∞ isn't a number. But if you use it in extended reals, you can make it one.

    I agree that generally using ∞ as a number is an abuse of notation; I was trying to show this by abusing the hell out of it. My point was that--outside of limits--you can't treat ∞ as a number/variable and expect to have any sort of consistency in your number system; a point I was trying to illustrate by treating it as a number in 1/∞=0.
    As for your first point, I agree that it depends on which number system you are using, so I will not debate this point further, it is a mere difference of opinion.

    (I swear this argument would be so much easier if it were in person and I were allowed to talk using my hands and a chalkboard. You would see that I agree with you on every point except n/∞=0. You can only do such abrupt things with ∞ if you are evaluating limits. This is because when you are evaluating limits, you aren't using "=" as "is equal to", you are using it as "approaches". Limits were invented because you can't actually allow ∞ to be used as a number.
    lim[x->∞](n/x)=n/∞=0
    but n/∞≠0)

    I generally get where you're at, and I agree there is a large amount of agreement here. My point is not that 1/∞=0 is a general rule for mathematics. Under standard calculus you're completely correct, this kind of thing only occurs when evaluating limits.

    My point is there is such a formulation (extended real number system) where you can use ∞ as a number and 1/∞=0 is a correct statement, and that system is not inconsistent.

    Edit: Quote pruning....
    *Wants to point out that "[The Extended Real Number Line] is useful in describing various limiting behaviors in calculus and mathematical analysis" and that the rules for arithmetic "are modeled on the laws for infinite limits." but doesn't want to appear argumentative.*

    The reason that the system works is because it defines rules for playing with infinity according to the laws of infinite limits. In fact, that is the primary use of the ERNL; infinite limits. You aren't allow to play the trick that I did with the ∞/∞=1, because ∞/∞ is still undefined.

    You are saying what I was trying to say, but was unable to get across because I had been up for far too long.
  • DarthCookieDarthCookie Moderator, Friendly, Helpful, Flagger, Conversationalist Posts: 5,558 Mod
    Zyzzyzus said:

    Zyzzyzus said:

    Yeah I know and I took a complex analysis class. :p

    The fact that you brought it up proves that. Most of that explanation was for anyone else who tries to wade through this thread. :p
    Zyzzyzus said:


    Another fun thing is lines being circles with an infinite radius.

    The Riemann mapping theorem is also great.
    It shows any simply connected region that is an open subset of C but not the whole of C can be mapped onto any other such region by a bijective and holomorphic function.

    Most of the functions though will be impossible to express algebraically. But they still exist.

    Edit:
    Plus Liouville's theorem which states that any bounded entire function is constant.

    *looks some of this stuff up*
    *Spends an hour browsing Wolfram MathWorld*
    *Realizes what time it is*

    You, sir, have stolen a good deal of my free time, and I would like it back! :)
    lol

    Complex analysis is probably my favourite branch of Mathematics.
    It is quickly becoming mine, as well.
  • ShylightShylight Moderator, Friendly, Helpful, Flagger Posts: 6,436 Mod
    A perfect gif for this thread.


    SparklebuttimagePurplesmart
  • [Deleted User][Deleted User] Posts: 4,188
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  • Deathranger999Deathranger999 Member Posts: 1,692 ✭✭✭
    What's this debate about now?
    "Monotheism is progress of a sort because they're getting nearer the true figure all the time." -C. Hitchens
    "After all, to the well-organized mind, death is but the next great adventure." -A. Dumbledore
    "If you want to know what a man's like, take a good look at how he treats his inferiors, not his equals." -S. Black
    “Confidence is ignorance. If you're feeling cocky, it's because there's something you don't know.” -Foaly the Centaur
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