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Thread: Can a numerical infinity be neutral?

  1. #1 Can a numerical infinity be neutral? 
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    I believe that infinity can be used differently respecting different areas of math.

    For example, in basic operations, to what power must 10 be raised to equal zero?
    10 ^ -3 = 10 / 10 / 10/ 10 / 10 = 0.001
    So 10 ^ n = x ... As n decreases by 1, x increases at a geometric ratio of 0.1 (or dividing by 10)
    That would mean 10 ^ -infinity = 0

    But in trigonometry, with the tangent function
    As x increases, y increases towards +infinity but comes back from -infinity.

    The same goes with the rational function f(x)=1/x

    Does this mean infinity, in certain uses, is somehow neutral?
    Also, is this idea the basis of the real projective line?
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    Quote Originally Posted by Excidium View Post
    For example, in basic operations, to what power must 10 be raised to equal zero?
    10 ^ -3 = 10 / 10 / 10/ 10 / 10 = 0.001
    So 10 ^ n = x ... As n decreases by 1, x increases at a geometric ratio of 0.1 (or dividing by 10)
    That would mean 10 ^ -infinity = 0
    I forgot to explain how this would contrast +infinity. The point is 10 ^ +infinity would certainly not be zero, but infinity. So in this situation, there would be a great contrast between + and - infinity.


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    Quote Originally Posted by Excidium View Post
    Quote Originally Posted by Excidium View Post
    For example, in basic operations, to what power must 10 be raised to equal zero?
    10 ^ -3 = 10 / 10 / 10/ 10 / 10 = 0.001
    So 10 ^ n = x ... As n decreases by 1, x increases at a geometric ratio of 0.1 (or dividing by 10)
    That would mean 10 ^ -infinity = 0
    I forgot to explain how this would contrast +infinity. The point is 10 ^ +infinity would certainly not be zero, but infinity. So in this situation, there would be a great contrast between + and - infinity.
    Personally I think you make some good points.

    It does not seem infinity has a consistent definition or outcome.

    For example, using mathematical induction, you prove for all n.

    But, induction is based on the axiom of infinity in ZFC.

    Then when thinking about a divergent sequence, we say it approaches infinity.


    When we say a set is infinite, then it is not finite.
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    Quote Originally Posted by chinglu View Post

    Personally I think you make some good points.

    It does not seem infinity has a consistent definition or outcome.

    For example, using mathematical induction, you prove for all n.

    But, induction is based on the axiom of infinity in ZFC.

    Then when thinking about a divergent sequence, we say it approaches infinity.


    When we say a set is infinite, then it is not finite.
    Thank you chinglu. I agree that infinity doesn't seem to be a concrete concept. So, if the set of all real numbers is infinite, it shouldn't really include a numerical infinity? I don't understand ZFC that well.

    As for infinity's neutrality, I think this explains why one cannot attribute +infinity or -infinity to 1/0. Therefore being neutral, as seen in the rational function, where at the x=0, infinity comes to and runs back from both infinities (so there really is no "asymptote")
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    Quote Originally Posted by Excidium View Post
    As for infinity's neutrality, I think this explains why one cannot attribute +infinity or -infinity to 1/0. Therefore being neutral, as seen in the rational function, where at the x=0, infinity comes to and runs back from both infinities (so there really is no "asymptote")
    This is because the function is not continuous at x = 0, i.e. it does not approach the same limit from the left as it does from the right. The limit of the function as x approaches zero from the left and from the right is negative infinity and infinity respectively. The asymptotic nature is a property of the function and not one of infinity.
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    Quote Originally Posted by wallaby View Post
    This is because the function is not continuous at x = 0, i.e. it does not approach the same limit from the left as it does from the right. The limit of the function as x approaches zero from the left and from the right is negative infinity and infinity respectively. The asymptotic nature is a property of the function and not one of infinity.
    That could be true, but I don't think there's anything to restrict either possibility. I see it as a neutral infinity due to: 1/x, as x gets closer to zero it gets closer to + or - infinity. And also how when 1/0, when graphed on a Cartesian plane, shows a line with neutral slope.

    I'm more open to the idea of infinity. Axiomatically 1/0 cannot be defined so that might be why most people immediately stop there, negating any further insights into infinity.

    Also, some people close out infinity due to its strange properties, ie infinity + 1 = infinity + 2
    Using the +/- property, 1=2 ... wouldn't an argument also apply to 0 x 1 = 0 x 2? Using the divisional property?
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    Quote Originally Posted by Excidium View Post
    That could be true, but I don't think there's anything to restrict either possibility. I see it as a neutral infinity due to: 1/x, as x gets closer to zero it gets closer to + or - infinity. And also how when 1/0, when graphed on a Cartesian plane, shows a line with neutral slope.

    I'm more open to the idea of infinity. Axiomatically 1/0 cannot be defined so that might be why most people immediately stop there, negating any further insights into infinity.

    Also, some people close out infinity due to its strange properties, ie infinity + 1 = infinity + 2
    Using the +/- property, 1=2 ... wouldn't an argument also apply to 0 x 1 = 0 x 2? Using the divisional property?
    The problem with an equation like, infinity + 1 = infinity + 2, is that infinity is not itself a quantity. You can't add it or multiply it to anything to obtain a meaningful result, which is the aim of mathematics. I do realise that in many places one will see things like, infinity / infinity, however this is meerly an abuse of notation. A more proper implication of such an expression can be illustrated by considering two functions, f(x) and g(x). Now suppose there is a third function, h(x), that is formed out of the quotient of these functions, f and g. If f(x) and g(x) both grow without bound, as the value of x does the same, then we denote the limit of h(x) by.

    What is this value? Does the value of h(x) grow without bound as well? The limit of h(x) cannot be determined by the quotient of the limits of f(x) and g(x), as such a value is not defined to have a value in the set of real numbers. Such is the issue when the limit of h(x) is 1/0 or 1/infinity.

    Basically i just made a longer than necessary post that simply says, you can't make algebraic manipulations of infinity.
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    [QUOTE=Excidium;287433]
    Quote Originally Posted by chinglu View Post

    Personally I think you make some good points.

    It does not seem infinity has a consistent definition or outcome.

    For example, using mathematical induction, you prove for all n.

    But, induction is based on the axiom of infinity in ZFC.

    Then when thinking about a divergent sequence, we say it approaches infinity.


    When we say a set is infinite, then it is not finite.
    Thank you chinglu. I agree that infinity doesn't seem to be a concrete concept. So, if the set of all real numbers is infinite, it shouldn't really include a numerical infinity?
    Pretty good. I struggled for a month trying to prove an infinte set was constructed inside another set. I found a way.

    But, your statement above is what I had to struggle with.

    We think of an infinite set as a "thing". In fact it is a process that goes on forever.

    In normal proof theory, an infinite set is treated as a thing. This is justified in ZFC that I mentioned which means set theory.

    But, I discovered it is a thing, but in a proof you must prove concepts as a process that does not end.

    So, you are struggling with a problem not many consider. Hats off.


    As for infinity's neutrality, I think this explains why one cannot attribute +infinity or -infinity to 1/0. Therefore being neutral, as seen in the rational function, where at the x=0, infinity comes to and runs back from both infinities (so there really is no "asymptote")
    The idea of negative infinite and positive infinity do have justifications in a way.

    Let's say you have the natural numbers, 0.....n...... They have a starting point, at 0.

    But, let's consider the real numbers, they go from 0 backward and 0 frontward or positive and negative.

    So, these symbols indicate a "direction".

    Now, when you consider the function 1/x on the real numbers, you are coming into 1/x from the negative direction and the positive direction with the domain being,

    (-∞, 0) U (0,∞).

    Then the range from both of these direction as they approach 0, 1/x approaches ∞.

    Had the function been -1/x, you would have a different range as negative numbers.

    Anyway, the idea of infinity is all over the place and I am surprised this is the case after all this time under the rules of math.
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