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Math Help - The idea of infinity...

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    Junior Member BariMutation's Avatar
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    The idea of infinity...

    I just got done with my calc III final, and this concept has always been somewhat of a ghost to me. I understand the concept of infinity; it's not a number so much as it is an action. That is, a function can't equal infinity, but it certainly can approach it - the action of the function. Now, after reading a recent post about limits, it got me thinking...

    Is it really mathematically true to say 1/infinity = 0? Yes, I know we accept it to be 0 when we're dealing with limits, integrals, and infinite number series, but is that an actually true equality? It just seems as if we're comparing a verb to a noun, if you know what I mean.
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    \displaystyle \frac{1}{10}=.10, \ \frac{1}{1000}=.001, \ \frac{1}{100000000000000......000}=.00000000.....0  0000001

    As the number in the numerator increases without bound, the value decreases according. Increasing to infinity means have trillions of zeros and then we have trillions more and never stop. The number is infinitesimally small and continues to get smaller and smaller.
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    Junior Member BariMutation's Avatar
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    No, I understand that perfectly. 1/x acts in this manner - for very large values of x, the number gets infinitely small. However, notice how everything you put is an actual equality. They are absolutely true without question. On the other hand, 1/infinity = 0 is comparing a behavior (of the function approaching x = infinity) to a true number (0). Now, I'd understand if we said "1/infinity approaches 0", but to say that they're actually equal seems far fetched too me.
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    Thinking about probability is \displaystyle \frac{1}{\infty} statistically different than 0?
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    Quote Originally Posted by BariMutation View Post
    I just got done with my calc III final, and this concept has always been somewhat of a ghost to me. I understand the concept of infinity; it's not a number so much as it is an action. That is, a function can't equal infinity, but it certainly can approach it - the action of the function. Now, after reading a recent post about limits, it got me thinking...

    Is it really mathematically true to say 1/infinity = 0? Yes, I know we accept it to be 0 when we're dealing with limits, integrals, and infinite number series, but is that an actually true equality? It just seems as if we're comparing a verb to a noun, if you know what I mean.
    It depends upon exactly what "system" you are talking about. In the real number system, the one you are using in Calculus, there is no number called "infinity". That is just a short hand for the limit. It would NOT be correct to say "1/infinity= 0" any more than it would be to say that "1/blue= 0"!

    Now, in some more advanced forms of mathematics, you can extend the real number system to include something called "infinity"- in fact, there are a number of different ways to do that that can give a number of different "infinities". In some of those systems it is correct to say "1/infinity= 0".
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