Apparently, it is ok to talk about numbers that are infinite in scope but not infinite in size. That is, numbers like the Reals that go on and on and on in their symbolic expression as long as that expression is on the right side of the decimal point not the left side. This makes sense if we can’t talk about two numbers being infinitely large as having different values. Certainly the number, if you could actually talk about such numbers, …$999999999.0$ where the $9$s grow to the left forever is inexpressible. More over, even though I can tell you the 9s grow forever and you can understand that I mean this number is infinite in value, it would be the same number as …$111111111.0$ because both numbers are infinite in that they grow without bounds and ultimately there is no distinction between them. However, we can’t really talk about those numbers being the same either. There is a list of operations for the use of Infinity and subtracting Infinity from Infinity is undefined, you can’t do it because Infinity does not have a size.

What we can talk about are numbers that grow without bounds as long as they grow on the right side of the decimal point. $0.999999999...$ Is not the same number as $0.111111111...$ . This is pretty easy to see, right? Here we are dealing with infinitesimals. As these numbers grow it is not so much the number getting bigger that is interesting, and they are getting bigger though not growing towards Infinity but the difference between the number $0.999...99$ and $0.999...999$ gets smaller without bounds as there is another $0.000...0009$ to add. That is, as the length of the number on the right side of the decimal point increases without bounds the added component that is represented by the next digit to the right is smaller and smaller with no limit to how small. Here is the odd part though, as a number grows without bounds it is indistinguishable from another number that grows without bounds **if** there is a sequence of numbers that starts at the decimal point (in the left most position) and grows without bounds that exactly matches the left most unbounded sequence in another number.

That might not have been very well said so I will try to explain what I mean with an example. If I were to take a sequence of numbers like $0.999888$ and compare it to another number that was close but not the same like $0.999777$, I could easily show they were different simply by subtracting one from the other and show the result is non-zero. Now if I, instead, take a function that creates numbers for some value $n$ where the size of that sequence can grow without bounds say as $n$ starts at $1$ then $n=2$ then $3$ and so on forever, I might get a series of numbers like $f(1) = 0.98, f(2) = 0.9988, f(3) = 0.999888, … f(n) = 0.9999...8888...$ where the length of the number grows without bounds. Then I could compare it to a function that generated a series like $0.91, 0.9911, 0.999111, … 0.9999...1111...$ . I can see that they aren’t the same number and the functions are not equal but once I let $n$ grow without bounds, even though we can’t really talk about numbers that grow without bounds to the left of the decimal point, I can also show that as $n =>$ Infinity these two function become more and more equal (not more equal, they are equal or not equal, but less and less different) and the only thing we can talk about is the first sequence of numbers that grow without bounds, the $9$s, in this case. It would not matter if the two functions generated sequences that were even more different either, like $f(n) = 0.998, 0.999988, 0.999999888, ...$ compared to $f(n) = 0.33, 0.93333, 0.99333333, 0.9933333333, …$ (here the leading $9$s only increase when $n=$a prime number but the $3$s appear in pairs with each Whole number increase of $n$). As long as some sequence no matter how small the proportion at the beginning of the number compared to the numbers at the end became a sequence that grew without bounds and was the same as another number‘s leading boundless sequence. All values after the boundless sequence become infinitesimal in value and disappear.

Below is a table of elementary operations that are given as undefined for Infinity. More about operations with Infinity can be found at these Web addresses: http://en.wikipedia.org/wiki/User:Caue.cm.rego/infinity , http://www.mathsisfun.com/numbers/infinity.html , http://www.mathworks.com/help/toolbox/mupad/stdlib/undefined.html .

Undefined Operations on Infinity |
---|

0 × ∞ |

0 × -∞ |

∞ + -∞ |

∞ - ∞ |

∞ / ∞ |

∞^{0} |

1^{∞} |

What is odd about this table is the fact that while we can not subtract Infinity from Infinity. Apparently it is ok, though, to add or subtract one boundless number from another: $a= 0.999... , b = - 0.999..., a + b = 0$. Is this really ok? What if we derived $a = 0.999...$ In one way (by some function $A())$ and $b = -0.999...$ By a completely different method (some function $B()$)? Do we really know they are equal but with opposite signs? Infinity has no value, it is just the property of being unbounded. If we knew by the nature of the two functions that somewhere, at any point we want to give $n$ a finite value and check the results, the ending sequence of digits would be different between those two functions. Subtracting one from the other we would get a non-zero answer. Such a set of functions could look like the ones we were looking at above: $0.999888 - 0.999111 = 0.000777$. On the other hand, what if we knew they were boundless. What if you tried to do the same thing in that case: $0.999...888... - 0.999...111...= 0.000...777...$ . This number is zero, right? The string of numbers to the immediate right of the decimal point is a boundlessly increasing number of $0$s Those $0.000...777...$s are lost. We can’t get them back because we can’t truly know their value. Are they infinitely small or not even there or is there no difference? The above table says we can’t even try to know because the operation of subtracting Infinity from Infinity is undefined. I can easily see the wisdom of that choice but again, what about subtracting $0.999...$ From $0.999...$ To get $0$? How do we know it is zero if Infinity has no value? We can not truely say there are the same number of $9$s in one sequence or the other, all we can say is that they are both unbounded in their sequence.