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| Math Central | Quandaries & Queries |
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Question from Justin, a student: Hello again, I just had one other question nagging question about infinity. I read this article on "Types of Infinity" on Paul Hawkins calculus website and he stated that one infinity cannot be divided by another or that the answer is inderterminate because fundamentally infinity comes in different sizes with respect to infinite sets and that this applies also to calculus. And so I was wondering (if this is true) is this why when you divide infinity by infinity (in the extended real number system) the answer is indeterminate since fundamentally one inifnity is larger than another like in infinite sets or is there another reason? Thanks sooo much for answering my question again! I greatly appreciate it! All the Best, Justin |
Justin,
The reason that in the usual extension of the real numbers by "infinity" and "minus infinity" you cannot divide one infinite quantity by another has nothing to do with different sizes of infinity. Rather, it is the same as the reason why you cannot divide zero by zero.
Division is meant to be an inverse to multiplication - that is, dividing 6 by 3 should be the same the same as answering the question "what do you multiply by 3 to get 6"? This is only meaningful if there is a unique correct answer!
As multiplying any finite number by 0 gives 0, the question "what do you multiply by 0 to get 0?" does not have a single answer - thus there is no good value to give to 0/0.
Similarly, multiplying any positive number by infinity gives infinity. Thus there is no good value to give to infinity/infinity.
In the much more complicated extension used in nonstandard calculus, if N is infinite and k is not 1, then kN is an infinite number but not equal to N; and infinite numbers _can_ be divided by other infinite numbers. The answer may be "standard", infinite, or infinitesimal.
-RD
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