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KnifeMissile · 05-19-2005, 01:10 PM

Quote:

Originally Posted by Slavakion

Someone brought this up on another forum I go to.

The limit of 1/x as x approaches 0 equals infinity. But 1/0 is undefined, right? Can 1/0 be considered infinity?

What about the fact that mathematic operations should be reversible? If I have 1/2 and multiply it by two, I now have 1. If I have 1/0 and multiply it by 0... what do I have? And what did I just multiply by zero?

Do you usually learn more about this kind of thing in upper-level math classes? (I'm still trying to figure out what they fit into 4 levels of calculus)

One of the things you misunderstand is that division is not really an operation.

When we say "t = x/y", what we're really saying is "t is equal to x multiplied by the multiplicative inverse of y, if such a thing exists." Sometimes, such a thing doesn't exist, in which case, this equation doesn't apply.

Finding a multiplicative inverse is not an "operation," in the sense that you're talking about...

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Another thing you misunderstand is that "infinity" is not a number. It's no more a number than "really big, " or "somewhere in between" are numbers. It's a property, or a description. The notion of it being a number doesn't really make any sense. For instance, if you were to define infinity as "a number greater than all other numbers," (a definition I think most lay-people can agree upon) then what do you make of the statement?

Code:

&infin; < &infin; + 1

I think even lay people can agree that for all real numbers (where real can be taken in both the literary and mathemtatical sense) x < x + 1. But then, infinity isn't the largest number, is it? Inifinity plus one is even larger and that's a contradiction, isn't it? If nothing else, the truth (math) is consistent. Even if you wanted to give infinity special properties like:

Code:

&infin; = &infin; + 1

Then what do you make of this?

Code:

&infin;     = &infin; + 1
      &dArr;
&infin; - &infin; = &infin; + 1 - &infin;
      &dArr;
    0 = 1

Another clear contradiction, this time not even including infinity! No matter how much you try to hack up such a "number system," it becomes quickly apparent that infinity makes no sense as a number. You simply accept that there are an infinite number of numbers, they are well ordered, and they are unbounded.

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In answer to your last question, yes, you do learn this stuff in "upper-level math classes." However, they're really not that upper level. Unless you're in some "applied" university course, like engineering or physics, they won't bother teaching you this stuff 'cause you're only using math as a tool for making sense of the world. However, if you're studying mathematics in university, they will quickly teach you the fundamentals of logic and reasoning, because that's what mathematics is really about...