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ANNOTATION:
Can Infinity Be the Inverse of Zero?
First of all, I must admit that I am still in high school and have no signifigant merit of education. I do however wonder.
Anyway, I would like to discuss the possibility that infinity could be the inverse of zero ( 1/0 ). There are several situations in mathematics that I believe support this hypothesis.
1. First, notice that as numbers increase in value and "approach" infinity their inverses approach zero. Obviously the numbers will never reach infinity as their inverses will never reach zero. One could by pure speculation guess that the quantities that they approach are zero and infinity, and they are each other's inverse.
2. In a coordinate, the slopes of lines have signifigant importance (for simplicity assume all lines pass through the origin). A line with slope x and a line with slope 1/x is for all lines bisected by a line of slope 1. A line of slope 0 is a horizontal and the vertical's slope is it's inverse (by the previous property). As lines approach vertical, their slopes get larger, as if approaching infinity.
Any insight that is complete (not simply, '1/0 is "undefined"') would be appreciated.
-Other Thoughts of Speculation-
The contents of this section are purely speculative and easily false. They do not have concrete mathematical basis and are just my personal thoughts.
1. 1/0 is not any specific element of our number system. It is a representation of any concept, including infinity, that has no specific value. It is almost as if it were the concept excluded from our number system. It is neither positive nor negative, even nor odd, nor is it comparable to any value.
2. Any number x that is not 0, 1, -1, or 1/0 to the power of 1/0 is either 0 or 1/0. (2^(1/0) = {0, 1/0}) 0, 1, and 1/0 raised to the power of 1/0 remain to be themselves. -1 is either 1 or -1 (I cannot tell since 1/0 is neither, according to these hypotheses, positive nor negative)
3. Any addition or multiplication of 1/0 results in 1/0.
Well, enjoy. Tell me what you think.
Author: Brenden Colling (brendencolling[ at ]hotmail.com)
Date: Apr 7, 2002REPLY:
I'll also admit that I'm a sophomore in high school and similarly have no significant merit of education. You first have to consider that infinity, unlike zero, is not a single value. As infinity can have different values, its inverse, therefore, would have different values. Infinite numbers make up only half of the set of hyperreals, and their inverses are numbers of the Infinitesimal set. That is, Numbers with absolute values greater than any positive real number (infinite numbers) have inverses which have absolute values smaller than any positive real number (infinitesimal numbers). They can be shown in sequences.
a=(1,2,3,4...n...)
b=(1/2,1/3,1/4...1/n...)
Although the terms of b approach 0, the end result is not 0, it is an infinitesimal.
That's the way I understand things, at least.
Author: Dulanjana Karunatilaka (hrmdtb[ at ]hotmail.com)
Date: Dec 19, 2002REPLY: My Idea
Well I am a high school student too. I dont know whether I am correct but I would like to comment on the concept opf 1/0. We know that 4/2 = 2 and 8/2 = 4, etc. The core meaning being that "how many 2's are to be added to sum upto 4"? We find the answer being 2. We need to add two 2's to come up with 4. the same can be thought of the 8/2. How many 2's do we need to add up to sum upto 8? In this case its 4. Now when we talk of 1/0 it is the same question. how many 0's are to be added to sum upto 1. One may think it is infinity, but adding up an infinite amount of 0's would still equal zero (0 + 0 + 0 + 0 + 0 + 0.... = 0). Even adding up more than an infinite amount of 0's would not add upto 1 (well.... "more than an infinite" doesnt sound right:)). So I guess asking the question 1/0 has no answer and thus is "undefined"
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