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The number $i$ is defined as a solution to the equation $x^2+1=0$.

How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$?

The purpose of course is to be able to solve different equations, such as $\frac{1}{x^2}=0,\frac{2}{x}=0$, etc.

Another way to look at this, is by using $j$ in order to define different (distinct) values of $\infty$.

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marked as duplicate by MJD, Umberto P., Davide Giraudo, Jyrki Lahtonen, M Turgeon Apr 30 at 14:10

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@TomCollinge $\infty$ is not a number. –  5xum Apr 30 at 10:02
    
I don't think you will get definitions of different infinities out of this. The definition of "division" for infinite cardinals is very restrictive and certainly won't cover a zero case such as yours. See for example the Division section in en.wikipedia.org/wiki/Cardinal_number#Division –  Tom Collinge Apr 30 at 10:19

2 Answers 2

up vote 8 down vote accepted

The problem with this is that it is simple to prove that $0\cdot x = 0$ for any number $x$. Thus, if $\frac 1x = 0$, that means that $x\cdot \frac1x = x\cdot 0$ meaning that $1=0$ which is not true.

This means that if there exists a number for which $\frac{1}{x}=0$, then some of the standard axioms of multiplication must not hold for such a number.

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Thanks for explaining that :) –  barak manos Apr 30 at 10:15
    
Every time you have $1_A=0_A$ this implies that a ring $A(+,\cdot)$ is consisting only of one element, namely $A=\{0\}$. –  7raiden7 Apr 30 at 10:22

The problem is that you can define all you want, but then you need to figure out what you can do with that number. For example, if j is the solution of 1/x = 0, what would be the solution of 2/x = 0? What are the properties of j?

Of course you can make this a bit clearer and instead of the name "j" you use the name "error", with the understanding that error + x = error - x = error * x = error / x = x - error = x / error = error, whatever real or complex number x is. That's just not very useful. And then you run into trouble when a correct calculation shows you for example that the solutions to a problem are all values x that solve x^2 = 1 or 1/x = 0. So instead of two solutions +1, -1 you suddenly have three solutions +1, -1 and error, which is quite rubbish.

Or, the answer in short: We don't do this because it is a rubbish idea that is in no way helpful.

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