# A solution to the equation $\frac{1}{x}=0$ [duplicate]

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 $\infty$.

## marked as duplicate by MJD, Umberto P., Davide Giraudo, Jyrki Lahtonen, M TurgeonApr 30 '14 at 14:10

• @TomCollinge $\infty$ is not a number. – 5xum Apr 30 '14 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 '14 at 10:19

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.
• 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 '14 at 10:22