Why does it matter to approach zero from the left or right in 1/0? I was surprised to see that 1/0 is undefined. One answer mentions that $1/0$ can be +$\infty$ or -$\infty$ depending on whether $0$ is approached by the left or the right:
                   
But why does this make a difference? Aren't both numbers equal by all measurable accounts?
Also, if zero is neither positive nor negative, why $1/0$ does not equal "unsigned infinity", which would be infinity in its own dimension (like imaginary numbers), and hence, $0$ in the real dimension?
Also, if we keep dividing 1 in parts of zero size, won't we have +$\infty$ number of parts?

Changed to an answer and expanded at asker's request.
The problems with extending the definition of division to make $\frac{1}{0}$ meaningful appear when you treat it like any other real number. The real numbers have certain operations defined on them, notably addition, subtraction, multiplication and (except when the denominator is $0$) division. These satisfy certain properties, such as $b\cdot \frac{a}{b}=a$ and $0\cdot a=0$ for any real numbers $a,b$ such that the expressions are defined. If we want these to still hold after defining $\frac{1}{0}$, we would have to let $1=0\cdot \frac 1 0 = 0$! This is absurd, so whatever we build by defining $\frac{1}{0}$ is very different from the real numbers, and lacks at least one of its fundamental properties.
