Why does it matter to approach zero from the left or right in 1/0?

Question

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?

Answer 1

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.

Answer 2

I think it is mainly due to the sign (positive or negative) of $1/x$ depending on whether $x$ is positive or negative.

There may be more "deeper" answers than the above, but for most practical purposes that is the reason.

Answer 3

You have to be careful here; you're poking at some advanced mathematics. Before delving into the discussion at hand, concerning your "infinity in [its] own dimension" remark, have a look at this. Now, onto why $\frac{1}{0}$ is (for the most part) problematic. We shall begin with a brief discussion of the concept of number and division.

We may only consider the concept of a non-negative integer, as the rest follows :) While there are a number of good, and technical, definitions of this type of mathematical object, it will suffice for us to interpret it as a set of sets. So, for example we may say $0 = \varnothing = \{\}, 1 = \{0\} = \{\{\}\},$ $2 = \{0, 1\} = \{\{\},\{\{\}\}\},$ and so on (Reference: Advanced Calculus, Shlomo Sternberg). Now, we may conveniently define $\mathbb{Z}^*$ (the set of non-negative integers) to be the set containing all objects describable in the form "$\{\{\}_1,\{\{\}\}_2,..., \{...\{\}...\}_{n - 1}\}$", where $n \in \mathbb{Z}^+$ and the indices have only been shown for convenience. Notice, that this disallows the number "$\infty$", since "$\infty$" is defined as (for our purposes) the object such that no number is greater than or equal to it. Indeed, this is problematic given our current framework, hence the link at the beginning. Anywho, we can now also define the binary operators "$+$", "$-$", "$\div$", and "$\times$". I'll leave the details to you, and only focus on "$\div$".

$$ \mathbf{\tag{1} Def:}\ \forall x, y \in \mathbb{Z}, \ {x} \div {y} = \frac{x}{y} = r \in \mathbb{Z}^* : (r \times y = x ) \iff (x \div y = r : y \text{ can be subtracted } r \text{ times from } x) \text{ (i.e., } x - r \times y = 0). $$

Certainly, $x \geq y$ and $y \neq 0$ or $x < y$ and $x,\ y \neq 0$ is a necessary condition for $x \div y$ to be defined. As a corollary to what we have covered so far, $\forall x \in \mathbb{Z}^*,\ x \div 0$ is undefined, for there is simply no integer, $r$, such that $r \times 0 = x$ if $x > 0$; if $x = 0$, $x \div 0$ is still undefined, though, for a subtly different reason which I'll allow you the pleasure of working out for yourself.

The fact that "$\lim_{x \rightarrow \infty} x^{-1} = \pm \infty$" is merely a subtlety of the real number system, the notion of limit, and mathematicians rejecting the idea of defining infinity with coarse machinery.

Answer 4

Another problem here lies in that there doesn't exist a single object like "unsigned infinity". When we write (2+3)=5, there exists a single, unique number 5 which (2+3) equals. But, if we let (1/0) equal unsigned infinity, we've let it equal both positive infinity and negative infinity which are not the same infinity. Multiple infinities don't just happen because we have both positive and negative numbers. If "0" indicates an infinitesimal number, which may or may not equal the real number 0, then (1/0) and (15/0) will equal different hyperreal infinite numbers.