TITLE: Why does division by zero not have an imaginary number "option"?
QUESTION [7 upvotes]: In regular math, you cannot get the square root of a negative number. Likewise, you cannot divide by zero. Both dividing by zero and taking the square of a negative have no place in real life.
However, we have systems in place to deal with the square root of a negative number (hence, imaginary numbers). Why does an option for divide by zero not exist?

REPLY [8 votes]: I tried this myself. If I understand correctly, you intend by your question to ask "Why can we not define a number (call it $q$), such that $q \cdot 0 = 1$?". Just a bit of algebra here: we have defined 0 as the additive identity. So we have $q \cdot 0 = q \cdot (0 + 0)$. Using the distributive law (which is always required in algebras, as far as I am aware), we have: $q \cdot 0 = q \cdot 0 + q \cdot 0)$. Now subtracting $q \cdot 0$ from each side, we have $0 = q \cdot 0$, which is a contradiction. Thus, such a number cannot exist. Notice, also, that we did not assume that our number was real; in fact, we assumed  nothing  about the number $q$. Thus it is the properties of $0$ that decide this, not the properties of our hypothesized $q$.

REPLY [2 votes]: We can define a quantity $i$ to have the property $i^2 = -1$, add $i$ to our set of numbers, and still have all of the rules of algebra work correctly.
Suppose we could define 1/0 to be a certain quantity, say z.  Then we have this problem:  If we start with the definition
$1/0 =z$, then multiply both sides by $0$, we get
$1 = z*0$, or
$1=0$, which is inconsistent.