How come there isn’t an imaginary system for dividing a number by 0, while there is one for taking the square root of a negative number?
In: Mathematics
There can be – the extended real lines add an infinity, and extend division by zero to equal infinity. This still leaves 0/0 undefined, but even that can be defined in a consistent way if we want.
The issue is simply how useful it is. In the complex numbers it is no longer true that sqrt(ab) = sqrt(a) * sqrt(b), which is an annoying thing to lose (along with some other exponent identities), but not that big of a deal. What we gain is enormously useful, since all sorts of cyclic phenomena are well modeled by complex numbers.
By extending into the lines with infinities, we lose core division identities – it’s no longer true that a * (1/a) = 1, because allowing 0 * (1/0) = 1 creates contradictions. The division operator makes more intuitive sense (since 1/0 certainly feels like Infinity), but it means division is not always the opposite of multiplication, so actually doing algebra becomes more difficult. This is still sometimes useful, but mostly only in analysis (higher level calculus) and a few very abstract physical models.
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