Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
You can. People have; one example is the projectively extended real line, which is more or less the real numbers and a point at infinity. In this number system, 1/0 = ∞. The real question is “why don’t we use a number system where we can divide by 0 in everyday life?”
Part of it is cultural; “we just don’t.” But there are good reasons for it. Number systems that allow division by 0 inevitably lose some useful properties of the real numbers (side note: the real numbers aren’t any more or less “real” than any other kind of number; mathematicians just suck at naming things). For instance, ∞ introduces all sorts of weirdness. ∞-∞, in the projectively extended real line, is undefined, just like 1/0 is in the reals. So, now you can say 1/x=∞, but you can’t say x-x=0! Generally, trying to allow division by 0 is more trouble than it’s worth for everyday purposes.
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