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. All the people saying it’s impossible are wrong. For example, you could add a value called ∞ to the real numbers, and say that x/0 = ∞ for any nonzero value of x, [turning the number line into more of a number circle](https://en.wikipedia.org/wiki/Projectively_extended_real_line).
However, this sort of thing is not as useful as the idea of imaginary/complex numbers, for two reasons. First, the resulting system isn’t that interesting: it’s basically just the real numbers with a single extra point, and that point has rather boring properties like ∞ + x = ∞ and ∞ – x = ∞ for any real number x. The complex numbers have far more interesting behavior. Second, and more importantly, the complex numbers provide additional insight into the real numbers, especially in calculus and related fields. The mathematician Jacques Hadamard once said that “the shortest and best way between two truths of the real domain often passes through the imaginary one”. Adding the point ∞ to the real numbers, on the other hand, doesn’t tell us much at all.
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