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 divide by zero if you are on the extended complex plane, which is the complex number (x i +y, so the combination of real and imaginary numbers ) and ∞( infinity). It is often represented as the [https://en.wikipedia.org/wiki/Riemann_sphere](https://en.wikipedia.org/wiki/Riemann_sphere)
There is only one ∞, not a +∞ or -∞ that you can get to in limes valuation for a real number.
A number on the complex plane is usually represented with z.
z/0 = ∞ and z/∞ =0 that is if z is not 0 or ∞
∞/0 = ∞ and 0/∞ =0
What is still undefined is ∞/∞ and 0/0
But if you use the extended complex plane you need to know what it lacks some stuff you are used to in for example real numbers.
One example is that numbers do not have a well-defined order. 4> 3 is well known but is i >1, i<1 or i=1? The answer is complex numbers do not have a single defined order, you can only compare the absolute value that is the distance from 0 and is represented by |z|
The result is that the distance to |i| and |1| both are 1. This is also why there is only one ∞ and not +∞ and -∞
This shows that if you add stuff like the ability to take the square root of -1 the resulting number will loo some other stuff like an absolute order.
This is true for complex numbers not just if you extend the plane and include ∞. But is a relatively simple example that if you gain stuff you loos stuff too. So you can divide by zero in some situations if you know what to do and what other consequences that have.
It has lots of practical applications and for example, https://en.wikipedia.org/wiki/Control_theory where https://en.wikipedia.org/wiki/Zeros_and_poles is a common tool to know how to control systems.
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