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
i is defined. It is the square root of -1. It is always that value. Because it has a defined value you can assign a value to it and do stuff with it.
Dividing by zero is undefined. The closest thing to what you’re talking about is Limits, where you investigate what happens as parts of a function tend towards either zero or infinity.
Eg: what does a (x^2 -4) / (2x-4) equal when x = 2? At exactly x = 2 the function is undefined, but you can take limits and see if it it tends towards a value, tends towards zero or tends towards infinity.
One of the critical issues of assigning a value to zero is that there are different natures of zeros. X^2 at x=0 is a different zero than x^3 at x=0. They both have the value of zero, but if you take x^2 / x^3 and compare it to x^3 / x^2 you are going to see very different behavior as you approach zero, even though the top and bottom of the equation are both zero – in one you can cancel to 1/0 that tends to infinity, in the other you can cancel to 0/1 that tends to zero.
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