What is $\lfloor i\rfloor$?

Question

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. However, I do not know what $\lfloor i\rfloor$ equals. Is there a clearly defined definition of $\lfloor i\rfloor$?

Answer 1

$\Bbb{C}$ is not totally ordered, and thus the floor function can't really be defined conventionally on it.

You could think of the floor of $z$ as $\lfloor \text{Re} \ z\rfloor+i*\lfloor \text{Im} \ z\rfloor$, which in the case of $z=i$, would be $i$, but in general, the floor is not defined on $\Bbb{C}$.

Answer 2

The floor function was intended to be used in the real numbers only.

However, it has been generalized to complex numbers, to be applied on the real part and the imaginary part separately.

For example, generalized floor($0.9+1.1i$) would equal $0+1i$.

Then, the floor of $i$ would be $i$.