Floor function in complex plane

Question

My question is very simple. How would one take the floor of a complex number and what is the floor of i?

Answer 1

The floor function is defined as the largest integer less than or equal to the real number given.

Since "less than or equal to" isn't defined for complex numbers you can't have a floor function for them.

Answer 2

Wolfram Alpha defines $\lfloor a+bi\rfloor$ as $\lfloor a\rfloor+\lfloor b]i$. With this convention you may say that $\lfloor i\rfloor = i$

Taka a look also here

Answer 3

Momo's answer is probably more official, but there could be another definition as follows:

$$\lfloor a + bi \rfloor = n(a+bi)$$ where $n$ is a positive real number $\le 1$ such that $$|\lfloor a+bi \rfloor | = \lfloor |a+bi| \rfloor$$

In other words, keep the same angle and reduce the absolute value to the next integer.