I was working on a problem which involved the usage of the Floor Function(Or the greatest integer function) and a question came to my mind,
Is the floor function defined for complex values?
More precisely, Is $$\lfloor x+iy \rfloor$$ defined?
WolframAlpha says that it equals $\lfloor x \rfloor + \lfloor y \rfloor i$ but I doubt if that's true
My thought was that the floor function isn't defined for the complex values because it returns an integer less than or equal to the number inside the floor so it isn't defined. Another question,
Can we define $\lfloor x + iy \rfloor = x+iy$ ?
Any hint/answer is appreciated
Thanks in advance
Best Answer
Inequality is indeed not defined for complex numbers.
A possible extension of the definition of the floor function, is to apply it to each of the components of the complex number. Or one step further, to apply it so that we get a Gaussian integer. Yet another alternative is to apply it to the magnitude.
In the first case we have $\lfloor 1.6+1.7i\rfloor = \lfloor 1.6\rfloor+\lfloor 1.7\rfloor i = 1+i$.
In the second case we might have $\lfloor 1.6+1.7i\rfloor = \frac 32+\frac 32 i$.
In the third case we get $\lfloor 1.6+1.7i\rfloor=\lfloor |1.6+1.7i|\rfloor=\lfloor\sqrt{1.6^2+1.7^2}\rfloor$.
Up to you how to define it, or leave it undefined.