A few days ago, I had a problem that looked like this: evaluate cot(pi) I know that on the unit circle, cot(pi) ends up as -1/0. In my precalc class, we say that this is undefined because you can't divide by zero. I decided to plug the problem into wolfram and it tells me that there is in fact an answer, that being complex infinity. I'm not sure what to make of this as I've never heard of complex infinity. Am I wrong to say that cot(pi), or any other number divided by zero is undefined, or is the correct answer complex infinite? Thanks!
The correct answer is undefined. When you take complex analysis, the correct answer is complex infinity :-) Here's a page that reveals all. http://en.wikipedia.org/wiki/Riemann_sphere
Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus. Thank you
Hey physicsdreams. A complex number is written in the form of z = a + bi where a and b are just real numbers. The infinite-complex number is just a number that has an infinite 'length'. We define the 'length' of a complex number to be SQRT(a^2 + b^2). Basically if you look at the Riemann-Sphere wiki that was posted above, this 'infinite' complex number is at the point where the 'north pole' is, and the complex number that is 'zero' (i.e. z = 0 + 0i = 0) is at the south pole.
You can visualize the complex numbers as a sphere with zero at one pole and "complex infinity," a symbolic extra point, at the other pole. When you do this, you can make sense of saying that the function 1/z takes the value complex infinity at z = 0. That's where Wolfram is getting its answer from. You're not really dividing by zero, but rather taking the limit of a complex function as the function's value approaches the north pole in the complex sphere. This is somewhat advanced math, typically taken by undergrad math majors after a couple of years of calculus and a class in real analysis. It would never be accurate to say you can divide by zero. Perhaps Wolfram should do a better job of explaining what they're doing so as not to confuse people who haven't taken a course in complex variables.
Thank you all for your explanations. Hopefully I'll gain a better understanding of this advance concept in the future.
It's undefined when working with the reals. I don't think there's any reason to get into anything more complex at this point (certainly not complex analysis).