Is the following complex number "finite"? This is my first question on this forum, so, forgive me in advanced if I make some type of syntax error...
I am working on applying a theorem which involves computing a definite integral, and showing that the number obtained from this definite integral is finite, i.e., "< $\infty$". 
However, the result I obtain, which I am quite sure of is ArcTanh[100] = 0.0100003 - 1.5708 I. My question is:
Is: 0.0100003 - 1.5708 i < $\infty$? 
My first inkling was that if the components a,b of a + bi are finite, then the complex number should be finite, but, I am not sure how the notion of finiteness applies in this case.
Thanks!
 A: Yes, of course that complex number is less than infinity.  Any complex number can be most easily compared to "infinity" (an arbitrarily large number) by simply calculating its magnitude, that is the square root of the sum of the squares of its $x$ and $y$ components. Of course, if you CAN calculate that magnitude, it is less than infinity.  On the other hand, it is easy to draw curves (I switch to a POLAR representation) which manifestly tend toward infinity as their angle $\theta$ grows large.  Such a curve would be, for example, $R = \theta$.  The limit of $R$ as $\theta$ approaches infinity IS infinity, and we can picture this simply as a spiral that keeps on going, and keeps on going...
A: All complex numbers are finite, and one way to gauge "how finite" a number is (in a manner of speaking) would be to look at its modulus, which measures the distance away from the origin.
For $z = 0.0100003 - 1.5708 i$, we have $|z| = \sqrt{0.0100003^2 + (-1.5708)^2} \approx 1.5708.$
That is to say, your number is about $1.5708$ units away from the origin. The fact that it is an almost purely imaginary number has nothing to do with its being finite.
