[Update: after you read this post, read my follow-up post.]
Reminds me of my favorite calculator trick.
Set your calculator to degree mode (NOT radians).
Type in a bunch of 5′s: 555555, or whatever.
Examine the mantissa of the result. Magic!
Well, if you give it a try you find out that
Surely it can’t be a coincidence that the significant digits look so much like . It isn’t.
So why does this work?
First, notice that . Thus, if (the -digit integer of all 5′s), then .
Also, you may remember that for close to zero, . Of course, this is only true if you are using radians. If you are using degrees, then for close to zero .
Putting this all together we see that .