Yesterday I wrote about a neat calculator trick that I had just learned.
We saw that if the calculator was set to degree mode, then times a high enough power of 10 is approximately .
A commenter named Robert suggested looking at the difference between this approximation for and itself. He remarked that the error is also approximately too (when multiplied by a sufficiently high power of 10)!
If we shift the decimal point over we get an approximation of :
Now we look at the error
which if we shift again is
Wow! Now why is this true?
The key observation is that
or in general,
where denotes the -digit integer of all 5’s. With a little algebra we see that
As I mentioned last time, if the calculator is in degree mode and , then . So,
This implies that
Now, the error in this approximation is
We can rewrite this as