[Update: after you read this post, read my follow-up post.]

I received an interesting comment on yesterday’s blog post from Nemo. It was a cool calculator trick that I’d never seen before. Nemo wrote:

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.

Press “1/x”.

Press “sin”.

Examine the mantissa of the result. Magic!

Well, if you give it a try you find out that

,

and similarly

.

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 .

Ta da!