The math behind a neat calculator trick

[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

${\sin(1/555555)=0.000000031415958=3.1415958\times 10^{-8}}$ ,

and similarly

${\sin(1/5555555555)=3.141592653903954\times 10^{-12}}$ .

Surely it can’t be a coincidence that the significant digits look so much like ${\pi}$ . It isn’t.

So why does this work?

First, notice that $\frac{1}{180}=0.005555\bar{5}$ . Thus, if ${n_{k}=555\cdots 5}$ (the ${k}$ -digit integer of all 5’s), then $\frac{1}{n_{k}}\approx 180\times 10^{-k-2}$ .

Also, you may remember that for ${x}$ close to zero, ${\sin(x)\approx x}$ . Of course, this is only true if you are using radians. If you are using degrees, then for ${x}$ close to zero ${\displaystyle\sin x\approx \frac{\pi}{180}x}$ .

Putting this all together we see that ${\displaystyle\sin(\frac{1}{n_{k}})=\sin(\frac{1}{555\cdots 5})\approx\frac{\pi}{180}(180\times 10^{-k-2})=\pi\times 10^{-k-2}}$ .

Ta da!

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