The math behind a neat calculator trick

Posted by: Dave Richeson | February 17, 2010

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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Posted in Math | Tags: calculator trick, pi, trigonometry

Responses

Now take your result, multiply it time 1 x 10^(-k-2) so you get almost pi and subtract it from the real pi. The difference is close to pi (with the decimal point shifted).
By: Robert on February 18, 2010
at 10:12 am

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- Thanks, Robert! That’s amazing. Your comment inspired a follow-up post.
  By: Dave Richeson on February 18, 2010
  at 11:32 pm
  
  Reply
[...] about the neat calculator trick Yesterday I wrote about a neat calculator trick that I had just [...]
By: More about the neat calculator trick « Division by Zero on February 18, 2010
at 11:31 pm

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[...] Why sin(11) is approximately -1 Math teachers at play carnival#23 Calculator trick [...]
By: Weekend miscellany — The Endeavour on February 19, 2010
at 2:58 pm

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- 11 radians you mean. That’s the answer
  By: Usov on April 27, 2010
  at 10:02 am
  
  Reply
[...] post es una traducción hecha de “The Math behind a neat calculator trick” post del blog Division by Zero del Professor Dave Richeson, reproducido y traducido con su [...]
By: Truco Matematico « Apuntes Matemáticos on February 22, 2010
at 10:38 am

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Nice. This was just small enough and interesting enough to fit into my working memory and give me a jolt of happy.
By: david on April 27, 2010
at 9:24 pm

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Very nice trick, will be using that one myself =] and also nice explanation of why
By: steve on April 28, 2010
at 12:18 am

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[...] 来源：http://divisbyzero.com/2010/02/17/the-math-behind-a-neat-calculator-trick/ Posted in Brain Storm Tags: 证明, 圆周率, 惊奇数学事实Trackback: http://www.matrix67.com/blog/archives/3155/trackback 我猜您可能还喜欢： Buffon投针实验：究竟为什么是pi？ [...]
By: Matrix67: My Blog » Blog Archive » 如果你的计算器上没有pi…… on April 28, 2010
at 1:21 am

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Can we get e using similar trick?
By: est on April 28, 2010
at 4:41 am

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[...] 来源：http://divisbyzero.com/2010/02/17/the-math-behind-a-neat-calculator-trick/ [...]
By: 如果你的计算器上没有pi…… « 四千卐格林丹的天然物语 on April 29, 2010
at 1:18 am

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