[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!

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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:

Roberton February 18, 2010at 10:12 am

Thanks, Robert! That’s amazing. Your comment inspired a follow-up post.

By:

Dave Richesonon February 18, 2010at 11:32 pm

[...] about the neat calculator trick Yesterday I wrote about a neat calculator trick that I had just [...]

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More about the neat calculator trick « Division by Zeroon February 18, 2010at 11:31 pm

[...] Why sin(11) is approximately -1 Math teachers at play carnival#23 Calculator trick [...]

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Weekend miscellany — The Endeavouron February 19, 2010at 2:58 pm

11 radians you mean. That’s the answer

By:

Usovon April 27, 2010at 10:02 am

[...] 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 [...]

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Truco Matematico « Apuntes Matemáticoson February 22, 2010at 10:38 am

Nice. This was just small enough and interesting enough to fit into my working memory and give me a jolt of happy.

By:

davidon April 27, 2010at 9:24 pm

Very nice trick, will be using that one myself =] and also nice explanation of why

By:

steveon April 28, 2010at 12:18 am

[...] 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？ [...]

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Matrix67: My Blog » Blog Archive » 如果你的计算器上没有pi……on April 28, 2010at 1:21 am

Can we get e using similar trick?

By:

eston April 28, 2010at 4:41 am

[...] 来源：http://divisbyzero.com/2010/02/17/the-math-behind-a-neat-calculator-trick/ [...]

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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, 2010at 1:18 am