Least operations and numbers possible!

You need to approxiamte π\pi using only the following operations on Integers :

  • Addition

  • Subtraction

  • Multiplication

  • Division

  • Exponentiation

  • Logarithm

  • Factorial

Some of which I found :

4214322π^4\sqrt{\frac{2143}{22}}\approx\pi 1047990+93648π^{10}\sqrt{\frac{47}{990}+93648}\approx\pi

Hope if you can find more accurate using least operations and numbers possible!

Note by Zakir Husain
3 months, 1 week ago

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1 vote

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Comments

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I found an another one

29808.1+(11033)29\sqrt[9]{29808.1+\left(1-\frac{10^{-3}}{3}\right)^{2}}

Wasi Husain - 3 months, 1 week ago

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Nice!

Jeff Giff - 3 months ago

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Woah! That second one is awesome!

I assume you are not looking for trivial approximations (such as 3141592653510000000000\frac{31415926535}{10000000000})

:)

David Stiff - 3 months, 1 week ago

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Similar to your second one, but not as precise:

62625+298099\sqrt[9]{\frac{62}{625} + 29809}

David Stiff - 3 months, 1 week ago

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Awesome !

Zakir Husain - 3 months, 1 week ago

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A longer one:
2865814513315\sqrt[15]{28658145-\frac{1}{33}}

Jeff Giff - 3 months, 1 week ago

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a shorter one: 306+52545\sqrt[5]{306+\dfrac{5}{254}}

Jeff Giff - 3 months, 1 week ago

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313\sqrt[3]{31} is 99.993%99.993\% accurate and its so simple

Jason Gomez - 3 months ago

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355113\frac{355}{113} is 99.999992%99.999992\% accurate

Jason Gomez - 3 months ago

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What about 2105334314167014872593.14159265358979323846238174277486\dfrac{21053343141}{6701487259}\approx \red{3.141592653589793238462}38174277486 it's accurate till 21 digits after the decimal point.

Zakir Husain - 3 months ago

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Hell no, this is the most accurate

Jason Gomez - 3 months ago

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@Jason Gomez My calculator could differentiate π and the approximation given by Wasi, for yours no

Jason Gomez - 3 months ago

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@Jason Gomez I found these rational approximations using python it's quit easy :)

That's why I wanted approximations other than those.

Zakir Husain - 3 months ago

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That’s correct to twenty two places, mind blown

Jason Gomez - 3 months ago

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@Jason Gomez It’s 99.9999999999999999999992% accurate, so that’s twenty two digits correct I think

Jason Gomez - 3 months ago

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@Jason Gomez You can divide it using this calculator and then can get digits of pi from here

Then you can check yourself...

Zakir Husain - 3 months ago

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@Zakir Husain Ok I realised my mistake, I said places instead of digits, I meant twenty two digits( includes the three)

Jason Gomez - 3 months ago

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@Jason Gomez Every nine is another digit

Jason Gomez - 3 months ago

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I am complying with the least operations and numbers rule more(I believe when they make problems involving π, rather than having to factor out 7, 113 should also be used, atleast occasionally)

Jason Gomez - 3 months ago

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ln(23.14+ln21000)\ln(23.14+\frac{\ln2}{1000}) is 99.9999993%99.9999993\% accurate

Jason Gomez - 3 months ago

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Nice! Check out the one by Wasi Husain which is so near pi my calculator says it equals pi :)

Jeff Giff - 3 months ago

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This is not good enough though :)

Jeff Giff - 3 months ago

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That one has accuracy 99.99999999999995%<x<100%99.99999999999995\%\lt x\lt 100\% as my calculator rounds the solution to the 13th place after the decimal point :)

Jeff Giff - 3 months ago

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The last digit is not 5 it’s 4

Jason Gomez - 3 months ago

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Not exactly the simplest, but it doesn't use any radicals. It's actually based off of the traditional approximation of π\pi, that being 227\frac{22}{7}:

220007002+(8175×104)106\dfrac{22000}{7002 + (8175 \times 10^{-4}) - 10^{-6}}

David Stiff - 3 months ago

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Why not everyone include the accuracy with their approximation

Jason Gomez - 3 months ago

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Calculated as (approximation)π×100\frac{\text(approximation)}{π}×100 and only the first digit where the deviation happens should be included, if it goes over pipi then subtract it from 200200 and give

Jason Gomez - 3 months ago

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Because everyone can find it whenever they wanted :)

well number of digits accuracy is more better I think

Zakir Husain - 3 months ago

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Then put accuracy or the number of digits?

Jason Gomez - 3 months ago

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@Jason Gomez Whatever you may like

Zakir Husain - 3 months ago

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3 is 95%95\% accurate

Jason Gomez - 3 months ago

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Not so close : πW1(e1111120)20+11111\pi\approx -W_{-1}(-e^{1111^{-1}-20})-20+1111^{-1} Where Wk(z)W_k(z) is the Lambert W function (Not following the rules)

Zakir Husain - 2 months, 2 weeks ago

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πln(6403203+744)163\pi\approx\frac{\ln\left(640320^{3}+744\right)}{\sqrt{163}}

Zakir Husain - 2 months, 2 weeks ago

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πln(6403203+7441968846403203+744)163\pi\approx\frac{\ln\left(640320^{3}+744-\frac{196884}{640320^{3}+744}\right)}{\sqrt{163}}

Zakir Husain - 2 months, 2 weeks ago

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π7.0255\pi\approx\dfrac{7.025}{\sqrt{5}}

Zakir Husain - 1 month, 3 weeks ago

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