# Least operations and numbers possible!

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

• Subtraction

• Multiplication

• Division

• Exponentiation

• Logarithm

• Factorial

Some of which I found :

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

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

Note by Zakir Husain
5 months, 3 weeks ago

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I found an another one

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

- 5 months, 3 weeks ago

Nice!

- 5 months, 3 weeks ago

Woah! That second one is awesome!

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

:)

- 5 months, 3 weeks ago

Similar to your second one, but not as precise:

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

- 5 months, 3 weeks ago

Awesome !

- 5 months, 3 weeks ago

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

- 5 months, 3 weeks ago

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

- 5 months, 3 weeks ago

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

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

Hell no, this is the most accurate

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

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

That's why I wanted approximations other than those.

- 5 months, 2 weeks ago

That’s correct to twenty two places, mind blown

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

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

Then you can check yourself...

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

Every nine is another digit

- 5 months, 2 weeks ago

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)

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

This is not good enough though :)

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

The last digit is not 5 it’s 4

- 5 months, 2 weeks ago

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

$\dfrac{22000}{7002 + (8175 \times 10^{-4}) - 10^{-6}}$

- 5 months, 3 weeks ago

Why not everyone include the accuracy with their approximation

- 5 months, 2 weeks ago

Calculated as $\frac{\text(approximation)}{π}×100$ and only the first digit where the deviation happens should be included, if it goes over $pi$ then subtract it from $200$ and give

- 5 months, 2 weeks ago

Because everyone can find it whenever they wanted :)

well number of digits accuracy is more better I think

- 5 months, 2 weeks ago

Then put accuracy or the number of digits?

- 5 months, 2 weeks ago

Whatever you may like

- 5 months, 2 weeks ago

3 is $95\%$ accurate

- 5 months, 2 weeks ago

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

- 5 months ago

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

- 5 months ago

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

- 5 months ago

$\pi\approx\dfrac{7.025}{\sqrt{5}}$

- 4 months, 2 weeks ago