mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Approximating π

 2022-03-14 
A few weekends ago, I visited Houghton-le-Spring to spend two days helping with an attempt to compute the first 100 decimal places of π by hand. You can watch Matt Parker's video about our calculation to find out about our method and how many correct decimal places we achieved.
One of my calculations
Spending two days computing an approximation of π led me to wonder how accurate calculations using various approximations of π would be.
One nice way to visualise this is to ask: what is the largest circle whose area can be correctly computed to the nearest mm² when using a chosen approximation of π? In this blog post, I'll answer this question for a range of approximations of π.

3

First up, how about the least accurate approximation we could possibly use: π = 3.
Using this approximation, the areas of circles with a radius of up to 1.88mm could be calculated correctly to the nearest mm². That's a circle about the size of an ant.

Pi Day: 3.14

Today is Pi Day, as in the date format M.DD, today's date is the first three digits of π. Using this approximation, circles with a radius of up to 17.7mm or 1.77cm can be calculated correctly to the nearest mm². That's a circle about the size of my thumb.

Pi Approximation Day: 22/7

In the date format DD/M, 22 July gives an approximation of π that is more accurate than 3.14. Using this approximation, circles with a radius of up to 19.8mm or 1.98cm can be calculated correctly to the nearest mm². That's a slightly bigger circle that's still about the size of my thumb.

Our approximation

In Houghton-le-Spring, our final computed value was 3.1415926535886829815214... The first 11 decimal places of this are correct.
Using this approximation, circles with a radius of up to \(6.71\times10^5\)mm or 671m can be calculated correctly to the nearest mm². That's a circle about the size of Regent's park.

The 100 decimal places we were aiming for

If we'd avoided any mistakes in Hougton-le-Spring, we would've obtained the first 100 decimal places of π. Using the first 100 decimal places of π, circles with a radius of up to \(7.8\times10^9\)mm or 7800km can be calculated correctly to the nearest mm². That's a circle just bigger than the Earth.

The 527 decimal places that William Shanks computed

In 1873, William Shanks computed 707 decimal places of π in Houghton-le-Spring. His first 527 decimal places were correct. Using his approximation, circles with a radius of up to approximately \(10^{263}\)mm or \(10^{244}\) light years can be calculated correctly to the nearest mm². The observable universe is only around \(10^{10}\) light years wide.
That's a quite big circle.
×3      ×3      ×3      ×3      ×3
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
When does "MM" give 14 for the month?
Steve Spivey
×3   ×4   ×4   ×3   ×4     Reply
I wonder if energy can be put into motion with pi, so that would be a lot of theoretical energy
Willem
×3   ×3   ×3   ×3   ×4     Reply
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "sixa-x" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

convergence python mathsjam logic geogebra approximation palindromes go determinants mathslogicbot puzzles signorini conditions gather town wool interpolation numerical analysis fractals flexagons game of life gaussian elimination video games a gamut of games menace numbers draughts standard deviation graph theory reddit rugby asteroids pascal's triangle mathsteroids triangles estimation map projections plastic ratio 24 hour maths advent calendar european cup gerry anderson chess craft platonic solids finite group dragon curves cambridge light crochet fonts pythagoras correlation youtube matrix of minors logs binary dates exponential growth curvature football chalkdust magazine inline code speed reuleaux polygons inverse matrices games people maths wave scattering royal baby news electromagnetic field tennis world cup talking maths in public folding tube maps trigonometry dinosaurs sobolev spaces ternary pi approximation day pizza cutting statistics mean data christmas card noughts and crosses national lottery london underground machine learning final fantasy golden ratio pi edinburgh game show probability finite element method error bars manchester science festival latex martin gardner programming guest posts fence posts errors php data visualisation big internet math-off folding paper matrices graphs oeis tmip arithmetic frobel harriss spiral javascript weak imposition anscombe's quartet coins preconditioning bempp simultaneous equations raspberry pi phd databet crossnumber propositional calculus cross stitch captain scarlet bubble bobble turtles boundary element methods hyperbolic surfaces hannah fry manchester matrix of cofactors bodmas stirling numbers polynomials recursion countdown london stickers zines sport logo newcastle runge's phenomenon braiding probability geometry hats matt parker sorting hexapawn ucl squares rhombicuboctahedron misleading statistics nine men's morris books accuracy computational complexity matrix multiplication chebyshev live stream realhats radio 4 quadrilaterals christmas dataset royal institution pac-man datasaurus dozen golden spiral the aperiodical weather station sound

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024