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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.
![](javascript:showlimage("pi2022calc.jpg"))
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.
![](javascript:showlimage("pi2022ant.png"))
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.
![](javascript:showlimage("pi2022thumb.jpg"))
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.
![](javascript:showlimage("pi2022thumb2.jpg"))
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.
![](javascript:showlimage("pi2022regents.png"))
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.
![](javascript:showlimage("pi2022earth.jpg"))
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.
![](javascript:showlimage("pi2022universe.png"))
That's a quite big circle.
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⭐ top comment (2022-08-15) ⭐
When does "MM" give 14 for the month?Steve Spivey
×3 ×4 ×4 ×3 ×4 I wonder if energy can be put into motion with pi, so that would be a lot of theoretical energy ×3 ×3 ×3 ×3 ×4
Willem
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2020-07-29
A week ago, it was 22 July: Pi Approximation Day.
22/7 (22 July in DD/M format) is very close to pi, closer in fact than 14 March's
approximation of 3.14 (M.DD).
During this year's Pi Approximation Day, I was wondering if there are other days that give good
approximations of interesting numbers. In particular, I wondered if there is a good 2π (or τ)
approximation day.
π is close to 22/7, so 2π is close to 44/7—but sadly there is no 44th July.
The best approximation day for 2π is 25th April, but 25/4 (6.25) isn't really close to
2π (6.283185...) at all. The day after Pi Approximation Day, however, is a good approximation of 2π-3 (as π-3 is
approximately 1/7). After noticing this, I realised that the next day would be a good approximation
of 3π-6, giving a nice run of days in July that closely approximate expressions involving pi.
After I tweeted about these three, Peter Rowlett suggested
that I could get a Twitter bot to do the work for me. So I made one:
@HappyApproxDay.
Since writing this post, Twitter broke @HappyApproxDay by changing their API, but the bot lives on on Mathstodon: @HappyApproxDay@mathstodon.xyz.
@HappyApproxDay is currently looking for days that approximate expressions involving
π, τ, e, √2 and √3, and approximate the chosen expression better than
any other day of the year. There are an awful lot of ways to combine these numbers, so @HappyApproxDay@mathstodon.xyz
looks like it might be tooting quite a lot...
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