2c:null 2d:["$","title",null,{"children":"It's pi day! 🥮 | Declan Naughton | Observable"}] 32:T634,Just **100** terms used in this approximation gives an approximation accurate to **6 decimal places**. To get the accuracy I sought however, I found **500k terms** does the trick. You can confirm that above - but **be warned** ⚠️ executing this might cause problems on a slow computer or a mobile. Although the error versus Math.PI goes to 0, thats really because Javascripts 64-bit float arithmetic doesn't capture the full precision needed. Along similar lines Javascript is able to introduce little but significant errors into the approximation calculation, therefore this model makes use of [a separate arithmetic library](https://github.com/MikeMcl/bignumber.js#readme).

### Summary Here are the significant digits [used by NASA](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/) that I was able to calculate using the Nilakantha series: Π ≈ 3.1415926535897932 Javascript float arithmetic can be problematic in usecases such as this, but in this example it could be managed. calculang is **not at** [the forefront](https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80) of pi calculation methods, but it [may](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/) still suffice for interplanetary navigation 🎉, and against some methods it also saves some chalk. 🙂 *For a different and more visual method to approximate pi using calculang, see [Monte Carlo Pi 🥮](https://observablehq.com/@declann/monte-carlo-pi?collection=@declann/calculang)* . . . ## Appendix33:[] 30:{"id":"9bef7297d02cc3a1","slug":"its-pi-day","title":"It's pi day! 🥮","thumbnail":"afa83e3eaaa81c93c6cd73b1cddcb502db1aa6951c3fbca1620a6927bd020c45","access_level":"public","create_time":"$D2023-03-14T22:27:22.937Z","update_time":"$D2024-03-05T11:22:08.665Z","view_time":null,"first_public_time":"$D2023-03-14T23:35:20.080Z","has_unpublished_edits":null,"fork":true,"stars":0,"starred":false,"owner":{"type":"team","id":"fa28dcfb9c2f53ce","login":"declann","name":"Declan Naughton","avatar_url":"https://avatars-next.observableusercontent.com/b36edfd0f90184ac4c0a8d8726969fe861a1d1fe9417e998413c8a47e6163983","tier":"starter_2024"},"version":510,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"fca367047a53a31f","login":"declann","name":"Declan Naughton","avatar_url":"https://avatars-next.observableusercontent.com/b36edfd0f90184ac4c0a8d8726969fe861a1d1fe9417e998413c8a47e6163983","tier":null}],"members":[{"type":"individual","id":"fca367047a53a31f","login":"declann","name":"Declan Naughton","avatar_url":"https://avatars-next.observableusercontent.com/b36edfd0f90184ac4c0a8d8726969fe861a1d1fe9417e998413c8a47e6163983","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"ba17c9710e3d4563","slug":"calculang","title":"calculang","owner":{"id":"fa28dcfb9c2f53ce","login":"declann","name":"Declan Naughton"}},"type":"notebook","body":{"titling":"auto","title":"It's pi day! 🥮","cells":[{"id":0,"value":"# It's pi day! 🥮","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"Today, March 14th (3.14), we celebrate a famous ratio between a circles diameter and it's circumference - a glorious, irrational, [even transcendental](https://en.wikipedia.org/wiki/Transcendental_number) number called [pi](https://pt.wikipedia.org/wiki/Pi).\n\n![Pi = 3.1415926535897932384626433832795028...](https://upload.wikimedia.org/wikipedia/commons/5/5d/PI_constant.svg)\n\n## Irrational\n\npi, symbolized by **π**, is **irrational** - it can't be expressed exactly by **any fraction**.\n\nBut that doesn't stop mathematicians from finding more alternative ways to express it.\n\nAn example given to us by [Leibniz](https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz) ([or Madhava or his followers](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80)), is this [alternating](https://en.wikipedia.org/wiki/Alternating_series), odd-number series:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"tex.block`1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\frac{1}{9}-\\cdots=\\frac{\\pi}{4}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"
\nWe can use expressions like this to **approximate** the irrational number,
#thanks Leibniz/Madhava/et al. 🙌\n

\n\n## π approximation using Leibniz formula\n\nAccording to the formula, if we sum some terms from the series on the left and multiply the result by 4 then we will get a number approaching pi.\n\nThere are many ways we can do this, for example:\n\n### 1. Calculation by hand\n\nOn this day some years ago Matt Parker of [Stand-Up Maths](https://standupmaths.com/) calculated a value by hand & to an absurd precision^*.\n\nThe video is an informative and entertaining *pi day treat*. Calculation results and interpretation are @ 14mins:\n\n^*For a chalkboard! & excluding 1 little arithmetic error 🙃","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":76,"value":"html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":102,"value":"
\n### 2. Calculation using calculang & Leibniz formula\n\nHere is code for a calculang model that calculates a `pi_approximation` based on the series above for a given # of terms:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":126,"value":"md`\n~~~js\n// alternating sign: +, -, +, -, ...\nexport const sign = () => Math.pow(-1, term_number() - 1);\n\n// denominator from sequence of odd numbers\nexport const denominator = () => (term_number() - 1) * 2 + 1;\n\n// input: number of current term (internal)\nexport const term_number = () => term_number_in; \n\n// one term:\nexport const term = () => sign() * (1 / denominator());\n\n// sum of terms (1 to terms input):\nexport const sum_of_terms = () => range(1, terms() + 1).reduce(\n (acc, term_number_in) => acc + term({ term_number_in }),\n 0);\n\n// pi approximation:\nexport const pi_approximation = () => sum_of_terms() * 4;\n\n// input: total number of terms to sum\nexport const terms = () => terms_in;\n\n~~~\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":232,"value":"#### calculang Leibniz approximation results","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":240,"value":"viewof terms_in = Inputs.select(new Map([...(_.range(1,10+1).map(d=>[d,d])), [\"100\",100], [\"1000 (1k)\",1000], [\"10000 (10k)\",10000], [\"100000 (100k)\",100000],[\"1000000 (1M)\",1000000]]), {label: \"# terms\", value: 1})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":235,"value":"md`Π ≈ ${leibniz.pi_approximation({terms_in})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"md`error versus Math.PI ≈ ${Math.PI - leibniz.pi_approximation({terms_in})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"Cycling through these results we can confirm what Matt remarked in the video: the series converges on pi **really slowly**!\n\nUsing **1M** terms gives an accuracy to just **5 decimal places**.\n\nExperimenting with even more iterations and averaging approaches I still failed to get an accuracy to 15 decimal places, which is desirable for [interplanetary navigation](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/) - and in this day and age who doesn't need that!\n

\n## π approximation using calculang & Nilakantha series\n\nNot content therefore, I developed a similar calculang model using the [Nilakantha series](http://www.maeckes.nl/Formule%20voor%20pi%20%28Nilakantha%29%20GB.html) which is known to converge on pi faster.\n\nYou can explore the results here (with caution for larger # terms ⚠️):","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":310,"value":"viewof nil_terms_in = Inputs.select(\n new Map([\n ..._.range(1, 10 + 1).map((d) => [d, d]),\n [\"100\", 100],\n [\"1000 (1k)\", 1000],\n [\"10000 (10k)\", 10000],\n [\"100000 (100k) [Caution ⚠️]\", 100000],\n [\"500000 (500k) [Caution ⚠️]\", 500000],\n [\"2000000 (1M)\", 2000000]\n ]),\n { label: \"# terms\", value: 1 }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":317,"value":"md`Π ≈ ${nilakantha.pi_approximation({terms_in:nil_terms_in})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":319,"value":"md`error versus Math.PI ≈ ${Math.PI - nilakantha.pi_approximation({terms_in:nil_terms_in})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":190,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":198,"value":"location1 = \"declann/calculang-miscellaneous-models\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":203,"value":"sha = \"2f2a\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":200,"value":"leibniz_location = \"/models/pi/odd-numbers-series\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":192,"value":"leibniz = require(`https://cdn.jsdelivr.net/gh/${location1}@${sha}${leibniz_location}.js`)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":280,"value":"nilakantha_location = \"/models/pi/nilakantha-series-bignumber\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":278,"value":"nilakantha = require(`https://cdn.jsdelivr.net/gh/${location1}@${sha}${nilakantha_location}.js`)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":502,"value":"## credits\n\nmade with 💖 and [calculang](https://github.com/calculang)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W33","stdlib":"1"}} 31:"$30:body"