2c:null 2d:["$","title",null,{"children":"An Infinite Series That Converges Quickly on Pi (π) | Michaël Guitton | Observable"}] 32:T73f,md`🦉 It is worth noting that using Liu Hui’s π algorithm, a Chinese mathematician and astronomer named Zu Chongzhi (429–500 AD) famously computed π to be between \`3.1415926\` and \`3.1415927\` and gave two rational approximations:

${a}`)}

… naming them respectively *Milü* (close ratio) and *Yuelü* (approximate ratio). An easy mnemonic helps memorize the first rational approximation by writing down each of the first three odd integers twice — e.g. “\`113355\`” — then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: - think “\`113355\`” - slash the middle: “\`113/355\`” - flip: “\`355/113\`” With a correct value for its six first decimal places, it remained the most accurate approximation of π available for the next 800 years.`33:[{"id":"fd58bc54e647e9ec5bcbc27990774bd877e0460b462eae9c31d5b7d3eb816233fe5062edbfacefd4e3c4c04abd4d1959bdf97bffeeb59287f579de4587ab5340","href":"https://static.observableusercontent.com/files/fd58bc54e647e9ec5bcbc27990774bd877e0460b462eae9c31d5b7d3eb816233fe5062edbfacefd4e3c4c04abd4d1959bdf97bffeeb59287f579de4587ab5340","name":"fred-moon-3NvRkNaiHtc-unsplash.jpg","type":"image/jpeg","lastModified":1663848518741.164,"size":694334}] 30:{"id":"58e9a998a9c54693","slug":"an-infinite-series-that-converges-quickly-on-pi","title":"An Infinite Series That Converges Quickly on Pi (π)","thumbnail":"00c5f188bf480b11a8d1c835f9ee84786728a2b6861b3b62b67059db7fc84641","access_level":"public","create_time":"$D2022-07-26T16:19:36.489Z","update_time":"$D2025-12-19T18:19:00.813Z","view_time":null,"first_public_time":"$D2022-09-23T16:15:34.918Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"4c1622bb5c37477f","login":"galopin","name":"Michaël Guitton","avatar_url":"https://avatars-next.observableusercontent.com/10e2fe999ac95c8686156fd07e96a686520149a4eef18a95f97795b7a7be1e29","tier":"starter_2024"},"version":64,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"950eb34948878085","login":"galopin","name":"Michaël Guitton","avatar_url":"https://avatars-next.observableusercontent.com/b59822ff41962e9448473fadd4a11819ac725b074ad3868aabf532bade0ac00d","tier":null}],"members":[{"type":"individual","id":"950eb34948878085","login":"galopin","name":"Michaël Guitton","avatar_url":"https://avatars-next.observableusercontent.com/b59822ff41962e9448473fadd4a11819ac725b074ad3868aabf532bade0ac00d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"4e6f7f31a6e99891","slug":"computing-digits-of-pi","title":"Computing Digits of Pi (π)","owner":{"id":"4c1622bb5c37477f","login":"galopin","name":"Michaël Guitton"}},"type":"notebook","body":{"titling":"auto","title":"An Infinite Series That Converges Quickly on Pi (π)","cells":[{"id":0,"value":"# An Infinite Series That Converges Quickly on Pi (π)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":57,"value":"

${Object.assign(await FileAttachment(\"fred-moon-3NvRkNaiHtc-unsplash.jpg\").image(), { alt: 'The Motion of the Stars', style: 'max-width: 1OO%; height: auto', title: 'Photo by Fred Moon (@fwed) on Unsplash', width: 2800, height: 1866 })}","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"TeX`\\begin{aligned}\n\\pi = 3 + \\frac{4}{2\\times3\\times4} - \\frac{4}{4\\times5\\times6} + \\frac{4}{6\\times7\\times8} - \\frac{4}{8\\times9\\times10} + \\cdots\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"This infinite series was published by the Indian mathematician and astronomer Nilakantha Somayaji (1445—1545 AD).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":32,"value":"md`The computed value of π after ${format(R.length)} iteration${\n R.length === 1 ? \"\" : \"s\"\n} is:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"html`

${R.slice(-1).pop().result}

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":40,"value":"viewof Q = Inputs.range([1000, 10000], {\n placeholder: \"1000–10000\",\n step: 100,\n value: 1000\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"Nowadays, series that converge even faster include [Machin’s series](/@galopin/yet-another-converging-series-for-pi) and [Chudnovsky’s series](/@galopin/the-chudnovsky-algorithm-for-calculating-pi).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":19,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":21,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":30,"value":"R = [...compute(pi, Q)]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"/*!\n * An infinite series that converges quickly on Pi (π) was published by\n * the Indian mathematician and astronomer Nilakantha Somayaji (1445-1545)\n *\n * π = 3 + (4 / 2 * 3 * 4) - (4 / 4 * 5 * 6) + (4 / 6 * 7 * 8) - (4 / 8 * 9 * 10) + …\n */\nfunction* pi() {\n function term(input, sign, divisor) {\n divisor = input * ++input * ++input;\n return (sign * 4) / divisor;\n }\n\n let pi = 3;\n let i = 1;\n\n while (true) {\n let quotient = term(i * 2, i & 1 ? 1 : -1);\n\n pi += quotient;\n\n let reset = yield {\n iterations: i++,\n term: quotient,\n result: pi\n };\n\n if (reset) {\n pi = 3;\n i = 1;\n }\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"function* compute(series, count = 20) {\n for (let value of series()) {\n if (--count < 0) break;\n yield value;\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"format = {\n return Intl && Intl.NumberFormat\n ? new Intl.NumberFormat(\"en-US\").format\n : d3.format(\",\");\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"TeX = (expr) =>\n html`

${tex(expr)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1"}} 31:"$30:body"