2c:null 2d:["$","title",null,{"children":"A Nifty Algorithm for Converting Positive Integers to Roman Numerals | Michaël Guitton | Observable"}] 32:T728,There is archaeological evidence that Romans could count way above `3,999`. In most circumstances, Romans would not go beyond `100,000` and would use words to denote big numbers instead of letters from the Latin alphabet. The oldest noteworthy inscription containing numerals representing very large numbers is on the *Columna Rostrata* — a monument erected in the *Forum Romanum* to commemorate the victory at sea of consul Gaius Duilius over the Carthaginians in 260 BCE during the First Punic War — which brags about a booty of gold and silver worth at least `2,000,000` sesterces. In this column, a glyph for `100,000` — which was an early form of `(((I)))` (or `ⅭⅭⅭⅠↃↃↃ`) — was repeated 23 times, making `2,300,000`. This is the highest known number written numerically and it illustrates not only the early Roman use of repeated symbols but also a custom that extended to modern times — that of using `(I)` for `1,000`, `((I))` for `10,000`, `(((I)))` for `100,000`, and `((((I))))` for `1,000,000`. The symbol `(I)` for `1,000` frequently appears in various other forms, including the cursive `∞`. - `1,000` = `(I)` = `ⅭⅠↃ` - `5,000` = `I))` = `ⅠↃↃ` - `10,000` = `((I))` = `ⅭⅭⅠↃↃ` - `50,000` = `I)))` = `ⅠↃↃↃ` - `100,000` = `(((I)))` = `ⅭⅭⅭⅠↃↃↃ` - `500,000` = `I))))` = `ⅠↃↃↃↃ` - `1,000,000` = `((((I))))` = `ⅭⅭⅭⅭⅠↃↃↃↃ` Toward the end of the Roman Republic, a bar — known as the *vinculum* or *virgula* — was placed over a number to multiply it by `1,000`. This bar also came to represent ordinal numbers. In the early Roman Empire, bars enclosing a number around the top and sides came to mean multiplication by `100,000`. The use of the single bar on top lasted into the Middle Ages, but the three bars did not.33:T439,md`Roman numerals from \`1\` to \`10\` are denoted as follows: ~~~ ${tablet.split(/\n/).slice(0, 1).pop()} ~~~ Multiply them by \`10\`: ~~~ ${tablet.split(/\n/).slice(1, 2).pop()} ~~~ And by \`100\`: ~~~ ${tablet.split(/\n/).slice(2, 3).pop()} ~~~ Then assemble the following matrix using the ${ matrix.match(/\w/gm).length } occurrences of the single-letter numerals above: ~~~ ${matrix} ~~~ And trim duplicates to obtain a table which lists the ${ symbols.length } unique Latin symbols in ascending order: ~~~JavaScript ["${symbols.join('", "')}"] ~~~ Finally, build this ${ lookup.length }-element lookup table which maps a decimal digit to a sequence of indices: ~~~ ${lookup.map((e, i) => i + ": [" + e.join(", ") + "]\n").join("")} ~~~ The formula to map a tuple (\`\`, \`\`) to a given Latin symbol is: \` = + ${Object.values(hash).pop()} × \` where \`\` and \`\` values are between ${Object.values( hash ).shift()} and ${Object.values(hash).pop()}.`34:T70d,romanizeDigit = { /** * Lookup map from decimal digits to Latin symbol offsets * * ^ ^ ^ 0 1 2 * 1-10: I II III IV V VI VII VIII IX X # 0 I V X * 10 × 1-10: X XX XXX XL L LX LXX LXXX XC C # 1 X L C * 100 × 1-10: C CC CCC CD D DC DCC DCCC DM M # 2 C D M * * @example * lookup[4] yields [0, 1] which corresponds to: * - ['I', 'V'] when power-of-ten exponent is 0 * - ['X', 'L'] ............................. 1 * - ['C', 'D'] ............................. 2 */ const lookup = [ [], [0], [0, 0], [0, 0, 0], [0, 1], [1], [1, 0], [1, 0, 0], [1, 0, 0, 0], [0, 2] ]; /** * Latin symbols in ascending order used to denote… */ const lo = /* numbers ≤ 3999 */ ["I", "V", "X", "L", "C", "D", "M"]; const hi = /* numbers ≥ 4000 */ lo.map((x) => x + /* vinculum */ "\u0305"); lo.push(hi.slice(1, 2).pop()); /** * Convert decimal digit to Roman numeral according to the digit's power-of-ten exponent * * @example * romanizeDigit(4, 1) yields 'XL' since digit 4 × 10^1 = 40 * * @param {number} digit - numeral from 0 to 9 * @param {number} exponent - power-of-ten exponent * @return {string} Roman numeral with value digit × 10^exponent */ return function romanizeDigit(digit, exponent) { if (String(+digit).charAt(0) != digit) { throw RangeError("Only numerals between 0 and 9 are adequate"); } if (exponent > 6 || (exponent === 6 && digit >= 4)) { throw Error("Uh-oh!"); } let numeral = lo; if (exponent > 3 || (exponent === 3 && (digit === 4 || digit === 9))) { numeral = hi; exponent -= 3; } return lookup[digit].map((x) => numeral[x + 2 * exponent]).join(""); }; }35:[{"id":"5d5fc97f6cea738611b0b9319b229bf3156c68c166c41803b76f14b146832d420aeaf1483234fbfa140caf19580d74f826d4330ea9e2838606eeb066d6af14b3","href":"https://static.observableusercontent.com/files/5d5fc97f6cea738611b0b9319b229bf3156c68c166c41803b76f14b146832d420aeaf1483234fbfa140caf19580d74f826d4330ea9e2838606eeb066d6af14b3","name":"code-points.csv","type":"text/csv","lastModified":1658757883237.764,"size":180}] 30:{"id":"2655c183f0241720","slug":"a-nifty-algorithm-for-converting-positive-integers-to-roman-numerals","title":"A Nifty Algorithm for Converting Positive Integers to Roman Numerals","thumbnail":"64bc58d8ee1e0a645f854a982501ae1835fbe90dfba167741584781b3bd04243","access_level":"public","create_time":"$D2022-07-25T14:04:09.117Z","update_time":"$D2022-09-01T14:40:41.126Z","view_time":null,"first_public_time":"$D2022-09-01T13:15:01.520Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"4c1622bb5c37477f","login":"galopin","name":"Michaël Guitton","avatar_url":"https://avatars-next.observableusercontent.com/10e2fe999ac95c8686156fd07e96a686520149a4eef18a95f97795b7a7be1e29","tier":"starter_2024"},"version":128,"publish_version":128,"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,"subscribed":null,"forks":0,"collection":{"id":"b4b517569c1f13f6","slug":"roman-numerals","title":"Roman Numerals","owner":{"id":"4c1622bb5c37477f","login":"galopin","name":"Michaël Guitton"}},"type":"notebook","body":{"title":"A Nifty Algorithm for Converting Positive Integers to Roman Numerals","cells":[{"id":0,"value":"# A Nifty Algorithm for Converting Positive Integers to Roman Numerals","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":77,"value":"Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet.\n\nThere are seven symbols in this system, each with a fixed integer value: `I`, `V`, `X`, `L`, `C`, `D`, `M` — `1`, `5`, `10`, `50`, `100`, `500`, `1000` respectively.\n\nThere is no Roman numeral for cardinal number zero as there was no need for a symbol to represent it. Either *nullum* or *nihilum* — Latin words meaning *none* and *nothing* — could be used aptly.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":79,"value":"The lowest number you can write in Roman numerals is number `I`, i.e. `1`. And the largest numeral is `MMMCMXCIX`, i.e. `3,999`.\n\nOn the other hand, using the *vinculum* — which is a mathematical symbol used for grouping —, you can multiply the numeral by `1,000`. For example, if you have to convert `1,000,000`, you can draw a line atop the numeral `M`, e.g. `M̅`.\n\nThe *vinculum* for marking multiples of `1,000` was conceived during the Imperial era and continued in use in the Middle Ages, though it became known more commonly as *titulus*. It was a convenient way to write numbers higher than `3,999`.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":113,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":81,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":70,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":75,"value":"The rules for Roman numerals are as follows:\n\n- Roman numerals are repeated to add value. \n \n > `III` is equivalent to `1 + 1 + 1 = 3`.\n\n- Only powers of `10` may be repeated in this way. \n \n > Thus, `VV` is invalid; `5 + 5` would instead be expressed as `X`.\n\n- No more than three repetitions of a numeral can be used. Five repetitions can be represented with a single, larger numeral. To represent four, use the next larger numeral, but precede it with a numeral to subtract from it — the numeral used for subtraction must be the largest power of `10` that is less than the numeral it precedes. \n\n > Thus, `IIII` is deemed to be invalid — even though there is a tradition favouring this representation on most Roman numeral clocks — and should instead be written as `IV`, i.e. one less than five. Likewise, `XL` represents `40` (`10` less than `50`), and `XC` represents `90` (`10` less than `100`).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":72,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":54,"value":" Here is an algorithm which cleverly exploits the repetitive nature of Roman numerals.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":56,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"Converting a positive integer between `0` and `3,999,999` to a Roman numeral takes the following steps:\n\n1. Firt of all, split this positive integer into digits;\n2. Then convert each decimal digit to the corresponding Roman numeral according to the digit’s power-of-ten exponent;\n3. Combine each computed Roman numeral.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":60,"value":"stringify(romanizeNumber)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"stringify(splitDigits)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"stringify(romanizeDigit)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"See also [Algorithms for Roman Numerals](https://www.hanshq.net/roman-numerals.html)…","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":89,"value":"md`For example, \\`romanizeNumber(2022)\\` returns the Roman numeral string \\`${year}\\` and \\`splitDigits(2022)\\` yields \\`[${digits.join(\n \", \"\n)}]\\`.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"year = romanizeNumber(2022)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":91,"value":"digits = splitDigits(2022)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":95,"value":"quad4 = romanizeNumber(4444)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":97,"value":"forty = romanizeDigit(4, 1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":86,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"

📎 Both upper and lower Roman numerals are available as the following Unicode code points:

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"viewof numerals = Inputs.table(codePoints, {\n maxWidth: 960,\n layout: \"fixed\",\n width: {\n Range: `${\"U+216x\".length * 1.25}ch`\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"

🦉 It is noteworthy that these symbols do have a specific meaning as numbers.

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"blob = Object.fromEntries(\n query.map((item) => [\n item[0],\n item[1].map(({ Range, ...rest }) => /* drop Range key */ rest)\n ])\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"query = d3.groups(codePoints, (row) => row.Range)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"codePoints = FileAttachment(\"code-points.csv\").csv()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":24,"value":"tablet = `I II III IV V VI VII VIII IX X\nX XX XXX XL L LX LXX LXXX XC C\nC CC CCC CD D DC DCC DCCC DM M`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"matrix = /* drop non single-letter numerals */ tablet.replace(/ \\w{2,}/gm, \"\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"hash = /* map symbols to indices */ Object.fromEntries(\n matrix\n .match(/\\w/gm)\n .slice(0, 3)\n .map((x, i) => [x, i])\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"lookup = /* build lookup map */ [[]].concat(\n tablet\n .split(/\\n/)\n .slice(0, 1)\n .pop()\n .split(\" \")\n .map((e) => e.split(\"\").map((e) => hash[e]))\n .slice(0, -1)\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"symbols = /* trim duplicates */ [...new Set(matrix.split(/\\s+/))]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"function stringify(fn) {\n function indent(code) {\n return /^[ ]{4}(?=[^ ])/.test(\n code.split(\"\\n\").slice(1)\n ) /* a flaky regex that fit the bill */\n ? code.replace(/^[ ]{2}([ ]{2}|\\})/gm, \"$1\")\n : code;\n }\n return md`~~~JavaScript\n${indent(fn + \"\")}\n~~~`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":21,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":39,"value":"splitDigits = {\n if (!Number.MAX_SAFE_INTEGER) {\n Number.MAX_SAFE_INTEGER = Math.pow(2, 53) - 1;\n }\n\n /**\n * Split positive integer < 2^53 into digits\n *\n * @param {number} n - positive integer < 2^53\n * @return {number[]} digits starting with higher power of ten\n */\n return function splitDigits(n) {\n if (n < 0 || n > Number.MAX_SAFE_INTEGER || n % 1 !== 0) {\n throw RangeError(\n \"Only positive integers <= \" +\n Number.MAX_SAFE_INTEGER +\n \" are supported\"\n );\n }\n\n return Array.from(String(n), Number);\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"/**\n * Convert positive integer to Roman numeral\n *\n * @param {number} n - integer in range [0, 3999999]\n * @return {string} Roman numeral with value n\n */\nfunction romanizeNumber(n) {\n if (n < 0 || n > 3999999) {\n throw RangeError(\"Only numerals between 0 and 3999999 are allowed\");\n }\n\n let digits = splitDigits(n);\n\n return digits\n .map((digit, i) =>\n digit === 0 ? \"\" : romanizeDigit(digit, digits.length - i - 1)\n )\n .join(\"\");\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":19,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 31:"$30:body"