34:["$","title",null,{"children":"On a certain observation on differences of numbers and their mirror images | Prathyush | Observable"}] 3c:[] 38:{"id":"641de9001586eb94","slug":"on-a-certain-observation-on-differences-of-numbers-and-thei","title":"On a certain observation on differences of numbers and their mirror images","thumbnail":"877330d470626c4a33b6d0bc70441ace94d074f78da97a6681b124ec28b8917c","access_level":"public","create_time":"$D2022-01-09T23:07:37.639Z","update_time":"$D2022-01-20T19:37:36.875Z","view_time":null,"first_public_time":"$D2022-01-10T00:08:33.336Z","has_unpublished_edits":null,"fork":false,"stars":2,"starred":false,"owner":{"type":"team","id":"f33b22c086eea27f","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":"starter_2024","bio":"Software Designer","followed_by_you":false,"follows_you":false},"version":1314,"publish_version":1314,"role":null,"authored":false,"authors":[{"type":"individual","id":"3b0d270e1b775ddf","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":null,"bio":"Software Designer","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"3b0d270e1b775ddf","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":null,"role":"editor"}],"comments":[{"id":"d58563784128544e","reply_to_id":null,"content":"If you restrict to three digit numbers, it turns out to be the factor of 11 too! But then from 1000 onwards, it progresses as something where you have n(n+x)n where the x value increases as you jump 10s. Verify this claim.","create_time":"$D2022-01-12T10:48:33.778Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"3b0d270e1b775ddf","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":null}}],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"On a certain observation on differences of numbers and their mirror images","cells":[{"id":0,"value":"# On a certain observation on differences of numbers and their mirror images","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"md`I happened to observe recently that if you take a number, do a mirror image of it and then subtract it from the original, what you end up with is a multiple of 9.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":441,"value":"## Mirror symmetry","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"md`Let us work through this. Pick any number:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"viewof chosen = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":67,"value":"md`Feel free to tweak it using this slider:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"viewof shift = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":76,"value":"html`

Your number is ${number}. Its reverse is ${numberReverse}

Their difference is: ${mirrorDifference}

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":119,"value":"md`Which is surprisingly a multiple of nine! **${mirrorDifference} = 9 × ${factor}**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":157,"value":"md`In short, subtracting mirror image of a number from itself is always a factor of 9:
**(${Math.max(number, numberReverse)} - ${Math.min(numberReverse, number)}) ÷ 9 = ${factor}**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":447,"value":"## Permutations","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":251,"value":"md`Fun thing is, this generalizes to not just mirror images, but to any permutation of a number! Try rotating the digits with this slider:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":266,"value":"viewof rotateTimes = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":334,"value":"md`Your chosen number is: **${number}**. Shifted number is: **${shiftedNumber}**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":308,"value":"md`Their difference is ${rotatedDifference}, which is again, a multiple of nine!\n
We have: **${rotatedDifference} = 9 × ${rotatedDifference / 9}**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":455,"value":"md`## Basis Shift`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":402,"value":"md`And now for something even funnier. This generalizes to any number system with the numeral in that base less than 1 acting out the role of nine!`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":473,"value":"md`Let us see this fact in action. Choose your number system base:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":480,"value":"viewof radix = md``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":484,"value":"The number you have chosen ${number} in base 10 is now ${basisShiftedNumber} in the new base ${radix}. Its reverse is ${basisShiftedReverse}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":556,"value":"md`Difference of these are ${basisShiftDifference}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":532,"value":"md`So thought in the new number system, we have: ${basisShiftLHS} - ${basisShiftRHS} = ${basisShiftDifferenceInBase10}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":579,"value":"md`And it turns out that role of 9 in base 10 is played by ${radix - 1} which is one less than your choice of the new basis ${radix}. And again, this works out as a factor of this difference! ${basisShiftDifference} = ${radix - 1} × ${basisShiftFactorInBase10.toString(radix)} in base ${radix}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":592,"value":"md`In effect:
\n**${basisShiftLHS.toString(radix)} - ${basisShiftRHS.toString(radix)} = ${basisShiftDifference} = ${radix - 1} × ${basisShiftFactor}**
\nwhich in base 10 is equivalent to:
\n**${basisShiftLHS} - ${basisShiftRHS} = ${basisShiftDifferenceInBase10} = ${(radix - 1).toString(10)} × ${basisShiftFactorInBase10}**`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":233,"value":"md`What is happening here? Let us find out in the subsequent posts. From how it looks to me right now, these will be the next editions in this series:\n\n1/ Introduction to modulo arithmetic\n\n2/ Numbers as polynomials\n\n3/ Congruence rules\n\n4/ Casting out nines and elevens\n- a) Use cases as check sums\n- Relate to the idea in homomorphisms\n- Detour to the case of binary\n- That idea of obtaining the significant digits by doing modulus on each index of the powers\n\n5/ Order reduction\n\n6/ Connection to polynomial remainder/factor theorem\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":195,"value":"md`----------`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":280,"value":"shiftedNumber = rotate(number, +rotateTimes);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":98,"value":"numberReverse = reverseNumber(number);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"reverseNumber = number => +[...(number + \"\")].reverse().join(\"\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"number = +chosen + shift;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":253,"value":"rotate = (n, shift = 1) => {\n let numeral = [...\"\" + n];\n return +(numeral.map((_, idx) => numeral[(idx + shift) % numeral.length]).join(\"\"));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"mirrorDifference = difference(number, numberReverse);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":327,"value":"rotatedDifference = difference(number, shiftedNumber);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":596,"value":"basisShiftedReverseInBase10 = parseInt(basisShiftedReverse, radix);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":123,"value":"difference = (n, m)=> Math.abs(n - m);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":128,"value":"factor = mirrorDifference / 9;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":557,"value":"basisShiftDifferenceInBase10 = difference(number, basisShiftedReverseInBase10);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":724,"value":"basisShiftDifference = basisShiftDifferenceInBase10.toString(radix)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":501,"value":"basisShiftedNumber = number.toString(radix);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":544,"value":"basisShiftedReverse = basisShiftedNumber.split(\"\").reverse().join(\"\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":613,"value":"basisShiftFactorInBase10 = basisShiftDifferenceInBase10 / parseInt(radix - 1, 10)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1205,"value":"basisShiftFactor = basisShiftFactorInBase10.toString(radix)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":593,"value":"basisShiftLHS = Math.max(number, basisShiftedReverseInBase10);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":604,"value":"basisShiftRHS = Math.min(number, basisShiftedReverseInBase10);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3c","stdlib":"1"}} 33:null