Yksikkömurtolukujen desimaalikehitelmien jakson pituuksista 1 – pohdittavaa, ongelmia

Olen jo jokusen vuoden ajan ollut ihastunut desimaalilukuihin, lähinnä rationaalilukujen desimaalikehitelmiin. Aloin pohdiskella jaksojen pituuksia. Yksinkertaisuuden vuoksi keskityn tässä yksikkömurtolukuihin, eli tyyppiä 1/1, ½, ⅓, ¼, … oleviin murtolukuihin. Jos olet riittävän vanha, niin olet jakanut lukuja jakokulmassa riittävän paljon muistaaksesi, että jakso toistuu, kun jakokulmaan tulee jakojäännökseksi aiemmin jakojäännöksenä ollut luku.

Yritin luoda algoritmin, jonka avulla olisin voinut tutkia yksikkömurtolukujen desimaalikehitelmien jaksojen pituuksia GeoGebralla. Omassa päässäni ja paperilla onnistuinkin, mutta en osaa tarpeeksi hyvin käyttää GeoGebraa, jotta saisin algoritmin toimimaan. Tämä johtuu lähinnä siitä syystä, että GeoGebralla virheiden etsintä pitkästä komentorivistä on erittäin hankalaa minun aivoilleni ja sormilleni.

Lopulta tein Python koodinpätkän, joka tuottaa minulle vaikkapa listan sadan tai 10000 ensimmäisen yksikkömurtoluvun jaksojen pituuksista.

Solmu lehden Porasen ja Laukkasen artikkelissa (linkki liitteissä) on hyviä ehdotuksia, miten tällaista aihetta voisi käsitellä koulumatematiikassa ala-asteelta lukioon.

määritelmiä/sopimuksia

Alla olevassa taulukossa on 15 ensimmäistä yksikkömurtolukua ja likiarvot 15 numeron tarkkuudella, GeoGebra katkaisee päättyvät desimaalit eikä näytä lopun nollia. Desimaalikehitelmät voidaan jakaa päättyviin ja päättymättömiin. 

Oheisessa taulukossa 1/1, ½, ¼, ⅕,  ⅛, ja 1/10 ovat päättyviä. Sovitaan, että niiden jakson pituus on nolla.

Päättymättömien desimaalilukujen jaksojen pituudet vaihtelevat. Ykkösen mittaisia saadaan n:n arvoilla 3, 6, 9, 12, 15. Kahden mittaisia n:n arvolla 11, kuuden mittaisia n:n arvoilla 7, 13 ja 14

Sovitaan, että funktio f(n) = yksikkömurtoluvun 1/ n desimaalikehitelmän jakson pituus. Jokainen jakolaskua harrastanut varmaankin osaa vastata oikein seuraaviin kysymyksiin pelkästään intuitiolla tai sitten voimallisella matemaattisella pohdiskelulla.

1. ongelma – päättyvät desimaalit

Millaiset n:n arvot tuottavat päättyviä desimaalilukuja eli millaisilla n:n arvoilla f(n) = 0?

2. ongelma – pisin mahdollinen jakso

Mikä on suurin mahdollinen jakson pituus jollain n:n arvolla. Onko se n, 2n, n^2, n-1, n/2 vai mitä?

Eli onko olemassa n:n arvoja, joilla f(n) = n^2, f(n) = 2n, f(n) = n, … Millainen n:n lauseke on mahdollinen yläraja?

2. ongelma – yleisin jakso

Jos tutkitaan 15 ensimmäistä f(n):n arvoa, niin saadaan seuraavan näköinen jakauma jaksojen pituuksien arvoista.

Mikä on yleisin jaksojen lukumäärä eli moodi, kun tutkitaan 100 ensimmäistä f(n):n arvoa, entä 1000 tai 10000? 

3. ongelma – koordinaatisto

Alla olevassa kuvaajassa on jakson pituus n:n funktiona.

Tuosta kuvaajasta ei  näe sen kummempaa kommentoitavaa, katsotaan tilannetta kun n on hieman isompi.

Kuvaajiin ilmestyy suoria. Millaisia ovat kyseisien suorien yhtälöt?

Mitä ihmeen lukuja ovat nuo eri väreillä merkityt?

Lähipäivinä lähetän lisää tarinaa tähän liittyen. Jos haluat itse leikkiä kyseisillä luvuilla, niin alla on 10000 ensimmäistä f(n):n arvoa. Jos haluat ne GeoGebran listaksi, niin korvaa hakasulkeet [] aaltosulkeiksi {}. Tämä lista on tuotettu sivun https://oeis.org/A051626 linkistä T. D. Noe, Table of n, a(n) for n = 1..10000 Käytän tätä testatakseni omaa algoritmiani.              

lähteet

Poranen, Haukkanen, Solmu. Jaksolliset desimaaliesitykset algebrallisesta
näkökulmasta https://matematiikkalehtisolmu.fi/2011/Poranen-Haukkanen.pdf

Repeating decimal Wikipediassa https://en.wikipedia.org/wiki/Repeating_decimal    

T. D. Noe, Table of n, a(n) for n = 1..10000  


oikeat = [0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1, 6, 22, 15, 46, 18, 1, 96, 42, 2, 0, 4, 16, 34, 6, 6, 13, 53, 3, 108, 2, 3, 6, 112, 18, 22, 28, 6, 58, 48, 1, 22, 60, 5, 15, 0, 6, 42, 0, 21, 6, 130, 2, 18, 33, 3, 16, 8, 22, 46, 6, 46, 35, 6, 1, 28, 8, 42, 3, 148, 1, 75, 18, 16, 6, 15, 6, 78, 13, 13, 0, 66, 9, 81, 5, 2, 41, 166, 6, 78, 16, 18, 21, 43, 28, 6, 2, 58, 44, 178, 1, 180, 6, 60, 22, 3, 15, 16, 46, 6, 18, 95, 1, 192, 96, 6, 42, 98, 2, 99, 0, 33, 4, 84, 16, 5, 34, 22, 6, 18, 6, 30, 13, 35, 53, 21, 3, 30, 108, 8, 2, 48, 3, 222, 6, 1, 112, 113, 18, 228, 22, 6, 28, 232, 6, 46, 58, 13, 48, 7, 1, 30, 22, 27, 60, 42, 5, 18, 15, 41, 0, 50, 6, 22, 42, 16, 0, 256, 21, 6, 6, 28, 130, 262, 2, 13, 18, 44, 33, 268, 3, 5, 16, 6, 8, 2, 22, 69, 46, 15, 6, 28, 46, 141, 35, 18, 6, 30, 1, 272, 28, 96, 8, 146, 42, 58, 3, 6, 148, 66, 1, 42, 75, 4, 18, 60, 16, 153, 6, 34, 15, 155, 6, 312, 78, 6, 13, 79, 13, 28, 0, 53, 66, 144, 9, 6, 81, 108, 5, 138, 2, 110, 41, 3, 166, 33, 6, 336, 78, 112, 16, 30, 18, 294, 21, 22, 43, 173, 28, 116, 6, 6, 2, 32, 58, 35, 44, 48, 178, 179, 1, 342, 180, 22, 6, 8, 60, 366, 22, 5, 3, 78, 15, 186, 16, 1, 46, 84, 6, 378, 18, 42, 95, 382, 1, 6, 192, 21, 96, 388, 6, 176, 42, 130, 98, 13, 2, 99, 99, 18, 0, 200, 33, 30, 4, 9, 84, 6, 16, 204, 5, 8, 34, 174, 22, 41, 6, 46, 18, 418, 6, 140, 30, 46, 13, 16, 35, 60, 53, 6, 21, 215, 3, 432, 30, 28, 108, 198, 8, 219, 2, 42, 48, 221, 3, 44, 222, 148, 6, 32, 1, 10, 112, 75, 113, 6, 18, 152, 228, 48, 22, 460, 6, 154, 28, 15, 232, 233, 6, 66, 46, 78, 58, 42, 13, 18, 48, 13, 7, 239, 1, 6, 30, 66, 22, 96, 27, 486, 60, 81, 42, 490, 5, 112, 18, 2, 15, 210, 41, 498, 0, 166, 50, 502, 6, 4, 22, 78, 42, 508, 16, 24, 0, 18, 256, 34, 21, 46, 6, 43, 6, 52, 28, 261, 130, 6, 262, 240, 2, 506, 13, 58, 18, 30, 44, 53, 33, 178, 268, 42, 3, 540, 5, 180, 16, 108, 6, 91, 8, 60, 2, 252, 22, 78, 69, 3, 46, 278, 15, 42, 6, 16, 28, 281, 46, 112, 141, 18, 35, 284, 18, 570, 6, 95, 30, 22, 1, 576, 272, 192, 28, 246, 96, 26, 8, 6, 146, 293, 42, 90, 58, 98, 3, 592, 6, 48, 148, 99, 66, 299, 1, 300, 42, 33, 75, 22, 4, 202, 18, 84, 60, 138, 16, 51, 153, 5, 6, 88, 34, 618, 15, 66, 155, 132, 6, 0, 312, 18, 78, 48, 6, 315, 13, 30, 79, 42, 13, 42, 28, 35, 0, 32, 53, 107, 66, 21, 144, 646, 9, 58, 6, 30, 81, 326, 108, 130, 5, 8, 138, 658, 2, 220, 110, 48, 41, 18, 3, 308, 166, 222, 33, 60, 6, 224, 336, 3, 78, 338, 112, 96, 16, 113, 30, 341, 18, 8, 294, 228, 21, 78, 22, 230, 43, 6, 173, 46, 28, 80, 116, 232, 6, 700, 6, 18, 2, 46, 32, 12, 58, 708, 35, 13, 44, 330, 48, 6, 178, 7, 179, 359, 1, 102, 342, 30, 180, 28, 22, 726, 6, 81, 8, 336, 60, 61, 366, 42, 22, 66, 5, 246, 3, 18, 78, 742, 15, 148, 186, 41, 16, 318, 1, 125, 46, 50, 84, 75, 6, 27, 378, 22, 18, 380, 42, 108, 95, 16, 382, 174, 1, 192, 6, 256, 192, 193, 21, 15, 96, 6, 388, 90, 6, 70, 176, 84, 42, 78, 130, 393, 98, 262, 13, 336, 2, 60, 99, 13, 99, 199, 18, 368, 0, 44, 200, 8, 33, 66, 30, 268, 4, 202, 9, 810, 84, 5, 6, 81, 16, 126, 204, 6, 5, 820, 8, 822, 34, 2, 174, 413, 22, 276, 41, 69, 6, 336, 46, 166, 18, 15, 418, 419, 6, 812, 140, 28, 30, 78, 46, 66, 13, 141, 16, 66, 35, 213, 60, 18, 53, 856, 6, 26, 21, 30, 215, 862, 3, 43, 432, 272, 30, 26, 28, 66, 108, 96, 198, 6, 8, 438, 219, 146, 2, 440, 42, 441, 48, 58, 221, 886, 3, 42, 44, 18, 222, 414, 148, 178, 6, 66, 32, 420, 1, 208, 10, 42, 112, 180, 75, 151, 113, 4, 6, 455, 18, 82, 152, 60, 228, 390, 48, 459, 22, 153, 460, 210, 6, 3, 154, 34, 28, 464, 15, 126, 232, 155, 233, 16, 6, 936, 66, 312, 46, 940, 78, 110, 58, 6, 42, 473, 13, 24, 18, 79, 48, 952, 13, 95, 7, 28, 239, 24, 1, 465, 6, 53, 30, 192, 66, 322, 22, 144, 96, 970, 27, 138, 486, 6, 60, 976, 81, 44, 42, 108, 490, 982, 5, 98, 112, 138, 18, 462, 2, 495, 15, 110, 210, 99, 41, 166, 498, 3, 0, 6, 166, 464, 50, 33, 502, 234, 6, 252, 4, 336, 22, 253, 78, 84, 42, 112, 508, 1018, 16, 1020, 24, 30, 0, 5, 18, 78, 256, 294, 34, 103, 21, 1032, 46, 22, 6, 240, 43, 519, 6, 173, 52, 444, 28, 18, 261, 116, 130, 524, 6, 1050, 262, 18, 240, 30, 2, 150, 506, 32, 13, 212, 58, 1062, 18, 35, 30, 96, 44, 1068, 53, 48, 33, 84, 178, 21, 268, 179, 42, 246, 3, 506, 540, 342, 5, 30, 180, 1086, 16, 22, 108, 1090, 6, 273, 91, 8, 8, 1096, 60, 78, 2, 366, 252, 1102, 22, 48, 78, 15, 69, 1108, 3, 4, 46, 78, 278, 222, 15, 558, 42, 186, 6, 522, 16, 561, 28, 1, 281, 462, 46, 564, 112, 84, 141, 34, 18, 113, 35, 378, 284, 528, 18, 162, 570, 42, 6, 228, 95, 15, 30, 382, 22, 575, 1, 1152, 576, 6, 272, 132, 192, 180, 28, 21, 246, 581, 96, 232, 26, 388, 8, 498, 6, 1170, 146, 176, 293, 46, 42, 106, 90, 130, 58, 1180, 98, 78, 3, 13, 592, 593, 6, 140, 48, 99, 148, 1192, 99, 7, 66, 18, 299, 108, 1, 200, 300, 200, 42, 30, 33, 560, 75, 30, 22, 258, 4, 202, 202, 27, 18, 1216, 84, 286, 60, 6, 138, 1222, 16, 42, 51, 204, 153, 1228, 5, 41, 6, 8, 88, 18, 34, 206, 618, 174, 15, 16, 66, 112, 155, 41, 132, 84, 6, 208, 0, 46, 312, 534, 18, 50, 78, 418, 48, 1258, 6, 96, 315, 140, 13, 22, 30, 180, 79, 138, 42, 15, 13, 198, 42, 16, 28, 638, 35, 639, 0, 60, 32, 641, 53, 256, 107, 6, 66, 92, 21, 1290, 144, 215, 646, 6, 9, 1296, 58, 432, 6, 1300, 30, 1302, 81, 28, 326, 653, 108, 48, 130, 198, 5, 12, 8, 262, 138, 219, 658, 659, 2, 55, 220, 42, 110, 13, 48, 1326, 41, 221, 18, 242, 3, 105, 308, 44, 166, 570, 222, 102, 33, 148, 60, 208, 6, 268, 224, 32, 336, 630, 3, 192, 78, 10, 338, 5, 112, 638, 96, 75, 16, 680, 113, 644, 30, 6, 341, 1366, 18, 111, 8, 152, 294, 686, 228, 2, 21, 144, 78, 294, 22, 1380, 230, 460, 43, 69, 6, 72, 173, 154, 46, 318, 28, 198, 80, 15, 116, 42, 232, 699, 6, 233, 700, 660, 6, 28, 18, 66, 2, 32, 46, 656, 32, 78, 12, 141, 58, 108, 708, 42, 35, 84, 13, 158, 44, 18, 330, 713, 48, 1428, 6, 39, 178, 1432, 7, 30, 179, 239, 359, 719, 1, 130, 102, 6, 342, 272, 30, 1446, 180, 66, 28, 290, 22, 726, 726, 96, 6, 690, 81, 162, 8, 486, 336, 18, 60, 146, 61, 81, 366, 336, 42, 735, 22, 490, 66, 58, 5, 30, 246, 112, 3, 740, 18, 247, 78, 6, 742, 1486, 15, 248, 148, 210, 186, 373, 41, 66, 16, 498, 318, 214, 1, 234, 125, 166, 46, 42, 50, 8, 84, 502, 75, 755, 6, 176, 27, 4, 378, 15, 22, 210, 18, 78, 380, 761, 42, 60, 108, 508, 95, 46, 16, 1530, 382, 24, 174, 153, 1, 364, 192, 18, 6, 66, 256, 1542, 192, 34, 193, 48, 21, 1548, 15, 46, 96, 1552, 6, 155, 388, 43, 90, 779, 6, 222, 70, 52, 176, 312, 84, 1566, 42, 261, 78, 1570, 130, 66, 393, 6, 98, 738, 262, 1578, 13, 240, 336, 1582, 2, 79, 60, 506, 99, 678, 13, 21, 99, 174, 199, 28, 18, 133, 368, 30, 0, 200, 44, 228, 200, 53, 8, 1606, 33, 201, 66, 178, 30, 403, 268, 144, 4, 42, 202, 1618, 9, 1620, 810, 540, 84, 6, 5, 271, 6, 180, 81, 696, 16, 770, 126, 108, 204, 409, 6, 148, 5, 91, 820, 195, 8, 138, 822, 60, 34, 96, 2, 42, 174, 252, 413, 110, 22, 552, 276, 78, 41, 150, 69, 1662, 6, 3, 336, 833, 46, 556, 166, 278, 18, 42, 15, 33, 418, 42, 419, 88, 6, 205, 812, 16, 140, 336, 28, 30, 30, 281, 78, 396, 46, 423, 66, 112, 13, 1696, 141, 566, 16, 54, 66, 390, 35, 30, 213, 284, 60, 1708, 18, 812, 53, 570, 856, 294, 6, 16, 26, 95, 21, 430, 30, 287, 215, 22, 862, 78, 3, 18, 43, 576, 432, 866, 272, 173, 30, 192, 26, 138, 28, 1740, 66, 246, 108, 116, 96, 291, 198, 26, 6, 272, 8, 584, 438, 6, 219, 150, 146, 879, 2, 293, 440, 105, 42, 32, 441, 90, 48, 420, 58, 66, 221, 98, 886, 35, 3, 1776, 42, 592, 44, 24, 18, 1782, 222, 48, 414, 893, 148, 1788, 178, 99, 6, 162, 66, 179, 32, 299, 420, 768, 1, 900, 208, 300, 10, 342, 42, 138, 112, 33, 180, 1810, 75, 42, 151, 22, 113, 286, 4, 848, 6, 202, 455, 1822, 18, 8, 82, 84, 152, 870, 60, 305, 228, 138, 390, 366, 48, 166, 459, 51, 22, 786, 153, 288, 460, 5, 210, 1846, 6, 903, 3, 88, 154, 432, 34, 78, 28, 618, 464, 78, 15, 1860, 126, 198, 232, 186, 155, 933, 233, 132, 16, 935, 6, 1872, 936, 1, 66, 938, 312, 313, 46, 18, 940, 804, 78, 84, 110, 48, 58, 118, 6, 60, 42, 315, 473, 378, 13, 30, 24, 30, 18, 380, 79, 86, 48, 42, 952, 953, 13, 902, 95, 42, 7, 1912, 28, 382, 239, 105, 24, 36, 1, 112, 465, 32, 6, 6, 53, 230, 30, 107, 192, 386, 66, 21, 322, 21, 22, 444, 144, 138, 96, 646, 970, 924, 27, 388, 138, 58, 486, 1948, 6, 195, 60, 30, 976, 176, 81, 306, 44, 326, 42, 39, 108, 150, 490, 130, 982, 84, 5, 178, 98, 24, 112, 986, 138, 13, 18, 658, 462, 1978, 2, 282, 495, 220, 15, 99, 110, 331, 210, 48, 99, 180, 41, 664, 166, 18, 498, 998, 3, 999, 0, 308, 6, 1001, 166, 200, 464, 222, 50, 210, 33, 670, 502, 60, 234, 30, 6, 2016, 252, 224, 4, 966, 336, 816, 22, 9, 253, 1013, 78, 2028, 84, 338, 42, 954, 112, 6, 508, 96, 1018, 1019, 16, 78, 1020, 113, 24, 204, 30, 44, 0, 341, 5, 438, 18, 342, 78, 8, 256, 176, 294, 140, 34, 228, 103, 2062, 21, 174, 1032, 78, 46, 2068, 22, 108, 6, 230, 240, 41, 43, 165, 519, 6, 6, 1040, 173, 1041, 52, 46, 444, 298, 28, 1044, 18, 80, 261, 66, 116, 418, 130, 232, 524, 2098, 6, 190, 1050, 700, 262, 140, 18, 42, 240, 18, 30, 1055, 2, 2112, 150, 46, 506, 56, 32, 162, 13, 12, 212, 192, 58, 16, 1062, 708, 18, 532, 35, 710, 30, 39, 96, 60, 44, 2136, 1068, 330, 53, 2140, 48, 2142, 33, 6, 84, 1008, 178, 306, 21, 7, 268, 2152, 179, 215, 42, 359, 246, 336, 3, 30, 506, 102, 540, 432, 342, 98, 5, 30, 30, 498, 180, 65, 1086, 28, 16, 930, 22, 2178, 108, 726, 1090, 174, 6, 198, 273, 243, 91, 198, 8, 312, 8, 336, 1096, 219, 60, 1014, 78, 61, 2, 105, 366, 1101, 252, 42, 1102, 2206, 22, 2162, 48, 66, 78, 553, 15, 221, 69, 246, 1108, 474, 3, 2220, 4, 18, 46, 44, 78, 1040, 278, 742, 222, 1056, 15, 84, 558, 148, 42, 1118, 186, 1119, 6, 123, 522, 1121, 16, 32, 561, 318, 28, 258, 1, 2250, 281, 125, 462, 10, 46, 60, 564, 50, 112, 144, 84, 120, 141, 75, 34, 1133, 18, 2268, 113, 27, 35, 2272, 378, 6, 284, 22, 528, 273, 18, 228, 162, 380, 570, 152, 42, 762, 6, 108, 228, 364, 95, 1146, 15, 48, 30, 2296, 382, 198, 22, 174, 575, 966, 1, 460, 1152, 192, 576, 2308, 6, 231, 272, 256, 132, 154, 192, 330, 180, 193, 28, 30, 21, 44, 246, 15, 581, 534, 96, 16, 232, 6, 26, 583, 388, 233, 8, 90, 498, 2338, 6, 2340, 1170, 70, 146, 66, 176, 1173, 293, 252, 46, 1175, 42, 180, 106, 78, 90, 1178, 130, 336, 58, 393, 1180, 368, 98, 42, 78, 262, 3, 374, 13, 2370, 592, 336, 593, 18, 6, 264, 140, 60, 48, 476, 99, 2382, 148, 13, 1192, 30, 99, 2388, 7, 199, 66, 184, 18, 239, 299, 368, 108, 1199, 1, 2058, 200, 132, 300, 6, 200, 1148, 42, 8, 30, 2410, 33, 126, 560, 66, 75, 2416, 30, 290, 22, 268, 258, 2422, 4, 96, 202, 202, 202, 1038, 27, 48, 18, 810, 1216, 486, 84, 1218, 286, 5, 60, 305, 6, 348, 138, 81, 1222, 2446, 16, 195, 42, 126, 51, 222, 204, 490, 153, 6, 1228, 2458, 5, 1166, 41, 820, 6, 112, 8, 137, 88, 822, 18, 96, 34, 2472, 206, 2, 618, 619, 174, 33, 15, 413, 16, 570, 66, 210, 112, 276, 155, 1170, 41, 598, 132, 69, 84, 498, 6, 226, 208, 336, 0, 60, 46, 278, 312, 166, 534, 1188, 18, 192, 50, 45, 78, 1074, 418, 502, 48, 419, 1258, 228, 6, 630, 96, 812, 315, 4, 140, 342, 13, 28, 22, 46, 30, 592, 180, 78, 79, 1218, 138, 2538, 42, 66, 15, 2542, 13, 508, 198, 141, 42, 2548, 16, 425, 28, 66, 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