Analysis of Occurrence of Digit 0 in Natural Numbers Less Than 10 n

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1 meican Intenational Jounal of Reseach in Fomal, pplied & Natual Sciences vailable online at ISSN (Pint): , ISSN (Online): , ISSN (CD-ROM): IJRFNS is a efeeed, indexed, pee-eviewed, multidisciplinay and open access jounal published by Intenational ssociation of Scientific Innovation and Reseach (ISIR), US (n ssociation Unifying the Sciences, Engineeing, and pplied Reseach) nalysis of Occuence of Digit in Natual Numbes Less Than 1 n Neeaj nant Pande ssociate Pofesso Depatment of Mathematics & Statistics, Yeshwant Mahavidyalaya (College), Nanded 43162, Mahaashta, INDI bstact: Fo any positive intege n, all positive integes less than 1 n ae analyzed fo occuences of digit, which is the vey fist digit symbol. Multiple occuences of s ae also consideed. The fomula fo the numbe of occuences of s is developed. The fist and last instances of multiple occuence of ae detemined and ae fomulated. Mathematics Subject Classification 21 : 11Y35, 11Y6, 11Y99. Keywods: ll occuences, digit, natual numbes I. Intoduction Natual numbes ae infinite sequence of positive integes 1, 2, 3, They ae extensively studied in banch of Mathematics called Numbe Theoy. They ae so ich in thei popeties that thei applications ae widespead in almost all banches of Mathematics. In fact, they occu even outside Mathematics the most. Evey discipline of study involves counting and wheneve thee is counting, it is these numbes that ae needed and ae amongst the concepts of Mathematics most widely used anywhee. In ou outine couse, we use decimal system, i.e., numbe system with base 1, which has 1 digits, viz.,, 1, 2, 3, 4, 5, 6, 7, 8, 9. In this wok, the tem numbe, used thoughout, means the natual numbe. dopting the standad that itself is not included in the set of Natual numbes N, we conside anges 1 1 n, excluding 1 n, fo n N. So fo any n N, in the ange 1 1 n, the numbes unde consideation ae m, with 1 m < 1 n. The last numbe 1 n is omitted as it falls in the next categoy of numbes having moe than n numbe of significant digits. The numbes in the ange 1 m < 1 n contain n o fewe decimal digits in thei decimal expansion. nd of couse, we have consideed only significant s; which means that s afte some non-zeo digit as any numbe of pe-fixed s don t have any additional meaning and sense to convey. II. Occuence of Digit s unpaalleled impotance is well-known in any numbe system having place value chaacte [1]. ll consecutive numbes m in the ange 1 m < 1 n have been citically analyzed fo occuence of digit 1, in fact, evey non-zeo digit d, in [2] and fo successive digits in [3]. That wok didn t cove which we do in pesent one. Consideing highe anges, we have detemined this count of occuence of single and also double s by using a compute Java pogam in numbes one less than one quintillion, i.e., 1 18, and ae as given below. Table I : Numbe of Numbes with Single and Double s in thei Digits Numbes Range Less Numbe of Numbes with Single Numbe of Numbes with Two s Than , ,62 4, ,847 7, ,512, , ,993,276 12,116, ,367,44 146,39, ,868,151,445 1,695,721,32 IJRFNS 16-34; 216, IJRFNS ll Rights Reseved Page 6

2 Numbes Range Less Than Numbe of Numbes with Single Numbe of Numbes with Two s ,735,995,455 19,129,643, ,927,651,154 21,92,785, ,773,82,88,926 2,282,52,719, ,817,337,857,23 24,311,556,565, ,92,432,226, ,621,346,943, ,445,459,413,646,392 2,657,684,554,713, ,93,782,435,275,848 27,364,62,46,71, ,65,871,329,67, ,375,366,89,922,192 In the fist ange 1 m < 1 1 = 1, just doesn t occu even once. This ange contains only 9 numbes which ae all othe nine digits except. In the second ange 1 m < 1 2 = 1, single occus 9. Its instances ae in numbes 1, 2, 3, 4, 5, 6, 7, 8, and 9, all at unit s places. This fist occuence in initial block is 2-1 C = 9. In the thid ange, 1 m < 1 3 = 1,, single occus 171. It occus in ealie 9 numbes 1, 2, 3,, 9, 11, 12, 13,, 19, 21, 22, 23,, 29,, 91, 92, 93,, 99, at unit s places and in 11, 12, 13,, 19, 21, 22, 23,, 29,, 91, 92, 93,, 99, at ten s places which is 3-1 C = = 162, totaling to = 171. In this ange, double zeos occu in 1, 2, 3,, 9, the count of which is 3-1 C = 9. In the fouth ange, 1 m < 1 4 = 1,, single occus in ealie 171 numbes 1, 2, 3,, 9, 11, 12, 13,, 19, 21, 22, 23,, 29,, 91, 92, 93,, , 12, 13,, 19, 21, 22, 23,, 29,, 91, 92, 93,, 99, 111, 112,, 119, 121, 122,, 129,, 191, 192,, 199, 211, 212,, 219, 221, 222,, 229,, 291, 292,, 299, 911, 912,, 919, 921, 922,, 929,, 991, 992,, 999, at unit s places, in 111, 112,, 119, 121, 122,, 129,, 191, 192,, 199, 211, 212,, 219, 221, 222,, 229,, 291, 292,, 299, 911, 912,, 919, 921, 922,, 929,, 991, 992,, 999, at ten s places and in 111, 112,, 119, 121, 122,, 129,, 191, 192,, 199, 211, 212,, 219, 221, 222,, 229,, 291, 292,, 299, 911, 912,, 919, 921, 922,, 929,, 991, 992,, 999, at hunded s places giving count 4-1 C = = 2,187, totaling to ,187 = 2,358. Now in this ange double occus in ealie 9 numbes 1, 2, 3,, 9, 11, 12,, 19, 21, 22,, 29,, 91, 92,, 99, 11, 12,, 19, 21, 22,, 29,, 91, 92,, 99, 11, 12,, 19, 21, 22,, 29,, 91, 92,, 99. They give count 4-1 C = = 243, totaling to = 252. ll numbes in above table get justified this way. Occuences of multiple s in these anges can be similaly explained. In an attempt to fomulate the appaent patten in the esulting figues, we have deived following theoem. n Notation : Fist we intoduce the notation O fo numbe of numbes less than 1 n with numbe of significant s. IJRFNS 16-34; 216, IJRFNS ll Rights Reseved Page 7

3 Theoem 1 : If and n ae positive integes with < n, then the numbe of numbes containing exactly numbe of significant digit s in the ange 1 m < 1 n is n n j j O C 9 j Poof. Let n and be positive integes with < n. We pove the esult by induction by n and. The cases of initial values of ae aleady discussed, wheein the fomula is seen to be valid. Fo any positive, the minimum value of n is + 1. In this case, thee is only one possible position fo all s, viz., last succession, and then thee ae exactly 9 occuences of s :, 2,, 9 lso 1 n j j 1 O C 9 C 9 C 9 9 j So, the fomula is tue in this case. The block occuences begin with 9. The next value of n is + 2. Now except the leading position, thee ae n 1, in this case + 1, possible positions that can be occupied by numbe of s. These combinations ae +1 C. We actually mention these in this case:, 2,, 9 1,,19, 2,, 29,, 9,, 99, (= 9 9) (= 9) 1,,19, 2,, 29,, 9,, 99, (= 9 9) 1,, 9, 2 1,, 2 9,, 9 1,, 9 9, (= 9 9) In addition to ealie 9 occuences of s in the fist ow, now thee ae 9 2 additional occuences each in +1 C ows of new possible occuences and so total occuences of s is given by Pevious + +1 n- and hence equals 9 n 2 2 j i j n 2 O asseting that the fomula is tue in this case. Fo the next value of n, i.e., + 3, the same scenaio will occu. With new availability figue of + 2 (except the leading) positions fo s, thee will be addition of +2 3 and hence equals O j j n 3 3 j j O j j 2 n Continuing this indefinitely, the fomula is poved fo all positive integes and n >. The table given above is now extended to highe occuences of s using this fomula. Numbe Table II : Numbe of Numbes with Multiple s in thei Digits Numbe of Numbes with 3 s Numbe of Numbes with 4 s Numbe of Numbes with 5 s , ,843 11, ,25,558 24,984 15, ,966,254 4,374, ,22 IJRFNS 16-34; 216, IJRFNS ll Rights Reseved Page 8

4 Numbe Numbe of Numbes with 3 s Numbe of Numbes with 4 s Numbe of Numbes with 5 s ,735,65 71,335,98 7,823, ,599,342,17 1,75,759,47 141,746, ,523,722,855 15,281,177,4 2,351,478, ,616,291,75 27,54,319,455 36,444,481, ,811,599,339,249 2,7,15,166,17 535,54,65, ,615,95,618,333 34,112,545,834,779 7,515,597,21, ,269,164,92,58,28 419,628,863,131,344 11,752,919,27, ,8,168,677,286,188 5,45,824,67,69,124 1,335,45,134,376, ,86,138,51,647,58 59,492,59,713,497,34 1,76,447,88,76,68 Numbe Numbe of Numbes with 6 s Numbe of Numbes with 7 s Numbe of Numbes with 8 s , ,112 26, ,972,42 814,221 33, ,498,144 2,3,391 1,116, ,677,961,5 441,21,663 3,345, ,546,134,736 8,648,776, ,39, ,241,969,863,23 156,385,122,939 15,77,565, ,693,325,79,28 2,649,435,969, ,83,215, ,992,851,139,8 42,538,249,517,94 5,278,184,98, ,765,34,794,627, ,837,96,792,926 9,41,913,697,212 Numbe Numbe of Numbes with 9 s Numbe of Numbes with 1 s Numbe of Numbes with 11 s Numbe of Numbes with 12 s , ,484,334 49, ,74,369 1,925,46 57,843 1, ,17,649,251 61,33,59 2,446,47 67, ,46,49,96 1,656,95,832 83,47,932 3,52, ,5,897,336 39,958,966,584 2,44,382,22 11,521, ,935,692,984, ,131,596,592 61,598,46,564 3,399,78,744 Numbe Numbe of Numbes with 13 s Numbe of Numbes with 14 s Numbe of Numbes with 15 s Numbe of Numbes with 16 s Numbe of Numbes with 17 s Numbe of Numbes with 18 s , ,688 1, ,751,848 88,74 1, ,288,468 4,55,184 1,449 1,386 9 III. Fist Occuence of Digit The fist numbe containing is natually 1. In fact, stats becoming significant fom 1 & onwads only. The status of 1 as fist containe of significant single is univesal fo all highe anges of numbes. Fo sufficiently lage anges, fo 2 s, the fist instance is 1, fo 3 it is 1 and so on. It is fomulated simply. Fomula 1 : If n and ae natual numbes, then the fist occuence of zeos in numbes in ange 1 m < 1 n is, if n f =., if n IV. Last Occuence of Digit It is moe inteesting to detemine the last occuences of. The last numbe in anges shows a specific patten. IJRFNS 16-34; 216, IJRFNS ll Rights Reseved Page 9

5 Numbe Range < Last Numbe with Table III : The Last Occuences of Multiple s in the Digits of Numbes ,99 99,99 999,99 9,999,99 99,999,99 999,999, s ,9 99,9 999,9 9,999,9 99,999,9 999,999, s , 99, 999, 9,999, 99,999, 999,999, 4. 4 s , 99, 9,99, 99,99, 999,99, 5. 5 s , 9,9, 99,9, 999,9, 6. 6 s ,, 99,, 999,, 7. 7 s ,, 99,, 8. 8 s ,, These have also been fomulated. Fomula 2 : If n and ae natual numbes, then the last occuence of zeos in numbes in ange 1 m < 1 n is, if n l = n., if n Inteesting intege sequences esulting hee deseve independent teatment owing to thei own popeties. Refeences [1] Neeaj nant Pande, Numeal Systems of Geat ncient Human Civilizations, Jounal of Science and ts, Yea 1, 2 (13) 21, pp [2] Neeaj nant Pande, nalysis of Occuence of Digit 1 in Natual Numbes Less Than 1 n, dvances in Theoetical and pplied Mathematics, Volume 11, Numbe 2, 216, pp [3] Neeaj nant Pande, nalysis of Successive Occuence of Digit 1 in Natual Numbes Less Than 1 n, meican Intenational Jounal of Reseach in Science, Technology, Engineeing and Mathematics,16(1), 216, pp [4] Nishit K. Sinha, Demystifying Numbe System, Peason Education, New Delhi, 21. cknowledgments The autho extends his thanks to the anonymous efeees of this pape. IJRFNS 16-34; 216, IJRFNS ll Rights Reseved Page 1