American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-349, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access ournal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Analysis of Successive Occurrence of Digit in Natural Numbers Less Than 0 n Neera Anant Pande Associate Professor, Department of Mathematics & Statistics, Yeshwant Mahavidyalaya (College), Nanded 43602, Maharashtra, INDIA Abstract: All positive integers less than 0 n are under consideration. is the first natural number as well as non-zero digit symbol. A detail analysis of successive occurrence of digit in numbers less than 0 n is done in this work. The formula for the number of successive occurrences of s is determined. The first instance of successive s is easily predictable. The last occurrence is also formulated. All the analysis is extended ahead to multiple number of successive occurrences of s. Finally, all results are generalized for successive occurrences of all non-zero digits. Mathematics Subect Classification 200: Y35, Y60, Y99 Keywords: Successive occurrences, digit, natural numbers I. Introduction Positive integers form infinite list, 2, 3, which are studied in branch of Mathematics called Number Theory. They are so basic that their applications are spread in every branch of Mathematics as well as other subects. Beginning nine members, 2, 3, 4, 5, 6, 7, 8, 9 are digits as well. One more digit in place value system is 0. In the present work, the word number is used to mean natural number. We use the new convention that 0 itself is not in the set of Natural numbers N and consider the ranges 0 n, except 0 n, for n N. The numbers under consideration are m, with m < 0 n. The last number 0 n is not taken as it begins next higher range of number of digits (n + ). II. Successive Occurrence of Digit The first natural number is the digit. It s one more property is that it is there in number systems with all bases []. Detail analysis of occurrences of digit, in fact, every digit d with d 9, is given in [2]. Here, the successive occurrence of digit is analyzed in the range of 0 n, except the last number 0 n, for all natural numbers n. We have determined counts of successive occurrence of single and 2 s in numbers ust one less than one quintillion, i.e., 0 8, by using a computer language Java program and they are as given below. Table I : with Single and Double Successive s in their Digits with Single (Successive) Numbers Range Less Than with two Successive s. 0 0 2. 0 2 8 3. 0 3 243 8 4. 0 4 2,96 243 5. 0 5 32,805 2,96 6. 0 6 354,294 32,805 7. 0 7 3,720,087 354,294 8. 0 8 38,263,752 3,720,087 9. 0 9 387,420,489 38,263,752 0. 0 0 3,874,204,890 387,420,489. 0 38,354,628,4 3,874,204,890 2. 0 2 376,572,75,308 38,354,628,4 3. 0 3 3,67,583,974,253 376,572,75,308 AIJRSTEM 6-3; 206, AIJRSTEM All Rights Reserved Page 37
Neera Anant Pande, American International Journal of Research in Science, Technology, Engineering & Mathematics, 6(), September- November, 206, pp. 37-4 Numbers Range Less Than with Single (Successive) with two Successive s 4. 0 4 35,586,2,596,606 3,67,583,974,253 5. 0 5 343,5,886,824,45 35,586,2,596,606 6. 0 6 3,294,258,3,54,384 343,5,886,824,45 7. 0 7 3,50,343,20,48,297 3,294,258,3,54,384 8. 0 8 300,89,270,593,998,260 3,50,343,20,48,297 In the first range m < 0 = 0, single occurs ust once itself as a number. It s count can be seen as -(-) C 9 - =. Single occurrence is qualified to be treated as successive by absence of non-successive character. In the second range m < 0 2 = 00, single comes 8 times. Its 9 occurrences are in numbers, 2, 3, 4, 5, 6, 7, 8, and 9, at unit s places and 9 are in numbers 0, 2, 3, 4, 5, 6, 7, 8, and 9, 2-(-) at ten s places. So, second block has it C 9 2- = 2 9 = 8 times. As stated earlier, it is considered to be successive. In this range, double comes only once in number. Clearly this is of successive type. It s count can be seen as 2-(2-) C 9 2-2 = = time. In the third range, m < 0 3 =,000, single comes 243 times in numbers, 2, 3,, 9, 20, 22,, 29, 30, 32,, 39,, 90, 92,, 99, at unit s places 2, 3, 4,, 9, 20, 22,, 29, 30, 32,, 39,, 90, 92,, 99, at ten s places and 00, 02, 03,, 09, 20, 22, 23,, 30, 32, 33,, 39,, 90, 92, 93,, 99 at hundred s places. This occurrence count in third block is 3-(-) C 9 3- = 3 9 2 = 3 8 = 243, all of them being considered successive. In this range, successive double s occur in 2, 3,, 9, at unit s and ten s places and in 0, 2, 3,, 9 at ten s and hundred s places. Their count is 3-(2-) C 9 3-2 = 2 9 = 8 times. Also successive triple s occur once in. We can explain all numbers in above table this way. There is similar explanation for successive occurrences of multiple s in these ranges. There is clearly a systematic pattern in the resulting figures. We have formulated it by introducing a notation first. S Notation : We introduce the notation O for number of numbers less than 0 n containing r number of n r successive s. Theorem : If r and n are positive integers with r n, then the number of numbers containing exactly r number of successive digit s in the range m < 0 n is S n n ( r ) 9 n r Or C. Proof. Let n and r be positive integers with r n. There are in total 0 digits, viz., 0,, 2, 3, 4, 5, 6, 7, 8, 9, available to occupy n places in all numbers in range m < 0 n. We want successive r places to be occupied by digit. The various choices for these successive r places for digit will be n-(r-) C in number. Now for each such choice, remaining n r places are to be occupied by any of the remaining 9 digits except and there are 9 n r choices for each of that. This totals to n-(r-) C 9 n r ( ) and hence S n n r 9 n r Or C. This completes the proof of the theorem. The table given above can be extended to higher occurrences of successive s by using this formula. Table II : with Multiple Successive s in their Digits with 3 with 4 Successive s Successive s with 5 Successive s. 0 3 0 0 2. 0 4 8 0 3. 0 5 243 8 4. 0 6 2,96 243 8 5. 0 7 32,805 2,96 243 6. 0 8 354,294 32,805 2,96 AIJRSTEM 6-3; 206, AIJRSTEM All Rights Reserved Page 38
Neera Anant Pande, American International Journal of Research in Science, Technology, Engineering & Mathematics, 6(), September- November, 206, pp. 37-4 with 3 Successive s with 4 Successive s with 5 Successive s 7. 0 9 3,720,087 354,294 32,805 8. 0 0 38,263,752 3,720,087 354,294 9. 0 387,420,489 38,263,752 3,720,087 0. 0 2 3,874,204,890 387,420,489 38,263,752. 0 3 38,354,628,4 3,874,204,890 387,420,489 2. 0 4 376,572,75,308 38,354,628,4 3,874,204,890 3. 0 5 3,67,583,974,253 376,572,75,308 38,354,628,4 4. 0 6 35,586,2,596,606 3,67,583,974,253 376,572,75,308 5. 0 7 343,5,886,824,45 35,586,2,596,606 3,67,583,974,253 6. 0 8 3,294,258,3,54,384 343,5,886,824,45 35,586,2,596,606 with 6 Successive s with 7 Successive s with 8 Successive s with 9 Successive s. 0 6 0 0 0 2. 0 7 8 0 0 3. 0 8 243 8 0 4. 0 9 2,96 243 8 5. 0 0 32,805 2,96 243 8 6. 0 354,294 32,805 2,96 243 7. 0 2 3,720,087 354,294 32,805 2,96 8. 0 3 38,263,752 3,720,087 354,294 32,805 9. 0 4 387,420,489 38,263,752 3,720,087 354,294 0. 0 5 3,874,204,890 387,420,489 38,263,752 3,720,087. 0 6 38,354,628,4 3,874,204,890 387,420,489 38,263,752 2. 0 7 376,572,75,308 38,354,628,4 3,874,204,890 387,420,489 3. 0 8 3,67,583,974,253 376,572,75,308 38,354,628,4 3,874,204,890 with 0 Successive s 4 Successive s with Successive s 5 Successive s with 2 Successive s 6 Successive s 7 Successive s with 3 Successive s. 0 9 0 0 0 0 2. 0 0 0 0 0 3. 0 8 0 0 4. 0 2 243 8 0 5. 0 3 2,96 243 8 6. 0 4 32,805 2,96 243 8 7. 0 5 354,294 32,805 2,96 243 8. 0 6 3,720,087 354,294 32,805 2,96 9. 0 7 38,263,752 3,720,087 354,294 32,805 0. 0 8 387,420,489 38,263,752 3,720,087 354,294 8 Successive s. 0 4 0 0 0 0 2. 0 5 8 0 0 0 3. 0 6 243 8 0 0 4. 0 7 2,96 243 8 0 5. 0 8 32,805 2,96 243 8 The nature of the formula giving these numbers is such that the columns are gradually shifted down. III. First Successive Occurrence of Digit The first number containing is obviously. Containing 2 successive s, the first number is, for 3 it is and so on. We have formulated numbers with first successive s. AIJRSTEM 6-3; 206, AIJRSTEM All Rights Reserved Page 39
Neera Anant Pande, American International Journal of Research in Science, Technology, Engineering & Mathematics, 6(), September- November, 206, pp. 37-4 Formula : If n and r are natural numbers, then the first occurrence of r number of successive s in numbers in range m < 0 n is, if r n r f 0, if r n 0 IV. Last Successive Occurrence of Digit The last successive occurrences of in our ranges are also determined. Table III : The Last Occurrences of Multiple Successive s in the Digits of Numbers Number Range < Last Number with Successive 0 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9. 9 99 9,99 99,99 999,99 9,999,99 99,999,99 999,999,99 2. 2 s - 9 9,9 99,9 999,9 9,999,9 99,999,9 999,999,9 3. 3 s - - 9, 99, 999, 9,999, 99,999, 999,999, 4. 4 s - - -, 9, 99, 9,99, 99,99, 999,99, 5. 5 s - - - -, 9, 9,9, 99,9, 999,9, 6. 6 s - - - - -, 9,, 99,, 999,, 7. 7 s - - - - - -,, 9,, 99,, 8. 8 s - - - - - - -,, 9,, 9. 9 s - - - - - - - -,, We have formulated them. Formula 2 : If n and r are natural numbers, then the last occurrence of r successive s in numbers in range m < 0 n is, if r n 0, if r n l r n 0 0 90,if r n r Some integer sequences occurring here are analogous to those in [2]. V. Extension to Other Non-zero Digits Finally, we conclude by mentioning an important thing that whatever discussion has been done for occurrences of successive digit s is parallely applicable for other non-zero digits 2 through 9. Denoting the non-zero digit of interest by d, where d 9, again the range under consideration is m < 0 n and r n. Notation : We generalize the notation S n do r for number of numbers less than 0 n with r number of successive digits d s. Theorem 2 : If r, n and d are positive integers with r n and d 9, then the number of numbers containing exactly r number of successive digit d s in the range m < 0 n is S n n( r ) 9 nr dor C Proof. The presence of each digit d, with d n, is same for all numbers in the total range m < 0 n, and hence the proof is same as that for Theorem. Formula 3 : If r, n and d are positive integers with d 9, then the first occurrence of r number of successive d s in numbers in range m < 0 n is in number, if r n r f d 0, if r n 0 AIJRSTEM 6-3; 206, AIJRSTEM All Rights Reserved Page 40
Neera Anant Pande, American International Journal of Research in Science, Technology, Engineering & Mathematics, 6(), September- November, 206, pp. 37-4 Formula 4 : If r, n and d are positive integers with d 9, then the last occurrence of r number of successive d s in numbers in range m < 0 n is in number, if r n 0, if r n l r n d 0 0 90,if r n r It is noteworthy that the all Formulae, 2, 3, and 4 provided here are exactly same as formulae with corresponding numbers in [2] as the first and last occurrences of any non-zero digit happen to be successive ones. References [] Neera Anant Pande, Numeral Systems of Great Ancient Human Civilizations, Journal of Science and Arts, Year 0, 2 (3) 200, pp 209-222. [2] Neera Anant Pande, Analysis of Occurrence of Digit in Natural Numbers Less Than 0 n, Advances in Theoretical and Applied Mathematics, Volume, Number 2, 206, pp. 99-04. [3] Nishit K Sinha, Demystifying Number System, Pearson Education, New Delhi, 200. Acknowledgments The author is grateful to the Java Programming Language Development Team and the NetBeans IDE Development Team, as their free software have been used while actually performing the calculation on huge range of numbers during this work. Thanks are also extended to the Development Team of Microsoft Office Excel, which was used to re-check the validity of the formulae derived here. The extensive use of the Computer Lab of Mathematics & Statistics Department of the affiliating college for several continuous months has a lot of credit in materializing the analysis aimed at. The power support extended generously by the Department of Electronics of the institute has helped execute the computer programs without interruption and is acknowledged. The author expresses thankfulness to the University Grants Commission (U.G.C.), New Delhi of the Government of India for funding a related research work about special natural numbers under a Research Proect (F. 47-748/3(WRO)). Last but not the least; the author also extends the thanks to the anonymous referee/referees of this paper. AIJRSTEM 6-3; 206, AIJRSTEM All Rights Reserved Page 4