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arXiv:1302.1479v5 [math.HO] 8 Jan 2014 Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9

Inder J. Taneja

1

Abstract.Natural numbers from 0 to 11111 are written in terms of 1 to 9 intwo different ways. The first one in increasing order of 1

to 9, and the second one in decreasing order. This is done by using the operations ofaddition, multiplication, subtraction, potentiation,

anddivision. In both the situations there are no missing numbers, exceptone, i.e., 10958 in the increasing case.

1.Introduction

Author [1, 2, 3] wrote the numbers from 44 to 11111 in terms of 1to 9 in two different ways, one is in increasing order and

another in decreasing order. Some comments on this work can be seen in [5, 6, 7, 8]. The operations used are onlyaddition,

multiplication, andpotentiation. The idea of brackets is also used, i.e., the following operations were used:

From the mathematical point of view, the brackets are understood ascomposite rule. The operations such assubtraction

anddivisionare also very important. In this work, the operations ofsubtractionanddivisionare also included. This is done

to find missing numbers not available in the previous versions. This work is done by using the following operations:

In the previous work [3], there were approximately 1250 numbers were missing in both the cases. Here, we have found almost

all the missing numbers from 0 to 11111, except one, i.e., 10958 in the increasing case. These missing numbers having either

subtractionand/ordivision, and are written initalicforms to identify. Still, there are more operations that canbe applied,

such as: By applying these operations, may be one can find the number 10958. This shall be dealt elsewhere.

The mathematical idea behind this work is based on simple combinations. If we have two different positive natural numbers

in a sequence, for example?and?, then we can write, ?+?? ? × ?? ??and??

We have only four ways of writing two numbers, for example if we have?= 2 and?= 3, then one can write 2+3, 2×3,

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3and 23 in the increasing order, and 3 + 2, 3×2, 32and 32 in decreasing order.

Again, let us consider three positive natural numbers,?,?and?with either? < ? < ?or? > ? > ?. Following the same

procedure for two numbers, here below are 23 possibilities of writing these three numbers: ??? ????(??)?? ??× ?? ? × ???(? × ?)??(??)?? ??+?? ?+??? ??+?? ??×??and (?+?)??

The expressions (??)?and??×?are the same. The expressions???and???give very big values except?= 1.

Imagine if these letters increases from 3 to 4, 5, ... to 9, onemay have millions of possibilities of writing these 9 letters

either in increasing or in decreasing orders. The above explanation is only for addition, multiplication and brackets.If we

allow more operations, such as subtraction, division, etc., these possibilities increases much more.

From first version to this, there is a gap of approximately oneyear. During this time, I came across, two historical books,

[9, 10], where these authors specified only the representation of number 100 in different ways including much more opeartions,

such as, factorial, decimal, square root etc.

1Formerly, Professor of Mathematics, Universidade Federalde Santa Catarina, 88.040-900 Florian´opolis, SC, Brazil.e-mail: ijtaneja@gmail.com.

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2CRAZY SEQUENTIAL REPRESENTATION - INDER J. TANEJA

2.Crazy Sequential Representation

This is the fifth version of previous works. Readers can see previous versions at [1, 2, 3, 4]. Here below arecrazy sequential

representationof natural numbers written in terms of 1 to 9 in increasing as well as decreasing orders. The first column

represents the increasing order and the second represent the decreasing. Numbers withsubtractionand/ordivisionare written

inItalicform.

Increasing order

0=12+34-56-7+8+9?

1 = 1

23456789?

2=123+4-56-78+9?

3=123-45-6-78+9?

4=12-34-56-7+89?

5=12-34+5-67+89?

6=12+34+56-7-89?

7=1+23-4+56-78+9?

8=1-23-45+6+78-9?

9 = 1

2345678×9?

10 = 1

2345678+ 9?

11=1+23+4+5+67-89?

12=123+45-67-89?

13=1-23+4-56+78+9?

14=12-3-45+67-8-9?

15=123-45+6-78+9?

16=1-2+34+5+67-89?

17 = 1

234567×8 + 9?

18 = 1

234567+ 8 + 9?

19=12+34-5+67-89?

20=12+3-45+67-8-9?

21=1-23-45+6-7+89?

22=1-23+4-56+7+89?

23=1+2-3+45+67-89?

24 = 1

23456×7 + 8 + 9?

25 = 1

23456+ 7 + 8 + 9?

26=12-3+4-56+78-9?

27=12-3-45-6+78-9?

28=12+3-4-5-67+89?

29=12+34+5+67-89?

30 = 1

2345×6 + 7 + 8 + 9?

31 = 1

2345+ 6 + 7 + 8 + 9?

32=12-3+45+67-89?

33=12+34+56-78+9?

34=123+4-5-6+7-89?

35 = 1

234×5 + 6 + 7 + 8 + 9?

36 = 1

234+ 5 + 6 + 7 + 8 + 9?

37=1+23-4-5-67+89?

38=12+3+45+67-89?

39 = 1

23×4 + 5 + 6 + 7 + 8 + 9?

40 = 1

23+ 4 + 5 + 6 + 7 + 8 + 9?

41=12-34-5+67-8+9?

42 = 1

2×3 + 4 + 5 + 6 + 7 + 8 + 9?

43 = 1

2+ 3 + 4 + 5 + 6 + 7 + 8 + 9?

44 = 1×2 + 3 + 4 + 5 + 6 + 7 + 8 + 9?

45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9?

46 = 1 + 2×3 + 4 + 5 + 6 + 7 + 8 + 9?

47 = 1×23+ 4 + 5 + 6 + 7 + 8 + 9?

48 = 1 + 2

3+ 4 + 5 + 6 + 7 + 8 + 9?

49 = 1×2 + 3×4 + 5 + 6 + 7 + 8 + 9?

50 = 1 + 2 + 3×4 + 5 + 6 + 7 + 8 + 9?

51 = 1

23+ 4×5 + 6 + 7 + 8 + 9?

52=12-3-45+6-7+89?

53 = 1

2×3 + 4×5 + 6 + 7 + 8 + 9?

54 = 12 + 3 + 4 + 5 + 6 + 7 + 8 + 9?

55 = 1×2 + 3 + 4×5 + 6 + 7 + 8 + 9?

56 = 1 + 2 + 3 + 4×5 + 6 + 7 + 8 + 9?

57 = 1 + 2×3 + 4×5 + 6 + 7 + 8 + 9?

58 = 1×23+ 4×5 + 6 + 7 + 8 + 9?

59 = 1×2×3×4 + 5 + 6 + 7 + 8 + 9?

60 = 1 + 2×3×4 + 5 + 6 + 7 + 8 + 9?Decreasing order0=98-7-6-54-32+1?

1=98-76-54+32+1?

2=9+87-65+4-32-1?

3=98-76-5+4+3-21?

4=98-7-65-43+21?

5=98-76+5-43+21?

6=98-7-65+4-3-21?

7=98-7-6-54-3-21?

8=9-8+76-5-43-21?

9=9+87-65-43+21?

10=98-7+6-54-32-1?

11=9+8-7+65-43-21?

12=987-654-321?

13=98-7-6-54+3-21?

14=98+7-6-54-32+1?

15=98-76-5-4+3-2+1?

16=98-7-6-5-43-21?

17=9+87-65+4+3-21?

18=98+7-65-43+21?

19=98-7+6-54-3-21?

20=98+7-65+4-3-21?

21=9+87-6-5-43-21?

22=9-87+65+4+32-1?

23=9+87-65-4-3-2+1?

24=98+7+6-54-32-1?

25=9+8+7+65-43-21?

26=98-7-6+5-43-21?

27=9-87+65+43-2-1?

28=98-7+6-5-43-21?

29=9-87+65+43-2+1?

30=98+7-6-5-43-21?

31=98-76-5-4-3+21?

32=98-7-65+4+3-2+1?

33=98+7+6-54-3-21?

34=9+8+76+5-43-21?

35=98-7-6-54+3+2-1?

36=98-7-6-5-43-2+1?

37=98-76-5-4+3+21?

38=98-7-6-5-43+2-1?

39=98-76-5+43-21?

40=98-7-65-4-3+21?

41=98-76+5-4-3+21?

42=98+7+6-5-43-21?

43=98-76+54-32-1?

44 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2×1?

45 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1?

46 = 9 + 8 + 7 + 6 + 5 + 4 + 3×2 + 1?

47=98-76+5-4+3+21?

48 = 9 + 8 + 7 + 6 + 5 + 4 + 3

2×1?

49 = 9 + 8 + 7 + 6 + 5 + 4×3 + 2×1?

50 = 9 + 8 + 7 + 6 + 5 + 4×3 + 2 + 1?

51=9+87-65-4+3+21?

52=98-76+54-3-21?

53=9+87-65+43-21?

54 = 9 + 8 + 7 + 6 + (5 + 4 + 3)×2×1?

55 = 9 + 8 + 7 + 6 + 5×4 + 3 + 2×1?

56 = 9 + 8 + 7 + 6 + 5×4 + 3 + 2 + 1?

57 = 9 + 8 + 7 + 6 + 5×4 + 3×2 + 1?

58=98-7-6-5-43+21?

59 = 9 + 8 + 7 + 6 + 5 + 4×3×2×1?

60 = 9 + 8 + 7 + 6 + 5×4 + 32+ 1?

CRAZY SEQUENTIAL REPRESENTATION - INDER J. TANEJA3

Increasing order

61 = 1

2×3 + 4 + 5×6 + 7 + 8 + 9?

62 = 1×23 + 4 + 5 + 6 + 7 + 8 + 9?

63 = 1 + 23 + 4 + 5 + 6 + 7 + 8 + 9?

64 = 1 + 2 + 3 + 4 + 5×6 + 7 + 8 + 9?

65 = 12 + 3 + 4×5 + 6 + 7 + 8 + 9?

66 = 1×23+ 4 + 5×6 + 7 + 8 + 9?

67 = 1 + 2

3+ 4 + 5×6 + 7 + 8 + 9?

68 = 1×2 + 3×4 + 5×6 + 7 + 8 + 9?

69 = 1 + 2 + 3×4 + 5×6 + 7 + 8 + 9?

70 = 1

2+ 34 + 5 + 6 + 7 + 8 + 9?

71 = 1×2 + 34 + 5 + 6 + 7 + 8 + 9?

72 = 1 + 2 + 34 + 5 + 6 + 7 + 8 + 9?

73 = 12 + 3 + 4 + 5×6 + 7 + 8 + 9?

74 = 1 + 2 + 3 + 4 + 5 + 6×7 + 8 + 9?

75 = 12×3 + 4 + 5 + 6 + 7 + 8 + 9?

76 = 1×23+ 4 + 5 + 6×7 + 8 + 9?

77 = 1

2+ 3×4 + 5 + 6×7 + 8 + 9?

78 = 12 + 3×4 + 5×6 + 7 + 8 + 9?

79 = 1 + 2 + 3×4 + 5 + 6×7 + 8 + 9?

80 = 1×2 + 3 + 45 + 6 + 7 + 8 + 9?

81 = 1 + 2 + 3 + 45 + 6 + 7 + 8 + 9?

82 = 1 + 2×3 + 45 + 6 + 7 + 8 + 9?

83 = 12 + 3 + 4 + 5 + 6×7 + 8 + 9?

84 = 1×2 + 3 + 4×5 + 6×7 + 8 + 9?

85 = 1 + 2 + 3 + 4×5 + 6×7 + 8 + 9?

86 = 1 + 2 + 3 + 4 + 5 + 6 + 7×8 + 9?

87 = 1 + 2×3 + 4 + 5 + 6 + 7×8 + 9?

88 = 12 + 3×4 + 5 + 6×7 + 8 + 9?

89 = 1×2 + 3 + 4 + 56 + 7 + 8 + 9?

90 = 12 + 3 + 45 + 6 + 7 + 8 + 9?

91 = 1 + 2 + 34 + 5×6 + 7 + 8 + 9?

92 = 1 + 23 + 4 + 5 + 6×7 + 8 + 9?

93 = 1 + 2 + 3×4×5 + 6 + 7 + 8 + 9?

94 = 1×2 + 3×4 + 56 + 7 + 8 + 9?

95 = 12 + 3 + 4 + 5 + 6 + 7×8 + 9?

96 = 1×2 + 3 + 4×5 + 6 + 7×8 + 9?

97 = 1 + 2 + 3 + 4×5 + 6 + 7×8 + 9?

98 = 1×23 + 45 + 6 + 7 + 8 + 9?

99 = 1 + 2 + 3 + 4 + 5 + 67 + 8 + 9?

100 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8×9?

101 = 1 + 2 + 34 + 5 + 6×7 + 8 + 9?

102 = 12 + 3×4×5 + 6 + 7 + 8 + 9?

103 = 1×2×34 + 5 + 6 + 7 + 8 + 9?

104 = 1 + 23 + 4 + 5 + 6 + 7×8 + 9?

105 = 1 + 2×3×4 + 56 + 7 + 8 + 9?

106 = 12 + 3 + 4×5 + 6 + 7×8 + 9?

107 = 1×23 + 4 + 56 + 7 + 8 + 9?

108 = 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9?

109 = 1 + 2×3 + 4 + 5 + 6 + 78 + 9?

110 = 12 + 34 + 5 + 6×7 + 8 + 9?

111 = 12×3 + 45 + 6 + 7 + 8 + 9?

112 = 1×2 + 3×4 + 5 + 6 + 78 + 9?

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