Single Digit Representations of Natural Numbers

[Pages:55]Single Digit Representations of Natural Numbers

Inder J. Taneja1

A

. In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same

number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2,

3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation, division.

arXiv:1502.03501v1 [math.HO] 12 Feb 2015

1. I Let be a single digit positive natural numbers, i.e, {1 2 3 4 5 6 7 8 9} We can always write

0= - ;

1= ; 2= + ; 3= + + ; 4= + + + ; 5= + + + + 6= + + + + + ; ...

We observe that as number increases, one need more digits to write. But it is not true, for example to write 10, we can write as 10 = 11 - 1 = 2 ? 2 ? 2 + 2. Here we need only 3 digits for 1 and four digits for 2.

Author [7] studied representations of natural numbers using the digits from 1 to 9 in increasing and decreasing orders. For comments see [1, 2, 5, 6]. Historical study of numbers and their properties can be found in [3, 4]. Study of numbers in little different way calling "selfie numbers" is given by author [8].

The aim of this work is to write natural numbers from 0 to 1000 in terms of each digits 1, 2, 3, 4, 5, 6, 7, 8 and 9, with "as less as possible digits", using only the basic operations:

[addition, subtraction, multiplication, division, potentiation].

Before proceeding further, here below ares some numbers that can be represented in a symmetric way using any of 9 digits from 1 to 9.

Let us consider a series:

2. S

R

then we can write

(10) = 10 + 10 -1 + + 102 + 10 + 100

where {1 2 3 4 5 6 7 8 9}.

(10) =

( +1)-

1Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florian?opolis, SC, Brazil. e-mail: ijtaneja@.

1

2

SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

In particular for = 3, we have

Then, And

3(10) = 103 + 102 + 10 + =

5=

1(10) - +

=

- +

55 =

2(10) - +

=

- +

6=

1(10) + +

=

+ +

56 = 2(10) - =

+

+

+

Equivalently, one can write the following symmetric representations

5

=

11 - 1 1+1

=

22 - 2 2+2

=

33 - 3 3+3

=

44 - 4 4+4

=

55 - 5 5+5

=

66 - 6 6+6

=

77 - 7 7+7

=

88 - 8 8+8

=

99 - 9 9+9

;

55

=

111 - 1 1+1

=

222 - 2 2+2

=

333 - 3 3+3

=

444 - 4 4+4

=

555 - 5 5+5

=

666 - 6 6+6

=

777 - 7 7+7

=

888 - 8 8+8

=

999 - 9 9+9

;

...

6

=

11 + 1 1+1

=

22 + 2 2+2

=

33 + 3 3+3

=

44 + 4 4+4

=

55 + 5 5+5

=

66 + 6 6+6

=

77 + 7 7+7

=

88 + 8 8+8

=

99 + 9 9+9

;

56

=

111 + 1 1+1

=

222 + 2 2+2

=

333 + 3 3+3

=

444 + 4 4+4

=

555 + 5 5+5

=

666 + 6 6+6

=

777 + 7 7+7

=

888 + 8 8+8

=

999 + 9 9+9

;

...

Following the same procedure, we have more symmetries, such as,

11

=

22 2

=

33 3

=

44 4

=

55 5

=

66 6

=

77 7

=

88 8

=

99 9

;

11

=

22 2

+ +

22 2

=

33 3

+ +

33 3

=

44 4

+ +

44 4

=

55 5

+ +

55 5

=

66 6

+ +

66 6

=

77 7

+ +

77 7

=

88 8

+ +

88 8

=

99 9

+ +

99 9

;

37

=

1

111 +1+

1

=

2

222 +2+

2

=

3

333 +3+

3

=

4

444 +4+

4

=

5

555 +5+

5

=

6

666 +6+

6

=

7

777 +7+

7

=

8

888 +8+

8

=

9

999 +9+

9;

100

=

111 - 11 1+1

=

222 - 22 2+2

=

333 - 33 3+3

=

444 - 44 4+4

=

555 - 55 5+5

=

666 - 66 6+6

=

777 - 77 7+7

=

888 - 88 8+8

=

999 - 99 9+9

;

101

=

1111 11

=

2222 22

=

3333 33

=

4444 44

=

5555 55

=

6666 66

=

7777 77

=

8888 88

=

9999 99

;

111

=

222 2

=

333 3

=

444 4

=

555 5

=

666 6

=

777 7

=

888 8

=

999 9

;

SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

3

925

=

11111 - 11 11 + 1

=

22222 - 22 22 + 2

=

33333 - 33 33 + 3

=

44444 - 44 44 + 4

=

55555 - 55 55 + 5

=

66666 - 66 66 + 6

=

77777 - 77 77 + 7

=

88888 - 88 88 + 8

=

99999 - 99 99 + 9

;

926

=

11111 + 11 + 1

1

=

22222 + 22 + 2

2

=

33333 + 33 + 3

3

=

44444 + 44 + 4

4

=

55555 + 55 + 5

5

=

66666 + 66 + 6

6

=

77777 + 77 + 7

7

=

88888 + 88 + 8

8

=

99999 + 99 + 9

9;

...

In some cases, one has different symmetries:

1

=

2 2

? +

2 2

2

=

4 4

? +

4 4

3

=

6 6

? +

6 6

4

=

8 8

? +

8 8

0

=

1

?1- 1+1

1

1

=

3

?3- 3+3

3

2

=

5

?5- 5+5

5

3

=

7

?7- 7+7

7

4

=

9

?9- 9+9

9

1

=

1

?1+ 1+1

1

2

=

3

?3+ 3+3

3

3

=

5

?5+ 5+5

5

4

=

7

?7+ 7+7

7

5

=

9

?9+ 9+9

9

222

=

111

+

111 1

333

=

222

+

222 2

444

=

333

+

333 3

555

=

444

+

444 4

666

=

555

+

555 5

777

=

666

+

666 6

888

=

777

+

777 7

999

=

888

+

888 8

212

=

111

+

1111 11

323

=

222

+

2222 22

434

=

333

+

3333 33

545

=

444

+

4444 44

656

=

555

+

5555 55

767

=

666

+

6666 66

878

=

777

+

7777 77

989

=

888

+

8888 88

The above symmetric expressions are beautiful to see, but the number of digits used are not necessarily minimum number, for example,

5=4+

4 4

=

44 - 4 4+4

and

6=4+

4+4 4

=

44 - 4 4+4

In case of number five, 4 is used three times, for number six, 4 is used four times, while symmetric representations uses five times 4 in each case.

The following section deals with representation of natural numbers from 0 to 100 in with minimum possible digits in each case.

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SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

3. R

N

N

UD

D

0=1-1 =2-2 =3-3 =4-4 =5-5 =6-6 =7-7 =8-8 =9-9

1=1 = 2/2 = 3/3 = 4/4 = 5/5 = 6/6 = 7/7 = 8/8 = 9/9

2=1+1 =2 = 3 - 3/3 = (4 + 4)/4 = (5 + 5)/5 = (6 + 6)/6 = (7 + 7)/7 = (8 + 8)/8 = (9 + 9)/9

3=1+1+1 = 2 + 2/2 =3 = 4 - 4/4 = 5 - (5 + 5)/5 = 6 ? 6/(6 + 6) = (7 + 7 + 7)/7 = 88/8 - 8 = (9 + 9 + 9)/9

4=1+1+1+1 =2+2 = 3 + 3/3 =4 = 5 - 5/5 = 6 - (6 + 6)/6 = 77/7 - 7 = 8 ? 8/(8 + 8) = (9 ? 9 - 9)/(9 + 9)

5 = (11 - 1)/(1 + 1) = 2 + 2 + 2/2 = 3 + 3 - 3/3 = 4 + 4/4 =5 = 6 - 6/6 = 7 - (7 + 7)/7 = (88 - 8)/(8 + 8) = (99 - 9)/(9 + 9)

10 = 11 - 1 =2?2?2+2 = 3 ? 3 + 3/3 = (44 - 4)/4 =5+5 = (66 - 6)/6 = (77 - 7)/7 = (88 - 8)/8 = 9 + 9/9

6 = (1 + 1) ? (1 + 1 + 1) =2+2+2 =3+3 = 4 + (4 + 4)/4 = 5 + 5/5 =6 = 7 - 7/7 = 8 - (8 + 8)/8 = (99 + 9)/(9 + 9)

11 = 11 = 22/2 = 33/3 = 44/4 = 55/5 = 66/6 = 77/7 = 88/8 = 99/9

7 = (1 + 1) ? (1 + 1 + 1) + 1 = 2 + 2 + 2 + 2/2 = 3 + 3 + 3/3 = 4 + 4 - 4/4 = 5 + (5 + 5)/5 = 6 + 6/6 =7 = 8 - 8/8 = 9 - (9 + 9)/9

12 = 11 + 1 = 2 ? (2 + 2 + 2) =3+3?3 =4+4+4 =6+6 = (55 + 5)/5 = (77 + 7)/7 = (88 + 8)/8 = (99 + 9)/9

8 = (1 + 1)(1+1+1) = 2 ? (2 + 2) = 3 ? 3 - 3/3 =4+4 = 5 + 5 - (5 + 5)/5 = 6 + (6 + 6)/6 = 7 + 7/7 =8 = 9 - 9/9

13 = 11 + 1 + 1 = 2 + 22/2 = 3 + 3 ? 3 + 3/3 = 4 + 4 + 4 + 4/4 = (55 + 5 + 5)/5 = 6 + 6 + 6/6 = 7 + 7 - 7/7 = (88 + 8 + 8)/8 = (9 + 99 + 9)/9

9 = 11 - 1 - 1 = (2 + 2/2)2 =3?3 = 4 + 4 + 4/4 = 5 + 5 - 5/5 = 6 + 6 ? 6/(6 + 6) = 7 + (7 + 7)/7 = 8 + 8/8 =9

14 = 11 + 1 + 1 + 1 = 2(2+2) - 2 = 3 + 33/3 = 4 + (44 - 4)/4 = 5 + 5 + 5 - 5/5 = 6 + 6 + (6 + 6)/6 =7+7 = 8 + 8 - (8 + 8)/8 = 9 + (99 - 9)/(9 + 9)

15 = 11 + 1 + 1 + 1 + 1 = 2 + 2 + 22/2 =3+3+3?3 = 4 + 44/4 =5+5+5 = 6 + 6 + 6 ? 6/(6 + 6) = 7 + 7 + 7/7 = 8 + 8 - 8/8 = 9 + (99 + 9)/(9 + 9)

16 = (1 + 1)(1+1+1+1) = 2(2+2) = 33 - 33/3 =4?4 = 5 + 55/5 = 6 + (66 - 6)/6 = 7 + 7 + (7 + 7)/7 =8+8 = 9 + 9 - (9 + 9)/9

17 = (1 + 1)(1+1+1+1) + 1 = 2(2+2) + 2/2 = 3 + 3 + 33/3 = 4 ? 4 + 4/4 = 5 + (55 + 5)/5 = 6 + 66/6 = 7 + (77 - 7)/7 = 8 + 8 + 8/8 = 9 + 9 - 9/9

18 = (1 + 1) ? (11 - 1 - 1) = 2(2+2) + 2 = 3 ? (3 + 3) = 4 ? 4 + (4 + 4)/4 = 5 + (55 + 5 + 5)/5 =6+6+6 = 7 + 77/7 = 8 + (88 - 8)/8 =9+9

19 = (1 + 1) ? (11 - 1) - 1 = 22 - 2 - 2/2 = 3 ? (3 + 3) + 3/3 = 4 + 4 + 44/4 = 5 ? 5 - 5 - 5/5 = 6 + 6 + 6 + 6/6 = 7 + (77 + 7)/7 = 8 + 88/8 = 9 + 9 + 9/9

20 = (1 + 1) ? (11 - 1) = 22 - 2 = 3 ? 3 + 33/3 =4+4?4 =5?5-5 = 6 + 6 + 6 + (6 + 6)/6 = 7 + 7 + 7 - 7/7 = 8 + (88 + 8)/8 = 9 + 99/9

21 = 11 + 11 - 1 = 22 - 2/2 = 3 ? (3 + 3) + 3 = 4 + 4 ? 4 + 4/4 = 5 + 5 + 55/5 = 6 ? (6 ? 6 + 6)/(6 + 6) =7+7+7 = (88 + 88 - 8)/8 = 9 + (99 + 9)/9

22 = 11 + 11 = 22 = (33 + 33)/3 = (44 + 44)/4 = (55 + 55)/5 = (66 + 66)/6 = (77 + 77)/7 = (88 + 88)/8 = (99 + 99)/9

23 = 11 + 11 + 1 = 22 + 2/2 = 33 - 3 - 3/3 = 4 + 4 + 4 + 44/4 = 5 ? 5 - (5 + 5)/5 = 6 + 6 + 66/6 = (77 + 77 + 7)/7 = 8 + 8 + 8 - 8/8 = (99 + 99 + 9)/9

24 = (1 + 1) ? (11 + 1) = 22 + 2 = 33 - 3 =4+4+4?4 = 5 ? 5 - 5/5 =6+6+6+6 = 7 + 7 + (77 - 7)/7 =8+8+8 = (99 + 99 + 9 + 9)/9

SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

5

25 = (1 + 1) ? (11 + 1) + 1 = 22 + 2 + 2/2 = 33 - 3 + 3/3 = 4 + 4 + 4 ? 4 + 4/4 =5?5 = 6 ? 6 - 66/6 = 7 + 7 + 77/7 = 8 + 8 + 8 + 8/8 = (9 + 9 - (9 + 9)/9)) + 9

26 = (1 + 1) ? (11 + 1 + 1) = 22 + 2 + 2 = 33 - 3/3 = 4 + 44 ? 4/(4 + 4) = 5 ? 5 + 5/5 = 6 ? 6 - (66 - 6)/6 = 7 + 7 + (77 + 7)/7 = 8 + 8 + (88 - 8)/8 = 9 + 9 + 9 - 9/9

27 = (1 + 1 + 1)(1+1+1) = 22 + 2 + 2 + 2/2 = 33 = 4 ? 4 + 44/4 = 5 ? 5 + (5 + 5)/5 = 6 ? 66/(6 + 6) - 6 = 77 - 7 ? 7 - 7/7 = 8 + 8 + 88/8 =9+9+9

28 = (1 + 1 + 1)(1+1+1) + 1 = 22 + 2 + 2 + 2 = 33 + 3/3 = 44 - 4 ? 4 = 5 ? 5 + 5 - (5 + 5)/5 = 6 + (66 + 66)/6 = 7 ? (77/7 - 7) = 8 + 8 + (88 + 8)/8 = 9 + 9 + 9 + 9/9

29 = (1 + 1 + 1) ? (11 - 1) - 1 = 22 + 2 + 2 + 2 + 2/2 = 3 + 33 - 3/3 = 44 - 4 ? 4 + 4/4 = 5 ? 5 + 5 - 5/5 = 6 ? 6 - 6 - 6/6 = 77 - 7 ? 7 + 7/7 = 8 + (88 + 88 - 8)/8 = 9 + 9 + 99/9

30 = (1 + 1 + 1) ? (11 - 1) = 22 + 2 ? (2 + 2) = 3 + 33 = 4 ? (4 + 4) - (4 + 4)/4 =5?5+5 =6?6-6 = 77 - 7 ? 7 + (7 + 7)/7 = 8 + (88 + 88)/8 = 999/9 - 9 ? 9

31 = (1 + 1 + 1) ? (11 - 1) + 1 = 22 + (2 + 2/2)2 = 3 + 33 + 3/3 = 4 ? (4 + 4) - 4/4 = 5 ? 5 + 5 + 5/5 = 6 ? 6 - 6 + 6/6 = 7 ? 7 - 7 - 77/7 = 8 + 8 + 8 + 8 - 8/8 = 9 + (99 + 99)/9

32 = 11 ? (1 + 1 + 1) - 1 = 2 ? 2(2+2) = 33 - 3/3 = 4 ? (4 + 4) = ((5 + 5)/5)5 = 6 ? 6 - 6 + (6 + 6)/6 = 7 + 7 + 7 + 77/7 =8+8+8+8 = 9 + (99 + 99 + 9)/9

33 = 11 ? (1 + 1 + 1) = 22 + 22/2 = 33 = 4 ? (4 + 4) + 4/4 = ((5 + 5)/5)5 + 5/5 = 6 ? 66/(6 + 6) = (77 + 77 + 77)/7 = 8 + 8 + 8 + 8 + 8/8 = 99 ? (9 + 9 + 9)/(9 ? 9)

34 = 11 ? (1 + 1 + 1) + 1 = 2 + 2 ? 2(2+2) = 33 + 3/3 = 44 - (44 - 4)/4 = 5 ? 5 + 5 - 5/5 + 5 = 6 ? 6 - (6 + 6)/6 = 777/7 - 77 = 8 + 8 + 8 + (88 - 8)/8 = ((9 + 9) ? (9 + 9) + 9)/9

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SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

35 = 11 ? (1 + 1 + 1) + 1 + 1 = 22 + 2 + 22/2 = 3 + 33 - 3/3 = 4 + 4 ? (4 + 4) - 4/4 =5?5+5+5 = 6 ? 6 - 6/6 =7?7-7-7 = 8 + 8 + 8 + 88/8 = 9 + 9 + 9 + 9 - 9/9

36 = (1 + 1 + 1) ? (11 + 1) = (2 + 2 + 2)2 = 3 + 33 = 4 + 4 ? (4 + 4) = 5 ? 5 + 55/5 =6?6 = 7 ? 7 - 7 - 7 + 7/7 = 88 ? 8/(8 + 8) - 8 =9+9+9+9

37 = 111/(1 + 1 + 1) = (2 + 2 + 2)2 + 2/2 = 3 + 33 + 3/3 = 4 + 4 ? (4 + 4) + 4/4 = 5 + ((5 + 5)/5)5 = 6 ? 6 + 6/6 = 777/(7 + 7 + 7) = 888/(8 + 8 + 8) = 999/(9 + 9 + 9)

38 = 111/(1 + 1 + 1) + 1 = (2 + 2 + 2)2 + 2 = 33 + 33/3 = 44 - 4 - (4 + 4)/4 = 5 + ((5 + 5)/5)5 + 5/5 = 6 ? 6 + (6 + 6)/6 = 7 ? 7 - 77/7 = 8 + 8 + (88 + 88)/8 = 9 + 9 + 9 + 99/9

39 = (1 + 1 + 1) ? (11 + 1 + 1) = 2 ? (22 - 2) - 2/2 = 3 + 3 + 33 = 44 - 4 - 4/4 = 55 - 5 - 55/5 = 6 + 6 ? 66/(6 + 6) = 7 ? 7 - (77 - 7)/7 = 8 ? 8 - 8 - 8 - 8 - 8/8 = 9 + 9 + 9 + (99 + 9)/9

40 = (1 + 1) ? (1 + 1) ? (11 - 1) = 2 ? (22 - 2) = 3 + 3 + 33 + 3/3 = 44 - 4 =5?5+5+5+5 = 6 ? 6 + 6 - (6 + 6)/6 = 7 ? 7 - 7 - (7 + 7)/7 = 8 ? (8 + 8) - 88 = (9 ? 9 ? 9 - 9)/(9 + 9)

41 = (1 + 1 + 1) ? (11 - 1) + 11 = 2 ? (22 - 2) + 2/2 = 3 + 33 + 33/3 = 44 - 4 + 4/4 = 5 ? 5 + 5 + 55/5 = 6 ? 6 + 6 - 6/6 = 7 ? 7 - 7 - 7/7 = 8 ? 8 - 8 - 8 - 8 + 8/8 = (9 ? 9 ? 9 + 9)/(9 + 9)

42 = (1 + 1) ? (11 + 11 - 1) = 2 ? 22 - 2 = 3 ? 3 + 33 = 44 - (4 + 4)/4 = 5 + 5 + ((5 + 5)/5)5 =6?6+6 =7?7-7 = 8 ? 8 - (88 + 88)/8 = 9 + 9 ? 99/(9 + 9 + 9)

43 = (1 + 1) ? (11 + 11) - 1 = 2 ? 22 - 2/2 = 3 ? 3 + 33 + 3/3 = 44 - 4/4 = 55 - (55 + 5)/5 = 6 ? 6 + 6 + 6/6 = 7 ? 7 - 7 + 7/7 = 8 + 8 + 8 + 8 + 88/8 = 9 ? 9 - 9 - 9 - 9 - 99/9

44 = (1 + 1) ? (11 + 11) = 2 ? 22 = 33 + 33/3 = 44 = 55 - 55/5 = 6 ? 6 + 6 + (6 + 6)/6 = 7 ? 7 - 7 + (7 + 7)/7 = 88 ? 8/(8 + 8) = 99 ? (9 - 9/9)/(9 + 9)

45 = (1 + 1) ? (11 + 11) + 1 = 2 ? 22 + 2/2 = 3 + 3 ? 3 + 33 = 44 + 4/4 = 55 - 5 - 5 = 666/6 - 66 = 7 ? 7 + 7 - 77/7 = 8 ? 8 - 8 - 88/8 =9+9+9+9+9

46 = (1 + 1) ? (11 + 11 + 1) = 2 ? 22 + 2 = 3 + 3 ? 3 + 33 + 3/3 = 44 + (4 + 4)/4 = 55 - 5 - 5 + 5/5 = 6 ? 6 + (66 - 6)/6 = 7 ? 7 - (7 + 7 + 7)/7 = 8 ? 8 - 8 - 8 - (8 + 8)/8 = 9 + 9 + 9 + 9 + 9 + 9/9

47 = (1 + 1) ? (11 + 11 + 1) + 1 = 2 ? 22 + 2 + 2/2 = 3 + 33 + 33/3 = 4 + 44 - 4/4 = 5 + 5 + 5 + ((5 + 5)/5)5 = 66 + 6 ? 6/6 = 7 ? 7 - ((7 + 7)/7) = 888/8 - 8 ? 8 = 9 + 9 + 9 + 9 + 99/9

48 = (1 + 1) ? (1 + 1) ? (11 + 1) = 2 ? (22 + 2) = 3 ? 33 - 33 = 4 + 44 = 55 - 5 - (5 + 5)/5 =6?6+6+6 = 7 ? 7 - 7/7 =8?8-8-8 = 9 + 9 - 9 ? 9 + 999/9

49 = (11 - 1)(1+1)/(1 + 1) - 1 = 2 ? (22 + 2) + 2/2 = 33 + 33 - 33/3 = 4 + 44 + 4/4 = 55 - 5 - 5/5 = 6 ? 6 + 6 + 6 + 6/6 =7?7 = 8 ? 8 - 8 - 8 + 8/8 = (9 ? 99 - 9)/(9 + 9)

SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

7

50 = (11 - 1)(1+1)/(1 + 1) = 2 ? (22 + 2) + 2 = 3 + 3 + 33 + 33/3 = 4 + 44 + (4 + 4)/4 = 5 ? (5 + 5) = 6 ? 6 + 6 + 6 + (6 + 6)/6 = 7 ? 7 + 7/7 = 8 ? 8 - 8 - 8 + (8 + 8)/8 = (9 ? 99 + 9)/(9 + 9)

51 = (11 - 1)(1+1)/(1 + 1) + 1 = 2 ? (22 + 2) + 2 + 2/2 = 33 + 33 - 3 = 4 + 4 + 44 - 4/4 = 55 - 5 + 5/5 = 6 - 66 + 666/6 = 7 ? 7 + (7 + 7)/7 = 8 ? 8 - 8 - 8 - 8 + 88/8 = (999 - 9 ? 9)/(9 + 9)

52 = (1 + 1) ? (1 + 1) ? (11 + 1 + 1) = 2 ? (22 + 2 + 2) = 33 + 33 - 3 + 3/3 = 4 + 4 + 44 = 55 - 5 + (5 + 5)/5 = ((6 + 6)/6)6 - 6 - 6 = 7 ? 7 + (7 + 7 + 7)/7 = 8 + 88 ? 8/(8 + 8) = 9 ? 9 - 9 - 9 - 99/9

53 = (111 - 1)/(1 + 1) - 1 - 1 = 2 ? (22 + 2 + 2) + 2/2 = 33 + 33 - 3/3 = (44 - 44)/4 = 55 - (5 + 5)/5 = 6 ? 6 + 6 + 66/6 = 7 ? 7 + 77/7 - 7 = 8 ? 8 - 88/8 = 9 ? 9 - 9 - 9 - 9 - 9/9

54 = (111 - 1)/(1 + 1) - 1 = 2 ? (22 + 2 + 2) + 2 = 3 ? 3 ? (3 + 3) = 44 + (44 - 4)/4 = 55 - 5/5 = 66 - 6 - 6 = 7 ? 7 + 7 - (7 + 7)/7 = 8 ? 8 - (88 - 8)/8 =9?9-9-9-9

55 = (111 - 1)/(1 + 1) = 2 ? 22 + 22/2 = 33 + 33 + 3/3 = 44 + 44/4 = 55 = 66 - 66/6 = 7 ? 7 + 7 - 7/7 = 8 ? 8 - 8 - 8/8 = (999 - 9)/(9 + 9)

56 = (111 + 1)/(1 + 1) = (222 + 2)/(2 + 2) = (333 + 3)/(3 + 3) = 44 + 4 + 4 + 4 = 55 + 5/5 = (666 + 6)/(6 + 6) =7?7+7 =8?8-8 = (999 + 9)/(9 + 9)

57 = (111 + 1)/(1 + 1) + 1 = 2 ? 22 + 2 + 22/2 = 33 + 33 + 3 = 4 + (44 - 44)/4 = 55 + (5 + 5)/5 = 66 + (6 + 6 - 66)/6 = 7 ? 7 + 7 + 7/7 = 8 ? 8 - 8 + 8/8 = (999 + 9)/(9 + 9) + 9/9

58 = (111 + 1)/(1 + 1) + 1 + 1 = (2 + 2 + 2)2 + 22 = (3 + 3/3)3 - 3 - 3 = (44 - 4 - 4)/4 - 4 = 55 + 5 - (5 + 5)/5 = ((6 + 6)/6)6 - 6 = 7 ? 7 + 7 + (7 + 7)/7 = 8 ? 8 - 8 + (8 + 8)/8 = 9 + (9 ? 99 - 9)/(9 + 9)

59 = (11(1+1) - 1)/(1 + 1) - 1 = (44 - 4)/4 - 4 = 2 ? (22 + 2) + 22/2 = 33 + 33 - 3/3 = 55 + 5 - 5/5 = 66 + 6 - 6/6 = 77 - 7 - 77/7 = 8 ? 8 - 8 - 8 + 88/8 = 9 + (9 ? 99 + 9)/(9 + 9)

60 = (11(1+1) - 1)/(1 + 1) = 2 ? (2 ? (2 + 2) + 22) = 33 + 33 = 4 ? 4 + 44 = 55 + 5 = 66 - 6 = 7 ? 7 + 77/7 = 8 + 8 + 88 ? 8/(8 + 8) = 9 ? 9 - 9 - (99 + 9)/9

61 = (11(1+1) + 1)/(1 + 1) = (3 + 3/3)3 - 3 = (44 + 4)/4 - 4 = 2(2+2+2) - 2 - 2/2 = 55 + 5 + 5/5 = 66 - 6 + 6/6 = 7 ? 7 + (77 + 7)/7 = 8 ? 8 + 8 - 88/8 = 9 ? 9 - 99/9 - 9

62 = (11(1+1) + 1)/(1 + 1) + 1 = 2(2+2+2) - 2 = 3 + 33 + 33 - 3/3 = (44 - 4 - 4)/4 = 55 + 5 + (5 + 5)/5 = 66 - 6 + (6 + 6)/6 = 777/7 - 7 ? 7 = 8 ? 8 - (8 + 8)/8 = 9 ? 9 - 9 - 9 - 9/9

63 = (1 + 1 + 1) ? (11 + 11 - 1) = 2(2+2+2) - 2/2 = 3 + 33 + 33 = (44 - 4)/4 = (5 + 55/5)/(5 + 5) = 66 - 6 ? 6/(6 + 6) = 77 - 7 - 7 = 8 ? 8 - 8/8 =9?9-9-9

64 = (1 + 1)((1+1)?(1+1+1)) = 2(2+2+2) =4?4?4 = (3 + 3/3)3 = 55 + 5 + 5 - 5/5 = ((6 + 6)/6)6 = 77 - 7 - 7 + 7/7 =8?8 = (9 - 9/9) ? (9 - 9/9)

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SINGLE DIGIT REPRESENTATIONS - INDER J. TANEJA

65 = (1 + 1)((1+1)?(1+1+1)) + 1 = 2(2+2+2) + 2/2 = (3 + 3/3)3 + 3/3 = (44 + 4)/4 = 55 + 5 + 5 = 66 - 6/6 = 77 - (77 + 7)/7 = 8 ? 8 + 8/8 = 9 ? 9 - 9 - 9 + (9 + 9)/9

66 = 11 ? (1 + 1) ? (1 + 1 + 1) = 2(2+2+2) + 2 = 33 + 33 = (44 + 4 + 4)/4 = 55 + 55/5 = 66 = 77 - 77/7 = 8 ? 8 + (8 + 8)/8 = 99 ? (99 + 9)/(9 ? (9 + 9))

67 = 11 ? (1 + 1) ? (1 + 1 + 1) + 1 = 2(2+2+2) + 2 + 2/2 = (3 + 3/3)3 + 3 = 4 + (44 - 4)/4 = 55 + (55 + 5)/5 = 66 + 6/6 = 77 - (77 - 7)/7 = 8 ? 8 - 8 + 88/8 = 9 ? 9 - (99 + 9 + 9 + 9)/9

68 = (1 + 1) ? (11 ? (1 + 1 + 1) + 1) = 2(2+2+2) + 2 + 2 = ((3 + 3)3 - 3)/3 - 3 =4+4?4?4 = 5 + (55/5 + 5)/(5 + 5) = 66 + (6 + 6)/6 = 77 - 7 - (7 + 7)/7 = 88 - 8 - (88 + 8)/8 = 9 ? 9 - (99 + 9 + 9)/9

69 = (1 + 1 + 1) ? (11 + 11 + 1) = (22 + 2/2) ? (2 + 2/2) = 33 + 33 + 3 = 4 + (44 + 4)/4 = 55 + 5 + 5 + 5 - 5/5 = 66 + 6 ? 6/(6 + 6) = 77 - 7 - 7/7 = 88 - 8 - 88/8 = 9 ? 9 - (99 + 9)/9

70 = (11 - 1 - 1)(1+1) - 11 = 2 ? (22 + 2) + 22 = (3/3 + 3)3 + 3 + 3 = 4 + (44 + 4 + 4)/4 = 55 + 5 + 5 + 5 = 6 + ((6 + 6)/6)6 = 77 - 7 = 8 ? 8 + 8 - (8 + 8)/8 = 9 ? 9 - 99/9

71 = (11 + 1)(1+1)/(1 + 1) - 1 = 2 ? (2 + 2 + 2)2 - 2/2 = ((3 + 3)3 - 3)/3 = 4 + 4 + (44 - 4)/4 = 55 + 5 + 55/5 = 66 + 6 - 6/6 = 77 - 7 + 7/7 = 8 ? 8 + 8 - 8/8 = 9 ? 9 - 9 - 9/9

72 = (11 + 1)(1+1)/(1 + 1) = 2 ? (2 + 2 + 2)2 = 3 ? (33 - 3) =4+4+4?4?4 = 55 + 5 + (55 + 5)/5 = 66 + 6 = 77 - 7 + (7 + 7)/7 =8+8?8 =9?9-9

73 = (11 + 1)(1+1)/(1 + 1) + 1 = 2 ? (2 + 2 + 2)2 + 2/2 = ((3 + 3)3 + 3)/3 = 4 + 4 + (44 + 4)/4 = 5 ? (5 + 5 + 5) - (5 + 5)/5 = 66 + 6 + 6/6 = 77 + 7 - 77/7 = 8 ? 8 + 8 + 8/8 = 9 ? 9 - 9 + 9/9

74 = (1 + 1) ? 111/(1 + 1 + 1) = 2 ? (2 + 2 + 2)2 + 2 = 3 + ((3 + 3)3 - 3)/3 = (44 + 44 - 4)/4 = 5 ? (5 + 5 + 5) - 5/5 = 66 + 6 + (6 + 6)/6 = 77 - (7 + 7 + 7)/7 = 8 ? 8 + 8 + (8 + 8)/8 = 9 ? 9 - (9 - (9 + 9)/9)

75 = (1 + 1) ? 111/(1 + 1 + 1) + 1 = 2(2+2+2) + 22/2 = 3 + 3 ? (33 - 3) = (44 + 44)/4 = 5 ? (5 + 5 + 5) = 666/6 - 6 ? 6 = 77 - (7 + 7)/7 = 8 ? 8 + 88/8 = 9 ? 9 - (99 + 9)/(9 + 9)

76 = (1 + 1) ? (111/(1 + 1 + 1) + 1) = 2 ? ((2 + 2 + 2)2 + 2) = 3 + (3 + (3 + 3)3)/3 = 44 + 4 ? (4 + 4) = 5 ? (5 + 5 + 5) + 5/5 = 6 + 6 + ((6 + 6)/6)6 = 77 - 7/7 = 88 - (88 + 8)/8 = 9 ? 9 - (99 - 9)/(9 + 9)

77 = 11 ? ((1 + 1) ? (1 + 1 + 1) + 1) = 2 ? 2 ? 22 - 22/2 = 3 ? 33 - 3 - 3/3 = (4 - 4/4)4 - 4 = 55 + (55 + 55)/5 = 66 + 66/6 = 77 = 88 - 88/8 = 9 ? 9 - (9 + 9 + 9 + 9)/9

78 = 111 - 11 ? (1 + 1 + 1) = 2 ? 2 ? (22 - 2) - 2 = 3 ? 33 - 3 = 4 + (44 - 4 + 44)/4 = 5 ? 5 + 55 - (5 + 5)/5 = 66 + 6 + 6 = 77 + 7/7 = 88 - (88 - 8)/8 = 9 ? 9 - (9 + 9 + 9)/9

79 = (11 - 1 - 1)(1+1) - 1 - 1 = (2 + 2/2)(2+2) - 2 = 3 ? 33 - 3 + 3/3 = 4 + (44 + 44)/4 = 5 ? 5 + 55 - 5/5 = 66 + 6 + 6 + 6/6 = 77 + (7 + 7)/7 = 88 - 8 - 8/8 = 9 ? 9 - (9 + 9)/9

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