examples of uniqueness of digital representation of small numbers in a few bases

examples of uniqueness of digital representation of small numbers in a few bases

While a string of digits is not unique to any base

b

, within each base

b

the base

b

representation

of an

integer

n

is unique. The following table gives the base 2 (binary) through base 10 (decimal) representations of

n

from 1 to 50. It will be seen how “10”

occurs in

several cells diagonally in the table, but in each column it corresponds to the case

n

=

b

only. Likewise, “11” also occurs diagonally, but in each column it corresponds to

n

=

b

+

1

only. One other generality to mention: the representation

n

is the same for all bases

b

>

n

.

b

=

2

3

4

5

6

7

8

9

10

1

1

1

1

1

1

1

1

1

10

2

2

2

2

2

2

2

2

11

10

3

3

3

3

3

3

3

100

11

10

4

4

4

4

4

4

101

12

11

10

5

5

5

5

5

110

20

12

11

10

6

6

6

6

111

21

13

12

11

10

7

7

7

1000

22

20

13

12

11

10

8

8

1001

100

21

14

13

12

11

10

9

1010

101

22

20

14

13

12

11

10

1011

102

23

21

15

14

13

12

11

1100

110

30

22

20

15

14

13

12

1101

111

31

23

21

16

15

14

13

1110

112

32

24

22

20

16

15

14

1111

120

33

30

23

21

17

16

15

10000

121

100

31

24

22

20

17

16

10001

122

101

32

25

23

21

18

17

10010

200

102

33

30

24

22

20

18

10011

201

103

34

31

25

23

21

19

10100

202

110

40

32

26

24

22

20

10101

210

111

41

33

30

25

23

21

10110

211

112

42

34

31

26

24

22

10111

212

113

43

35

32

27

25

23

11000

220

120

44

40

33

30

26

24

11001

221

121

100

41

34

31

27

25

11010

222

122

101

42

35

32

28

26

11011

1000

123

102

43

36

33

30

27

11100

1001

130

103

44

40

34

31

28

11101

1002

131

104

45

41

35

32

29

11110

1010

132

110

50

42

36

33

30

11111

1011

133

111

51

43

37

34

31

100000

1012

200

112

52

44

40

35

32

100001

1020

201

113

53

45

41

36

33

100010

1021

202

114

54

46

42

37

34

100011

1022

203

120

55

50

43

38

35

100100

1100

210

121

100

51

44

40

36

100101

1101

211

122

101

52

45

41

37

100110

1102

212

123

102

53

46

42

38

100111

1110

213

124

103

54

47

43

39

101000

1111

220

130

104

55

50

44

40

101001

1112

221

131

105

56

51

45

41

101010

1120

222

132

110

60

52

46

42

101011

1121

223

133

111

61

53

47

43

101100

1122

230

134

112

62

54

48

44

101101

1200

231

140

113

63

55

50

45

101110

1201

232

141

114

64

56

51

46

101111

1202

233

142

115

65

57

52

47

110000

1210

300

143

120

66

60

53

48

110001

1211

301

144

121

100

61

54

49

110010

1212

302

200

122

101

62

55

50

Title

examples of

uniqueness of digital representation

of small numbers in a few bases

Canonical name

ExamplesOfUniquenessOfDigitalRepresentationOfSmallNumbersInAFewBases

Date of creation

2013-03-22 18:52:25

Last modified on

2013-03-22 18:52:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Classification

msc 11A05

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