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BASES de NUMÉRATION

 

Débutants

Nombres

Les BASES

 

Glossaire

Nombres

 

 

INDEX

 

Numération

 

Chiffres

 

Index bases

Introduction

Maximum

Toutes les bases

Mêmes chiffres en base b

 

Sommaire de cette page

>>> Mêmes chiffres en base 10 et base b

>>> Base 10 et base b – exactement

>>> Base b et base c – exactement

 

 

 

 

Mêmes chiffres en décimal

et en base b

 

Le nombre et sa conversion en base b présente les mêmes chiffres, en même quantité ou non.

100 + 50 + 8 = 81 + 72 + 5

 

 

 

Mêmes chiffres – Décimal et base b

 

 

Nombres qui comportent les mêmes chiffres dans leur conversion en base b, base allant de 2 à n – 2.

On note de nombreux cas triviaux avec la base 110.

 

 

Exemple

10 en décimal vaut 1010 en binaire, les deux sont composés de {0 et 1}, mais pas forcément en même quantité.

 

Autres exemples

2110 = 1219

12210 =  111123

 

 

10, 1010, 2

10, 101, 3

13, 31, 4

20, 202, 3

21, 12, 19

23, 32, 7

31, 133, 4

31, 13, 28

41, 14, 37

42, 24, 19

43, 34, 13

46, 64, 7

51, 15, 46

53, 35, 16

61, 16, 55

62, 26, 28

63, 36, 19

71, 17, 64

73, 37, 22

81, 18, 73

82, 28, 37

83, 38, 25

84, 48, 19

86, 68, 13

91, 19, 82

93, 39, 28

100, 1100100, 2

101, 1100101, 2

102, 10210, 3

110, 1101110, 2

112, 12, 110

113, 13, 110

114, 14, 110

115, 15, 110

116, 16, 110

117, 17, 110

118, 18, 110

119, 19, 110

121, 21, 60

121, 12, 119

122, 11112, 3

122, 12, 120

123, 1323, 4

131, 13, 128

133, 31, 44

133, 13, 130

141, 41, 35

141, 14, 137

144, 14, 140

151, 51, 30

151, 15, 146

155, 15, 150

158, 185, 9

161, 16, 155

166, 16, 160

171, 17, 164

 

 

177, 17, 170

181, 18, 173

188, 18, 180

191, 119, 13

191, 19, 182

196, 169, 11

199, 91, 22

199, 19, 190

200, 20, 100

201, 21110, 3

202, 20, 101

203, 3023, 4

210, 21210, 3

211, 21211, 3

211, 21, 105

211, 12, 209

212, 21212, 3

212, 112, 14

212, 12, 210

213, 1323, 5

220, 20, 110

221, 21, 110

221, 12, 219

222, 22, 110

223, 23, 110

224, 24, 110

225, 25, 110

226, 26, 110

227, 272, 9

227, 27, 110

228, 28, 110

229, 29, 110

231, 3213, 4

233, 32, 77

233, 23, 115

242, 42, 60

242, 24, 119

244, 24, 120

252, 52, 50

255, 25, 125

261, 126, 15

262, 26, 128

265, 526, 7

266, 62, 44

266, 26, 130

277, 27, 135

282, 82, 35

282, 28, 137

283, 238, 11

288, 28, 140

299, 92, 33

299, 29, 145

300, 30, 100

303, 30, 101

311, 13, 308

313, 31, 104

 

313, 13, 310

316, 631, 7

323, 32, 107

323, 23, 160

330, 30, 110

331, 1311, 6

331, 31, 110

331, 13, 328

332, 32, 110

333, 33, 110

334, 34, 110

335, 35, 110

336, 36, 110

337, 37, 110

338, 38, 110

339, 39, 110

343, 43, 85

343, 34, 113

353, 53, 70

353, 35, 116

363, 63, 60

363, 36, 119

366, 36, 120

370, 307, 11

371, 173, 16

373, 37, 122

383, 38, 125

393, 39, 128

399, 93, 44

399, 39, 130

400, 40, 100

404, 40, 101

411, 14, 407

414, 14, 410

422, 42, 105

422, 24, 209

424, 224, 14

424, 24, 210

433, 34, 143

440, 40, 110

441, 144, 19

441, 41, 110

441, 14, 437

442, 42, 110

442, 24, 219

443, 43, 110

444, 44, 110

445, 544, 9

445, 45, 110

446, 46, 110

447, 47, 110

448, 48, 110

449, 49, 110

454, 554, 9

454, 54, 90

466, 64, 77

 

466, 46, 115

484, 84, 60

484, 48, 119

488, 48, 120

499, 94, 55

500, 50, 100

505, 50, 101

511, 115, 22

511, 51, 102

511, 15, 506

515, 15, 510

518, 185, 19

522, 52, 104

525, 25, 260

533, 53, 106

533, 35, 176

544, 455, 11

544, 54, 108

545, 45, 135

550, 50, 110

551, 155, 21

551, 51, 110

551, 15, 546

552, 52, 110

553, 53, 110

554, 54, 110

555, 55, 110

556, 56, 110

557, 57, 110

558, 58, 110

559, 59, 110

566, 56, 112

577, 57, 114

588, 58, 116

599, 95, 66

599, 59, 118

600, 60, 100

606, 60, 101

611, 16, 605

614, 1146, 8

616, 16, 610

622, 26, 308

626, 62, 104

626, 26, 310

633, 63, 105

633, 36, 209

636, 336, 14

636, 36, 210

646, 64, 107

646, 46, 160

656, 56, 130

660, 60, 110

661, 61, 110

661, 16, 655

662, 62, 110

662, 26, 328

 

663, 63, 110

663, 36, 219

664, 64, 110

665, 65, 110

666, 66, 110

667, 67, 110

668, 68, 110

669, 69, 110

686, 86, 85

686, 68, 113

699, 96, 77

699, 69, 115

700, 70, 100

707, 70, 101

711, 17, 704

717, 17, 710

727, 27, 360

733, 37, 242

747, 47, 185

757, 57, 150

770, 70, 110

771, 71, 110

771, 17, 764

772, 72, 110

773, 73, 110

774, 477, 13

774, 74, 110

775, 75, 110

776, 76, 110

777, 77, 110

778, 78, 110

779, 79, 110

782, 278, 18

799, 97, 88

800, 80, 100

801, 1080, 9

808, 80, 101

811, 18, 803

818, 18, 810

822, 28, 407

825, 258, 19

828, 28, 410

833, 38, 275

834, 438, 14

844, 84, 105

844, 48, 209

848, 448, 14

848, 48, 210

858, 58, 170

866, 68, 143

880, 80, 110

881, 81, 110

881, 18, 873

882, 288, 19

882, 82, 110

882, 28, 437

 

883, 83, 110

884, 84, 110

884, 48, 219

885, 85, 110

886, 86, 110

887, 87, 110

888, 88, 110

889, 89, 110

899, 98, 99

900, 90, 100

909, 90, 101

910, 190, 26

911, 191, 26

911, 19, 902

912, 219, 21

912, 192, 26

913, 391, 16

913, 193, 26

914, 194, 26

915, 195, 26

916, 196, 26

917, 197, 26

918, 198, 26

919, 199, 26

919, 91, 102

919, 19, 910

929, 92, 103

929, 29, 460

933, 39, 308

939, 93, 104

939, 39, 310

949, 94, 105

949, 49, 235

958, 589, 13

959, 95, 106

959, 59, 190

961, 169, 28

969, 96, 107

969, 69, 160

979, 97, 108

989, 98, 109

990, 90, 110

991, 91, 110

991, 19, 982

992, 92, 110

993, 93, 110

993, 39, 328

994, 94, 110

995, 95, 110

996, 96, 110

997, 97, 110

998, 98, 110

999, 99, 110

1000, 1111101000, 2

1000, 1101001, 3

 

 

Mêmes chiffres exactement – Décimal base b

 

Nombres qui comportent exactement les mêmes chiffres dans leur conversion en base b, base allant de 2 à n – 2.

Ils sont 62 jusqu'à 1000.

En italique, chiffres inversés.

 

 

Exemple

13 en décimal vaut 31 en base 4, les deux sont composés de {0 et 1} en même quantité.

 

Autres exemples

4610 = 647

96110 = 16928

 

13, 31, 4

21, 12, 19

23, 32, 7

31, 13, 28

41, 14, 37

42, 24, 19

43, 34, 13

46, 64, 7

51, 15, 46

53, 35, 16

61, 16, 55

 

 

62, 26, 28

63, 36, 19

71, 17, 64

73, 37, 22

81, 18, 73

82, 28, 37

83, 38, 25

84, 48, 19

86, 68, 13

91, 19, 82

93, 39, 28

 

158, 185, 9

191, 119, 13

196, 169, 11

212, 112, 14

227, 272, 9

261, 126, 15

265, 526, 7

283, 238, 11

316, 631, 7

370, 307, 11

 

371, 173, 16

424, 224, 14

441, 144, 19

445, 544, 9

454, 554, 9

511, 115, 22

518, 185, 19

544, 455, 11

551, 155, 21

636, 336, 14

 

774, 477, 13

782, 278, 18

825, 258, 19

834, 438, 14

848, 448, 14

882, 288, 19

910, 190, 26

911, 191, 26

912, 219, 21

912, 192, 26

 

913, 391, 16

913, 193, 26

914, 194, 26

915, 195, 26

916, 196, 26

917, 197, 26

918, 198, 26

919, 199, 26

958, 589, 13

961, 169, 28

 

Ils sont 90 de 1 000 à 10 000.

La base ne dépasse pas 20.

 

 

Exemples

113510 = 51316

123410 = 34127

 

1030, 3001, 7

1045, 4501, 6

1135, 5131, 6

1234, 3412, 7

1236, 1623, 9

1273, 2371, 8

1366, 3661, 7

1380, 1803, 9

1431, 4113, 7

1454, 4145, 7

1653, 3165, 8

1723, 1327, 11

2027, 2702, 9

2060, 6002, 7

2116, 6112, 7

2154, 4152, 8

2193, 1329, 12

2315, 3152, 9

2534, 3425, 9

3160, 6130, 8

3161, 6131, 8

3162, 6132, 8

3163, 6133, 8

3164, 6134, 8

3165, 6135, 8

3166, 6136, 8

3167, 6137, 8

3226, 6232, 8

4063, 3064, 11

4100, 1004, 16

4621, 2146, 13

5141, 1415, 16

5180, 1805, 15

5270, 7205, 9

5272, 2527, 13

5412, 1524, 16

5567, 7565, 9

5637, 7653, 9

5657, 7675, 9

5783, 7835, 9

5812, 2851, 13

5920, 2905, 13

6053, 3605, 12

6182, 1826, 16

6572, 2576, 14

7018, 1087, 19

7140, 4170, 12

7141, 4171, 12

7142, 4172, 12

7143, 4173, 12

7144, 4174, 12

7145, 4175, 12

7146, 4176, 12

7147, 4177, 12

7148, 4178, 12

7149, 4179, 12

7243, 4237, 12

7364, 3476, 13

7507, 5705, 11

7513, 1537, 18

7587, 5778, 11

7721, 1277, 19

7834, 3748, 13

7873, 3778, 13

7941, 4719, 12

8200, 2008, 16

8461, 1486, 19

8591, 1895, 18

8665, 6568, 11

9143, 3491, 14

9167, 1679, 19

9186, 1689, 19

9241, 2419, 16

9304, 4309, 13

9343, 4339, 13

9413, 4391, 13

9471, 1749, 19

9730, 3790, 14

9731, 3791, 14

9732, 3792, 14

9733, 3793, 14

9734, 3794, 14

9735, 3795, 14

9736, 3796, 14

9737, 3797, 14

9738, 3798, 14

9739, 3799, 14

9813, 1839, 19

9825, 5829, 12

9851, 1859, 19

 

 

Mêmes chiffres exactement – Base b et base c

 

Nombres dont la valeur dans deux bases comprend les mêmes chiffres exactement.

Les occurrences sont très nombreuses. On retrouve les cas où l'une des base est 10.

 

Exemples

    710 = 213  = 125      = 2x3+1 = 1x5+2

  2310 = 2111 = 1221    = 2x11+1 = 1x21+2

10010 = 9111 = 1991    = 9x11+1 = 1x91+9

 

7, 21, 3, 12, 5

9, 21, 4, 12, 7

11, 21, 5, 12, 9

13, 31, 4, 13, 10

13, 21, 6, 12, 11

15, 21, 7, 12, 13

16, 31, 5, 13, 13

17, 32, 5, 23, 7

17, 21, 8, 12, 15

19, 31, 6, 13, 16

19, 21, 9, 12, 17

21, 41, 5, 14, 17

21, 21, 10, 12, 19

22, 211, 3, 112, 4

22, 42, 5, 24, 9

22, 31, 7, 13, 19

23, 32, 7, 23, 10

23, 21, 11, 12, 21

25, 221, 3, 121, 4

25, 41, 6, 14, 21

25, 31, 8, 13, 22

25, 21, 12, 12, 23

26, 42, 6, 24, 11

27, 21, 13, 12, 25

28, 130, 4, 103, 5

28, 31, 9, 13, 25

29, 41, 7, 14, 25

29, 32, 9, 23, 13

29, 21, 14, 12, 27

30, 42, 7, 24, 13

31, 51, 6, 15, 26

31, 43, 7, 34, 9

31, 31, 10, 13, 28

31, 21, 15, 12, 29

33, 41, 8, 14, 29

33, 21, 16, 12, 31

34, 42, 8, 24, 15

34, 31, 11, 13, 31

35, 32, 11, 23, 16

35, 21, 17, 12, 33

36, 51, 7, 15, 31

37, 211, 4, 122, 5

37, 52, 7, 25, 16

37, 41, 9, 14, 33

37, 31, 12, 13, 34

37, 21, 18, 12, 35

38, 53, 7, 35, 11

38, 42, 9, 24, 17

39, 21, 19, 12, 37

40, 31, 13, 13, 37

41, 51, 8, 15, 36

41, 41, 10, 14, 37

41, 32, 13, 23, 19

41, 21, 20, 12, 39

42, 42, 10, 24, 19

43, 61, 7, 16, 37

43, 43, 10, 34, 13

43, 31, 14, 13, 40

43, 21, 21, 12, 41

44, 62, 7, 26, 19

45, 63, 7, 36, 13

45, 41, 11, 14, 41

45, 21, 22, 12, 43

46, 141, 5, 114, 6

 

46, 64, 7, 46, 10

46, 51, 9, 15, 41

46, 42, 11, 24, 21

46, 31, 15, 13, 43

47, 52, 9, 25, 21

47, 32, 15, 23, 22

47, 21, 23, 12, 45

49, 61, 8, 16, 43

49, 54, 9, 45, 11

49, 41, 12, 14, 45

49, 31, 16, 13, 46

49, 21, 24, 12, 47

50, 62, 8, 26, 22

50, 42, 12, 24, 23

51, 201, 5, 102, 7

51, 63, 8, 36, 15

51, 51, 10, 15, 46

51, 21, 25, 12, 49

52, 310, 4, 103, 7

52, 31, 17, 13, 49

53, 53, 10, 35, 16

53, 41, 13, 14, 49

53, 32, 17, 23, 25

53, 21, 26, 12, 51

54, 42, 13, 24, 25

55, 313, 4, 131, 6

55, 61, 9, 16, 49

55, 43, 13, 34, 17

55, 31, 18, 13, 52

55, 21, 27, 12, 53

56, 62, 9, 26, 25

56, 51, 11, 15, 51

57, 71, 8, 17, 50

57, 63, 9, 36, 17

57, 52, 11, 25, 26

57, 41, 14, 14, 53

57, 21, 28, 12, 55

58, 64, 9, 46, 13

58, 42, 14, 24, 27

58, 31, 19, 13, 55

59, 32, 19, 23, 28

59, 21, 29, 12, 57

61, 61, 10, 16, 55

61, 51, 12, 15, 56

61, 41, 15, 14, 57

61, 31, 20, 13, 58

61, 21, 30, 12, 59

62, 62, 10, 26, 28

62, 42, 15, 24, 29

63, 63, 10, 36, 19

63, 21, 31, 12, 61

64, 71, 9, 17, 57

64, 31, 21, 13, 61

65, 72, 9, 27, 29

65, 41, 16, 14, 61

65, 32, 21, 23, 31

65, 21, 32, 12, 63

66, 2110, 3, 1002, 4

66, 231, 5, 123, 7

66, 51, 13, 15, 61

66, 42, 16, 24, 31

67, 74, 9, 47, 15

67, 61, 11, 16, 61

 

67, 52, 13, 25, 31

67, 43, 16, 34, 21

67, 31, 22, 13, 64

67, 21, 33, 12, 65

68, 152, 6, 125, 7

68, 62, 11, 26, 31

68, 53, 13, 35, 21

69, 63, 11, 36, 21

69, 54, 13, 45, 16

69, 41, 17, 14, 65

69, 21, 34, 12, 67

70, 64, 11, 46, 16

70, 42, 17, 24, 33

70, 31, 23, 13, 67

71, 71, 10, 17, 64

71, 65, 11, 56, 13

71, 51, 14, 15, 66

71, 32, 23, 23, 34

71, 21, 35, 12, 69

73, 2201, 3, 1021, 4

73, 81, 9, 18, 65

73, 73, 10, 37, 22

73, 61, 12, 16, 67

73, 41, 18, 14, 69

73, 31, 24, 13, 70

73, 21, 36, 12, 71

74, 82, 9, 28, 33

74, 62, 12, 26, 34

74, 42, 18, 24, 35

75, 63, 12, 36, 23

75, 21, 37, 12, 73

76, 84, 9, 48, 17

76, 51, 15, 15, 71

76, 31, 25, 13, 73

77, 52, 15, 25, 36

77, 41, 19, 14, 73

77, 32, 25, 23, 37

77, 21, 38, 12, 75

78, 71, 11, 17, 71

78, 42, 19, 24, 37

79, 72, 11, 27, 36

79, 61, 13, 16, 73

79, 43, 19, 34, 25

79, 31, 26, 13, 76

79, 21, 39, 12, 77

80, 62, 13, 26, 37

81, 81, 10, 18, 73

81, 63, 13, 36, 25

81, 51, 16, 15, 76

81, 41, 20, 14, 77

81, 21, 40, 12, 79

82, 82, 10, 28, 37

82, 75, 11, 57, 15

82, 64, 13, 46, 19

82, 42, 20, 24, 39

82, 31, 27, 13, 79

83, 83, 10, 38, 25

83, 53, 16, 35, 26

83, 32, 27, 23, 40

83, 21, 41, 12, 81

84, 84, 10, 48, 19

85, 71, 12, 17, 78

85, 61, 14, 16, 79

 

85, 41, 21, 14, 81

85, 31, 28, 13, 82

85, 21, 42, 12, 83

86, 86, 10, 68, 13

86, 62, 14, 26, 40

86, 51, 17, 15, 81

86, 42, 21, 24, 41

87, 322, 5, 223, 6

87, 63, 14, 36, 27

87, 52, 17, 25, 41

87, 21, 43, 12, 85

88, 31, 29, 13, 85

89, 81, 11, 18, 81

89, 54, 17, 45, 21

89, 41, 22, 14, 85

89, 32, 29, 23, 43

89, 21, 44, 12, 87

90, 82, 11, 28, 41

90, 42, 22, 24, 43

91, 331, 5, 133, 8

91, 91, 10, 19, 82

91, 61, 15, 16, 85

91, 51, 18, 15, 86

91, 43, 22, 34, 29

91, 31, 30, 13, 88

91, 21, 45, 12, 89

92, 332, 5, 232, 6

92, 84, 11, 48, 21

92, 71, 13, 17, 85

92, 62, 15, 26, 43

93, 93, 10, 39, 28

93, 85, 11, 58, 17

93, 72, 13, 27, 43

93, 63, 15, 36, 29

93, 41, 23, 14, 89

93, 21, 46, 12, 91

94, 163, 7, 136, 8

94, 73, 13, 37, 29

94, 64, 15, 46, 22

94, 42, 23, 24, 45

94, 31, 31, 13, 91

95, 74, 13, 47, 22

95, 32, 31, 23, 46

95, 21, 47, 12, 93

96, 51, 19, 15, 91

97, 81, 12, 18, 89

97, 76, 13, 67, 15

97, 61, 16, 16, 91

97, 52, 19, 25, 46

97, 41, 24, 14, 93

97, 31, 32, 13, 94

97, 21, 48, 12, 95

98, 82, 12, 28, 45

98, 62, 16, 26, 46

98, 53, 19, 35, 31

98, 42, 24, 24, 47

99, 201, 7, 120, 9

99, 71, 14, 17, 92

99, 63, 16, 36, 31

99, 21, 49, 12, 97

100, 91, 11, 19, 91

100, 84, 12, 48, 23

100, 31, 33, 13, 97

 

 

 

 

Suite

*    Numération – Spécificités et bizarreries

*    Autres en haut de page

Voir

*    Binaire

*    ChiffresGlossaire

*    Tables de calculs en base 2 à 16

*    Romain

*    Étymologie

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