# 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 ExamplesOfUniquenessOfDigitalRepresentationOfSmallNumbersInAFewBases 2013-03-22 18:52:25 2013-03-22 18:52:25 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Example msc 11A63 msc 11A05