### More place-value systems

Octal system | Base 8 |

Duo-decimal system | Base 12 |

Sexagesimal system | Base 60 |

Base64 | Base 64 |

Radix32 | Base 32 |

Vigesimal system | Base 20 |

Octal system | Base 8 |

Duo-decimal system | Base 12 |

Sexagesimal system | Base 60 |

Base64 | Base 64 |

Radix32 | Base 32 |

Vigesimal system | Base 20 |

For a place-value system, the value of a digit results from the position of the digit within the number. The value of the number is determined by the sum of the digits, each digit being multiplied by the digits place value. The place value is the position-dependent power of the base value b of the place value system. The numbers are written as a sequence of digits from left to right, the left is started with the highest place value, and the place value is then reduced to the next position by one (b-adic representation) in the exponent. The transition to negative exponents of the base value is indicated by a ".". The number of different digit symbols required is equal to the base value b.

Structure of a number in the place-value system with the digits ** a _{ n } a _{ n-1 } ... a _{ 0 }. a _{ 1 } ... a _{ -m } **. The value is calculated according to

$${a}_{n}{b}^{n}+{a}_{\mathrm{n-1}}{b}^{\mathrm{n-1}}+\dots +{a}_{1}{b}^{1}+{a}_{0}{b}^{0}+{a}_{-1}{b}^{-1}+\dots +{a}_{\mathrm{-m}}{b}^{\mathrm{-m}}=\sum _{i=0}^{\mathrm{m+n}}{a}_{\mathrm{i-m}}{b}^{\mathrm{i-m}}$$

Base value b = 10

Digits 0, 1, 2, ..., 9

$$1265.42=\sum _{i=0}^{6}{a}_{\mathrm{i-2}}{10}^{\mathrm{i-2}}$$

$$=2\cdot {10}^{-2}+4\cdot {10}^{-1}+5\cdot {10}^{0}+6\cdot {10}^{1}+2\cdot {10}^{2}+1\cdot {10}^{3}$$

$$=\frac{2}{100}+\frac{4}{10}+5+6\cdot 10+2\cdot 100+1\cdot 1000$$

Base value b = 2

Digits 0, 1

$${101101}_{2}=\sum _{i=0}^{6}{a}_{i}{2}^{i}$$

$$=1\cdot {2}^{0}+0\cdot {2}^{1}+1\cdot {2}^{2}+1\cdot {2}^{3}+0\cdot {2}^{4}+1\cdot {2}^{5}$$

$$=1+4+8+32={45}_{10}$$

Base value b = 16

Digits 0, 1, 2, ...,9, A, B, C, D, E, F

$${\mathrm{10FE1A}}_{16}=\sum _{i=0}^{6}{a}_{i}{16}^{i}$$

$$=\mathrm{A}\cdot {16}^{0}+1\cdot {16}^{1}+\mathrm{E}\cdot {16}^{2}+\mathrm{F}\cdot {16}^{3}+0\cdot {16}^{4}+1\cdot {16}^{5}$$

$$=10+16+3584+61440+1048576={1113626}_{\mathrm{10}}$$

The starting point of the iteration is the decimal number. The number is divided by the base and the integer part of the division is used for the next iteration. The remainder of the division is the digit for the new base. The iteration is repeated until the remainder is 0.

The calculation of the decimal places is made by multiplication with the base. The number before the decimal point is the next digit. The decimal part is multiplied by the base until the remainder is 0 or the maximum number of desired digits is reached. Here after 10 positions is the calculation aborted.