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The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base.In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten.
The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. [2] "Table of Pythagoras" on Napier's bones [3] The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC).
For example, decimal 365 (10) or senary 1 405 (6) corresponds to binary 1 0110 1101 (2) (nine bits) and to ternary 111 112 (3) (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27).
A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
For q = 2 2 = 4, it can be checked case by case using the above multiplication table that all four elements of F 4 satisfy the equation x 4 = x, so they are zeros of f. By contrast, in F 2, f has only two zeros (namely 0 and 1), so f does not split into linear factors in this smaller field.
This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33. Decimal: Binary: 11 3 1011 11 5 6 101 110 2 12 10 1100 1 24 1 11000 —— —————— 33 100001 Describing the steps explicitly: 11 and 3 are written at the top
Like the decimal base 10, the base is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number , many of the arguments made in favor of the duodecimal system also apply to the senary system.