Senary
Senary, also known as base-6 or seximal, is a positional numeral system that employs six as its radix and utilizes the digits 0 through 5 to represent values.[1][2] In this system, each position represents a power of six, such that the rightmost digit denotes 6^0 (units), the next 6^1 (sixes), and so forth; for example, the senary number 10 equals 6 in decimal, while 125 equals 53 in decimal.[1] The term "senary" derives from Latin roots meaning "of or relating to six," reflecting its foundation in groupings of six units or parts.[3] Historically, senary systems have been adopted independently by a limited number of cultures, with the most well-documented examples occurring among languages in southern New Guinea, particularly in the Morehead-Maro region.[4] These include Yam languages such as Yei, Ngkolmpu, and Arammba, as well as non-Yam languages like Ndom, where senary counting is tied to practical needs such as tallying yams—a staple crop organized into piles for storage and distribution.[5] In these contexts, higher powers of six (e.g., 36 as ptae in Ngkolmpu or 216 as tarumpao) serve as monomorphemic terms for large quantities, with 1296 (6^4) representing the approximate annual yam yield to feed a family.[5] Senary structures have also been hypothesized in ancient systems, including Sumerian base-60 (with auxiliary base-6 elements), certain Niger-Congo languages, Proto-Finno-Ugric, and Utian languages of California, though evidence remains reconstructive and debated.[6] Notable aspects of senary include its mathematical efficiency for division, as six is evenly divisible by both 2 and 3, facilitating equitable sharing of small groups without remainders—unlike decimal's challenges with thirds.[5] This property, combined with body-part tallying (e.g., using fingers or a closed fist as "one" up to six), likely motivated its development in resource-limited environments.[6] While rare in modern global use, senary persists in linguistic studies.[7]Fundamentals
Definition
Senary, also known as base-6 or seximal, is a positional numeral system with six as its radix, employing digits ranging from 0 to 5 to represent numerical values. In this system, the position of each digit signifies a successive power of 6, beginning with $6^0 = 1 for the rightmost place and increasing to the left. This structure enables the encoding of any non-negative integer through combinations of these digits, similar to how decimal (base-10) uses powers of 10 but adapted to a smaller set of symbols.[1] The value of a senary number expressed as digits d_n d_{n-1} \dots d_1 d_0, where each d_i is an integer from 0 to 5, is calculated by the formula: \sum_{i=0}^{n} d_i \cdot 6^i = d_n \cdot 6^n + d_{n-1} \cdot 6^{n-1} + \dots + d_1 \cdot 6^1 + d_0 \cdot 6^0 This summation reflects the weighted contribution of each digit based on its positional exponent, a fundamental principle of positional numeral systems.[8] The term "senary" originates from the Latin "senarius," meaning "consisting of six," highlighting its foundation in groupings of six units. As a standard radix system, senary utilizes only non-negative digits, distinguishing it from unary (base-1) or binary (base-2) systems in scale and from balanced variants that include negative digits for symmetric representations.[3]Notation and Digits
In the senary numeral system, the digits are limited to 0, 1, 2, 3, 4, and 5, as these suffice to represent all values from 0 to 5 in each positional place.[9] Unlike numeral systems with bases greater than 9, such as hexadecimal, no special symbols beyond the standard Arabic numerals are needed, simplifying representation within the system's constraints.[10] To distinguish senary numbers from those in other bases, particularly decimal, several notation conventions are employed. The most common is the use of a subscript "6" following the number, as in $123_6, which clearly indicates the base.[11] Alternatively, the number may be rendered in boldface for visual clarity in text where subscripts are unavailable. These approaches align with general practices for non-decimal bases, ensuring unambiguous interpretation.[11] Converting a decimal integer to senary involves repeated division by 6, recording the remainders as digits from the least significant (rightmost) to the most significant (leftmost). For instance, to convert the decimal number 10 to senary: divide 10 by 6 to get quotient 1 and remainder 4; then divide 1 by 6 to get quotient 0 and remainder 1. Reading the remainders upward yields $14_6.[12] This process leverages the base-6 structure, where each remainder is a valid digit between 0 and 5.[12] The reverse conversion, from senary to decimal, uses the positional value system: multiply each digit by $6 raised to the power of its position (starting from 0 for the rightmost digit) and sum the results. Continuing the example, $14_6 = 1 \times 6^1 + 4 \times 6^0 = 6 + 4 = 10_{10}.[12] This method directly computes the decimal equivalent by expanding the senary representation according to its place values.[12]Mathematical Properties
Integer Operations
Arithmetic operations on senary integers, which use digits from 0 to 5, follow positional methods analogous to those in base 10 but adjusted for the base-6 structure, where carries and borrows are based on powers of 6.[13] These operations ensure consistent results independent of the numeral system, as the underlying numerical values remain the same.[13]Addition
Addition in senary is performed column by column from right to left, summing digits and any incoming carry; if the sum is 6 or greater, a carry of 1 is propagated to the next column, and the remainder (sum modulo 6) is written in the current position.[14] For instance, consider adding $15_6 and $24_6:- Units column: $5 + 4 = 9_{10} = 1 \times 6 + 3, write 3, carry 1.
- Sixes column: $1 + 2 + 1 = 4 < 6, write 4.
Subtraction
Subtraction proceeds column by column from right to left, borrowing when the top digit is smaller than the bottom; borrowing subtracts 1 from the next higher column and adds 6 to the current digit.[14] A worked example is $52_6 - 34_6:- Units column: $2 < 4, borrow 1 from sixes (5 becomes 4), units become $2 + 6 = 8_{10} - 4 = 4, write 4.
- Sixes column: $4 - 3 = 1, write 1.
Multiplication
Multiplication uses the standard long multiplication algorithm, relying on a base-6 multiplication table derived from repeated addition; partial products are shifted and added according to place values. For single digits, notable patterns include $5_6 \times 5_6 = 41_6, as $5 + 5 + 5 + 5 + 5 = 25_{10} = 4 \times 6 + 1. In multi-digit cases, each digit of the multiplicand is multiplied by the multiplier, with carries applied as in addition. The base-6 table facilitates this, where products exceeding 5 require conversion (e.g., $5_6 \times 4_6 = 32_6).Division
Long division in senary divides the divisor into portions of the dividend, using the base-6 multiplication table to determine quotient digits and subtracting partial products; remainders less than the divisor are handled similarly to base 10. An example is $42_6 \div 2_6:- Divisor $2_6 into first digit 4: quotient digit 2 ($2_6 \times 2_6 = 4_6), subtract to get 0.
- Bring down 2: $2_6 into 2, quotient digit 1 ($2_6 \times 1_6 = 2_6), subtract to get 0 remainder.