12addition

12addition refers to addition performed in the base-12 (duodecimal) number system. In base-12, numbers are written with twelve distinct digits, typically 0–9 plus two additional symbols representing ten and eleven. The two most common conventions for those symbols are T and E, or sometimes X and E, but the key idea is that there are twelve digits in the numeral alphabet. Addition uses the same digit-by-digit method as in base-10, but carries occur once a column reaches the value twelve.

Formal description: To add two base-12 numbers, write them aligned by least significant digit, add corresponding digits from right to left, and carry a value of 1 to the next position whenever a column sum reaches or exceeds twelve. The resulting digit is the remainder modulo twelve. For example, 9 plus 3 equals 10 in base-12, written as 10 (one carry to the next position and a zero in the units place). Another example is 7 plus 8, which equals 15 decimal, or 13 in base-12 (one carry and a remainder of three).

Conversion and properties: Base-12 addition is binary-commutative and associative, and it shares the identity element 0 with decimal addition. To convert a base-12 result to decimal, multiply each digit by twelve raised to the appropriate power and sum. Conversely, decimal numbers can be converted to base-12 by repeated division by twelve.

Context and usage: Base-12 arithmetic has historical interest due to its factorability (divisible by 2, 3, 4, and 6), and it appears in some old measurement systems and discussions of alternative numeration bases. See also duodecimal system and base-n arithmetic.