Senary

Senary is a base-6 numeral system. It uses six digits: 0, 1, 2, 3, 4, and 5. When six is reached, a new place value begins to the left, analogous to how base-10 works when ten is reached.

Each place value to the left represents a power of six: 1, 6, 36, 216, and so on. For example, the senary numeral '25' equals two sixes plus five, which in base-10 is seventeen. Similarly, the number ten in base-10 is written as '14' in senary.

Mathematical properties

Six is composite, being the product of the first two prime numbers, two and three. This means that halves and thirds can be expressed as terminating fractions in senary: one half is 0.3 and one third is 0.2. In base-10 systems, one third is a repeating decimal.

Another property is that primes greater than three, when written in senary, always end in 1 or 5. This is because such numbers are not divisible by two or three, so their remainder when divided by six must be one of these two values.

Use and examples

Senary is not widely used in modern arithmetic. However, base-6 counting systems have been documented in spoken languages in Australia and Papua New Guinea. Additionally, a pair of six-sided dice used in games can produce results that may be interpreted using senary notation, though the faces of such dice are typically numbered from 1 rather than from 0.

Relationship to other systems

Senary is directly related to base-36 and base-216 (six squared and six cubed), since these are all powers of six. Base-12 (duodecimal) shares the advantage that both two and three divide evenly into the base, which facilitates division.