Balanced numeral systems

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Balanced numeral systems are place-value numeral systems with an odd base $2k+1,\,k\geq 1,$ such that $k$ digits are "negative digits," one digit is 0, and $k$ digits are "positive digits." With a balanced numeral system

• we do not need a minus sign to represent negative numbers (they have a leading "negative digit");
• negation amounts to negating each "digit";
• truncation is rounding (this has the advantage of minimizing truncation error accumulation in calculations).

Balanced ternary numeral system

The smallest base thus gives the balanced ternary numeral system, where the "balanced ternary digits," which may be represented as

{ 1, 0, 1 } or { −, 0, + } or { ↓, |, ↑ } or { ∨, o, ∧ },

have value −1, 0, and 1 respectively.

Balanced quinary numeral system

The second smallest base gives the balanced quinary numeral system, where the "balanced quinary digits," which may be represented as

{ 2, 1, 0, 1, 2 },

have value −2, −1, 0, 1 and 2 respectively.