Signed digit representation

A signed base (also signed digit representation or signed digit base)is a variant of a positional number system where, while the radix remains a natural number, the digits are changed to represent both negative and positive values.

Properties
Signed bases are similar to negative bases in that they can express both positive and negative values without a sign. The value of a number can be deduced the same way as any positional number system:

Construction
A signed digit base is constructed by picking the mandatory digits of -1, 0 and 1, and subsequently adding any amount of positive or negative further digits. For example, a signed base with radix 5 can have digit values of {-3, -2, -1, 0, 1}, {-2, -1, 0, 1, 2} (Balanced quinary), {-1, 0, 1, 2, 3}.

For every radix, odd or even, there can be a total of radix - 2 total signed digit number systems. This is because in every system, there must exist digits -1, 0, 1; if 0 doesn't exist, the number system becomes bijective, if -1 doesn't exist, the system turns into a standard positional number system (in other words, negative numbers cannot be represented), and if 1 doesn't exist, positive numbers can no longer be represented (note that this is different from a negative base, as negative bases still have positive digits, it is more equivalent to a direct opposite of a standard numeral system)

It is not possible to construct signed binary, or any signed-digit base 2 system.

Principal signed base
The principal signed base is the most useful signed base for a given radix and is defined as follows:


 * For an odd radix ≥ 3, the principal signed base is the corresponding balanced base. For example, with 7, it will be balanced septenary, with digits {-3, -2, -1, 0, 1, 2, 3}.
 * For an even radix ≥ 4, the principal signed base has the digit formula $$1-\frac{r}{2}$$to $$\frac{r}{2}$$. For example, with 6 the digit values would be {-2, -1, 0, 1, 2, 3}.

Balanced base
A balanced base is a special kind of signed digit base where the amount of negative and positive digits is the same (and zero is also used). Because of this, only balanced bases with odd radices exist. Balanced bases are the principal signed bases for their respective radix.

A balanced base with an odd radix r will have digit values ranging from $$\frac{1-r}{2}$$to $$\frac{r-1}{2}$$. This ensures the uniqueness of representation in each of these systems.