In a standard positional numeral system, the base is a positive integer, and different numerals are used to represent all non-negative integers. Each numeral represents one of the values up to but the value also depends on the position of the digit in a number. The value of a digit string in base is given by the polynomial form
The numbers written in superscript represent the powers of the base used. For example, in hexadecimal using the digit string means
A negative base may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative here, i.e. the base is equal to for some positive integer Denoting the base as every integer can be written uniquely as
where each digit is an integer from to and the leading digit is (unless ). The base expansion of is then given by the string Negative-base systems may thus be compared to signed-digit representations, such as balanced ternary, where the radix is positive but the digits are taken from a partially negative range.
Some numbers have the same representation in base as in base For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly,
and is represented by 1010001 in binary and 1010001 in negabinary.
Express the decimal number 119 as a negaternary number.
The base expansion of a number can be found by repeated division by recording the non-negative remainders of and concatenating those remainders, starting with the last. Note that if with remainder then For this example, in negaternary
Reading the remainders backward, we obtain the negaternary expression of which is
Express the negabinary number 101101 as a decimal number.
We can convert the negabinary number 101101 into a decimal number as follows:
Note: The negative-base system can represent both positive and negative numbers without the use of a minus sign.