Negative Integer Number Base

In a standard positional numeral system, the base $b$ is a positive integer, and $b$ different numerals are used to represent all non-negative integers. Each numeral represents one of the values $0, 1, 2, \ldots$ up to $b-1,$ but the value also depends on the position of the digit in a number. The value of a digit string $d_3d_2d_1d_0$ in base $b$ is given by the polynomial form

$(d_3 \times b^3)+(d_2 \times b^2)+(d_1 \times b)+(d_0).$

The numbers written in superscript represent the powers of the base used. For example, in hexadecimal $(b=16),$ using $A=10, B=11, \ldots,$ the digit string $2C5A$ means

$(2 \times 16^3)+(12 \times 16^2)+(5 \times 16)+(10).$

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 $b$ is equal to $-r$ for some positive integer $r \,(r \geq 2).$ Denoting the base as $-r,$ every integer $a$ can be written uniquely as

$a = \sum_{i=0}^{n}d_{i}(-r)^{i},$

where each digit $d_k$ is an integer from $0$ to $r - 1$ and the leading digit $d_n$ is $> 0$ (unless $n=0$ ). The base $-r$ expansion of $a$ is then given by the string $d_n d_{n-1} \ldots d_1 d_0.$ 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 $-r$ as in base $r.$ For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly,

$81=2^6+2^4+2^0=(-2)^6+(-2)^4+(-2)^0$

and is represented by 1010001 in binary and 1010001 in negabinary.

Express the decimal number 119 as a negaternary $(\text{base}=-3)$ number.

The base $-r$ expansion of a number can be found by repeated division by $-r,$ recording the non-negative remainders of $0,1,\ldots,r-1,$ and concatenating those remainders, starting with the last. Note that if $\frac ab =c$ with remainder $d,$ then $bc+d=a.$ For this example, in negaternary

$\begin{array}{rrrl} \frac{119}{-3} &= &-39 &\text{ remainder } 2 \\\\ \frac{-39}{-3} &= &13 &\text{ remainder } 0 \\\\ \frac{13}{-3} &= &-4 &\text{ remainder } 1 \\\\ \frac{-4}{-3} &= &2 &\text{ remainder } 2 \\\\ \frac{2}{-3} &= &0 &\text{ remainder } 2. \end{array}$

Reading the remainders backward, we obtain the negaternary expression of $119,$ which is $22102. \ _\square$

Express the negabinary $(\text{base}=-2)$ number 101101 as a decimal number.

We can convert the negabinary number 101101 into a decimal number as follows:

$\begin{aligned} 101101_{\text{negabinary}} &= 1 \times (-2)^5+0 \times (-2)^4+1 \times (-2)^3+1 \times (-2)^2+0 \times (-2)^1+1 \times (-2)^0 \\ &= -32+0-8+4+0+1 \\ &= -35. \ _\square \end{aligned}$

Note: The negative-base system can represent both positive and negative numbers without the use of a minus sign.