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:
\[\begin{align} 101101_{\text{negabinary}} &= 1 \times (-2)^5+0 \times (-2)^4+1 \times (-2)^3+1 \times (-2)^2+0 \times (-2)^1+1 \times (-2)^0 \\ &= -32+0-8+4+0+1 \\ &= -35. \ _\square \end{align}\]
Note: The negative-base system can represent both positive and negative numbers without the use of a minus sign.