Negative base

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A negative base (or negative radix) 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—that is to say, the base $\scriptstyle b$ is equal to $\scriptstyle -r$ for some natural number $\scriptstyle r$ (r ≥ 2).

Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a minus sign (or, in computer representation, a sign bit); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent.

The common names for negative-base positional numeral systems are formed by prefixing nega- to the name of the corresponding positive-base system; for example, negadecimal (base −10) corresponds to decimal (base 10), negaternary (base −3) to ternary (base 3), and negabinary (base −2) to binary (base 2).^[1]

Example

Consider what is meant by the representation 12,243 in the negadecimal system, whose base $\scriptstyle b$ is −10:

multiples of $\scriptstyle b^{4}$ (i.e., 10,000)	multiples of $\scriptstyle b^{3}$ (i.e., −1,000)	multiples of $\scriptstyle b^{2}$ (i.e., 100)	multiples of $\scriptstyle b^{1}$ (i.e., −10)	multiples of $\scriptstyle b^{0}$ (i.e., 1)
1	2	2	4	3

Since 10,000 + (−2,000) + 200 + (−40) + 3 = 8,163, the representation 12,243 in negadecimal notation is equivalent to 8,163 in decimal notation.

History

Negative numerical bases were first considered by Vittorio Grünwald in his work Giornale di Matematiche di Battaglini, published in 1885. Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak and A. Wakulicz in 1959^{[citation needed]}.

Negabinary was implemented in the early Polish computer BINEG, built 1957–59, based on ideæ by Z. Pawlak and A. Lazarkiewicz from the Mathematical Institute in Warsaw.^[2] Implementations since then have been rare.

Notation and use

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 $\scriptstyle d_{k}$ is an integer from 0 to $\scriptstyle r-1$ and the leading digit $\scriptstyle d_{n}$ is $\scriptstyle >0$ (unless $\scriptstyle n=0$ ). The base $\scriptstyle -r$ expansion of $\scriptstyle a$ is then given by the string $\scriptstyle d_{n}d_{{n-1}}\ldots d_{1}d_{0}$ .

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 $\scriptstyle -r$ as in base $r$ . For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly,

$17=2^{4}+2^{0}=(-2)^{4}+(-2)^{0}$

and is represented by 10001 in binary and 10001 in negabinary.

Some numbers with their expansions in a number of positive and corresponding negative bases are:

Decimal	Negadecimal	Binary	Negabinary	Ternary	Negaternary
−15	25	−1111	110001	−120	1220
:	:	:	:	:	:
−5	15	−101	1111	−12	21
−4	16	−100	1100	−11	22
−3	17	−11	1101	−10	10
−2	18	−10	10	−2	11
−1	19	−1	11	−1	12
0	0	0	0	0	0
1	1	1	1	1	1
2	2	10	110	2	2
3	3	11	111	10	120
4	4	100	100	11	121
5	5	101	101	12	122
6	6	110	11010	20	110
7	7	111	11011	21	111
8	8	1000	11000	22	112
9	9	1001	11001	100	100
10	190	1010	11110	101	101
11	191	1011	11111	102	102
12	192	1100	11100	110	220
13	193	1101	11101	111	221
14	194	1110	10010	112	222
15	195	1111	10011	120	210
16	195	10000	10000	121	211
17	197	10001	10001	122	212

Note that the base $\scriptstyle -r$ expansions of negative integers have an even number of digits, while the base $\scriptstyle -r$ expansions of the non-negative integers have an odd number of digits.

Calculation

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

${\begin{aligned}146&~/~-3=&-48,&~{\mbox{remainder}}~2\\-48&~/~-3=&16,&~{\mbox{remainder}}~0\\16&~/~-3=&-5,&~{\mbox{remainder}}~1\\-5&~/~-3=&2,&~{\mbox{remainder}}~1\\2&~/~-3=&0,&~{\mbox{remainder}}~2\\\end{aligned}}$

Therefore, the negaternary expansion of 146 is 21,102.

Note that in most programming languages, the result (in integer arithmetic) of dividing a negative number by a negative number is rounded towards 0, usually leaving a negative remainder. In such a case we have $\scriptstyle a=(-r)c+d=(-r)c+d-r+r=(-r)(c+1)+(d+r)$ . Because $\scriptstyle |d|<r$ , $\scriptstyle (d+r)$ is the positive remainder. Therefore, to get the correct result in such case, computer implementations of the above algorithm should add 1 and $r$ to the quotient and remainder respectively (shown below in the Python programming language):

def negaternary(i):
    digits = ''
    if not i:
        digits = '0'
    else:
        while i != 0:
            i, remainder = divmod(i, -3)
            if remainder < 0:
                i, remainder = i + 1, remainder + 3
            digits = str(remainder)+ digits
    return digits

C# implementation:

static string negatenary(int value)
{
	string result = string.Empty;
 
	while (value != 0)
	{
		int remainder = value % -3;
		value = value / -3;
 
		if (remainder < 0)
		{
			remainder += 3;
			value += 1;
		}
 
		result = remainder.ToString() + result;
	}
 
	return result;
}

Common Lisp implementation:

(defun negaternary (i)
  (if (zerop i)
      "0"
      (let ((digits "")
	    (rem 0))
	(loop while (not (zerop i)) do
	     (progn
	       (multiple-value-setq (i rem) (truncate i -3))
	       (when (minusp rem)
	         (incf i)
	         (incf rem 3))
	       (setf digits (concatenate 'string (write-to-string rem) digits))))
	digits)))

Arithmetic operations

The following describes the arithmetic operations for negabinary; calculations in larger bases are similar.

Addition

To add two negabinary numbers, start with a carry of 0, and, starting from the least significant bits, add the bits of the two numbers plus the carry. The resulting number is then looked up in the following table to get the bit to write down as result, and the next carry:

Number	Bit	Carry	Comment
−2	0	1	−2 occurs only during subtraction.
−1	1	1
0	0	0
1	1	0
2	0	−1
3	1	−1	3 occurs only during addition.

The second row of this table, for instance, expresses the fact that −1 = 1 + 1 × −2; the fifth row says 2 = 0 + −1 × −2; etc.

As an example, to add 1010101 (1 + 4 + 16 + 64 = 85) and 1110100 (4 + 16 − 32 + 64 = 52),

carry:          1 −1  0 −1  1 −1  0  0  0
first number:         1  0  1  0  1  0  1
second number:        1  1  1  0  1  0  0 +
               --------------------------
number:         1 −1  2  0  3 −1  2  0  1
bit (result):   1  1  0  0  1  1  0  0  1
carry:          0  1 −1  0 −1  1 −1  0  0

so the result is 110011001 (1 − 8 + 16 − 128 + 256 = 137).

Another Method

While adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit. Consider same example as above

extra carry:       1  1  0  1  0  0  0     
carry:          1  0  1  1  0  1  0  0  0
first number:         1  0  1  0  1  0  1
second number:        1  1  1  0  1  0  0 +
               --------------------------
Answer:         1  1  0  0  1  1  0  0  1

Subtraction

To subtract, multiply each bit of the second number by −1, and add the numbers, using the same table as above.

As an example, to compute 1101001 (1 − 8 − 32 + 64 = 25) minus 1110100 (4 + 16 − 32 + 64 = 52),

carry:          0  1 −1  1  0  0  0
first number:   1  1  0  1  0  0  1
second number: −1 −1 −1  0 −1  0  0 +
               --------------------
number:         0  1 −2  2 −1  0  1
bit (result):   0  1  0  0  1  0  1
carry:          0  0  1 −1  1  0  0

so the result is 100101 (1 + 4 −32 = −27).

To negate a number, compute 0 minus the number.

Multiplication and division

Shifting to the left multiplies by −2, shifting to the right divides by −2.

To multiply, multiply like normal decimal or binary numbers, but using the negabinary rules for adding the carry, when adding the numbers.

first number:                   1  1  1  0  1  1  0
second number:                  1  0  1  1  0  1  1 *
              -------------------------------------
                                1  1  1  0  1  1  0
                             1  1  1  0  1  1  0

                       1  1  1  0  1  1  0
                    1  1  1  0  1  1  0

              1  1  1  0  1  1  0                   +
              -------------------------------------
carry:        0 −1  0 −1 −1 −1 −1 −1  0 −1  0  0
number:       1  0  2  1  2  2  2  3  2  0  2  1  0
bit (result): 1  0  0  1  0  0  0  1  0  0  0  1  0
carry:           0 −1  0 −1 −1 −1 −1 −1  0 −1  0  0

For each column, add carry to number, and divide the sum by −2, to get the new carry, and the resulting bit as the remainder.

Fractional numbers

Base $\scriptstyle -r$ representation may of course be carried beyond the radix point, allowing the representation of non-integral numbers.

As with positive-base systems, terminating representations correspond to fractions where the denominator is a power of the base; repeating representations correspond to other rationals, and for the same reason.

Non-unique representations

Unlike positive-base systems, where integers and terminating fractions have non-unique representations (for example, in decimal 0.999… = 1) in negative-base systems the integers have only a single representation. However, there do exist rationals with non-unique representations; for example, in negaternary,

$0.(02)\ldots _{{(-3)}}={\frac {1}{4}}=1.(20)\ldots _{{(-3)}}$ .

Such non-unique representations can be found by considering the largest and smallest possible representations with integral parts 0 and 1 respectively, and then noting that they are equal. (Indeed, this works with any integral-base system.) The rationals thus non-uniquely expressible are those of form

${\frac {ar+1}{b(r+1)}}$ .

Imaginary base

Main article: Complex base systems

Just as using a negative base allows the representation of negative numbers without an explicit negative sign, using an imaginary base allows the representation of Gaussian integers. Donald Knuth proposed the quater-imaginary base (base 2i) in 1955.^[3]

Imaginary-base arithmetic is not much different from negative-base arithmetic, since an imaginary-base number may be considered as the interleave of its real and imaginary parts; using INTERCAL-72 notation,

x_(2i) + (2i)y_(2i) = x_(2i) ¢ y_(2i).

Notes

↑ Knuth 1998 and Weisstein each refer to the negadecimal system. In the index Knuth 1998 refers to the negabinary system, as does Weisstein. The negaternary system is discussed briefly in Petkovšek, Marko (1990), "Ambiguous numbers are dense", The American Mathematical Monthly 97 (5): 408–411, doi:10.2307/2324393, ISSN 0002-9890, MR 1048915 .
↑ Marczynski, R. W., "The First Seven Years of Polish Computing", IEEE Annals of the History of Computing, Vol. 2, No 1, January 1980
↑ D. Knuth. The Art of Computer Programming. Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems"

References

Vittorio Grünwald. Giornale di Matematiche di Battaglini (1885), 203-221, 367
A. J. Kempner. (1936), 610-617
Z. Pawlak and A. Wakulicz Bulletin de l'Academie Polonaise des Scienses, Classe III, 5 (1957), 233-236; Serie des sciences techniques 7 (1959), 713-721
L. Wadel IRE Transactions EC-6 1957, 123
N. M. Blachman, Communications of the ACM (1961), 257
IEEE Transactions 1963, 274-276
Computer Design May 1967, 52-63
R. W. Marczynski, Annotated History of Computing, 1980, 37-48
Knuth, Donald (1998), The Art of Computer Programming, Volume 2 (3rd ed.), pp. 204–205 .
Weisstein, Eric W., "Negabinary", MathWorld.
Weisstein, Eric W., "Negadecimal", MathWorld.

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