Negabinary Converter (Base −2)
Convert whole numbers to and from negabinary, base −2, where negative numbers need no sign. Any negative base from −2 to −16 works too, with the division steps and place values shown.
How to Use
- Choose decimal to negative base or the reverse.
- Set the base: −2 for negabinary, −10 for negadecimal, or any base down to −16.
- Type a whole number, positive or negative, or a string of digits in that base.
- Read the result, the decimal value and the number of digits.
- The drawing shows how the places alternate between adding and subtracting.
Worked Example
10 in negabinary. Divide by −2, keeping each remainder at 0 or 1: 10 = (−2)(−5) + 0; −5 = (−2)(3) + 1; 3 = (−2)(−1) + 1; −1 = (−2)(1) + 1; 1 = (−2)(0) + 1. Reading upwards: 11110. Check with the place values 16, −8, 4, −2, 1: 16 − 8 + 4 − 2 + 0 = 10.
−10 in negabinary works the same way and gives 1010: −8 + 0 − 2 + 0 = −10, with no minus sign anywhere.
The common mistake: a negative remainder. Ordinary integer division in most programming languages gives −5 ÷ −2 = 2 remainder −1. A remainder of −1 is not a negabinary digit; add 2 to make it 1 and add 1 to the quotient, giving 3.
Show Work
Formulas
Bases Below Zero
Vittorio Grünwald described negative bases in 1885, working through negadecimal arithmetic, and the idea was rediscovered several times, including by Aubrey Kempner in 1936 and Zdzisław Pawlak and Alfred Wakulicz in 1957. In Poland it became hardware: the experimental BINEG computer of 1959 and the UMC-1 used base −2.
The attraction is the same as balanced ternary’s: no sign bit and no special rules for negative numbers. The price is that carries can go either way, so addition is more involved, and two’s complement, which needs only an ordinary binary adder, won in practice.
About This Calculator
This converter turns whole numbers of any size into negabinary or any negative base from −2 to −16, and digit strings in those bases back into decimal. It shows each division step with its corrected remainder, the place values that alternate between adding and subtracting, and the ordinary positive-base form for comparison.
Everything runs in your browser; nothing is sent anywhere.
Related calculators: Balanced Ternary Converter, Number Base Converter, and Arbitrary Base Converter.
Frequently Asked Questions
What is negabinary?
Base −2. The place values are 1, −2, 4, −8, 16, −32 and so on, alternating in sign, and the digits are just 0 and 1. Every integer, positive or negative, has one representation with no minus sign: 10 is 11110 (16 − 8 + 4 − 2) and −10 is 1010 (−8 − 2).
How do I convert a number to negabinary?
Divide by −2 repeatedly, but always choose a remainder of 0 or 1: if the remainder comes out negative, add 2 to it and 1 to the quotient. 10 = (−2)(−5) + 0; −5 = (−2)(3) + 1; 3 = (−2)(−1) + 1; −1 = (−2)(1) + 1; 1 = (−2)(0) + 1. Reading the remainders upwards gives 11110.
How can you tell if a negabinary number is negative?
By its length: positive numbers have an odd number of digits (their top digit is at an even, positive place) and negative numbers an even number. 3 = 111 has three digits; −1 = 11 has two.
Is negabinary ever used?
Rarely in practice, but it was. The Polish computers BINEG (1959) and UMC-1 used negabinary arithmetic, and it appears in algorithms, puzzles and courses as an example of a base that needs no sign. Negadecimal (base −10) was described by Vittorio Grünwald in 1885.
Can bases be negative and also fractional?
Yes, though they get stranger. Negative bases work for any integer below −1. Bases such as the golden ratio (phinary) or complex bases such as 2i (quater-imaginary, proposed by Donald Knuth) can also represent every number of their kind without a sign.
How do I use the Negabinary Converter (Base −2)?
Just type or paste your value. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Coursework
10 = 11110 and −10 = 1010 in base −2, with each division shown.
Negative numbers without a sign
−1 = 11, −2 = 10 and −3 = 1101 in negabinary.
Negadecimal
In base −10, 10 is 190 (100 − 90) and −1 is 19.
Puzzles and programming
Check answers to the classic “convert to base −2” exercise.
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