Negabinary converter

A negabinary converter rewrites a signed whole number between everyday base-10 and base −2, the positional system whose radix is negative two and whose only digits are zero and one. Because successive powers of −2 alternate between positive and negative values, every integer—positive, negative, or zero—has a unique representation without a dedicated sign bit or two’s-complement width. Students of computer arithmetic, puzzle authors, and engineers exploring exotic number systems can paste a decimal value or a bit string, choose encode or decode, and receive a deterministic result with the cleaned decimal integer, the negabinary string, the digit count, and the fixed alphabet. Invalid digits, empty input, and non-integers fail with a clear invalid-input error instead of a silent wrong answer.

How negabinary (base −2) encoding and decoding work

Negabinary is positional notation with radix minus two. Each place is a power of −2: the units place is (−2) to the zero, which is one; the next place is −2; then four; then −8; then sixteen; and so on, alternating in sign as the exponent grows. The only digit symbols are zero and one—the same alphabet as ordinary binary—but the alternating place signs mean a string of ones and zeros can evaluate to a negative total without any leading minus character.

Classic checks include two encoding to one-one-zero (because four minus two equals two), four encoding to one-zero-zero, and negative one encoding to one-one (because minus two plus one equals minus one). To encode a signed decimal integer, repeatedly take the remainder when dividing by −2 and adjust so every remainder is non-negative: if the remainder would be −1 under truncating division, add two to the remainder and carry one into the next quotient.

Digits are collected least-significant first and reversed for the usual left-to-right string, with leading zeros stripped except for the number zero itself. To decode, walk the string from left to right with a Horner-style accumulation: multiply the running total by −2 and add the next digit. An optional 0nb or 0NB prefix is stripped the same way binary tools accept 0b, so scripts can tag base-minus-two literals clearly.

An invalid character such as a space, plus or minus sign inside the bit string, digit two or higher, letter, or punctuation mark aborts with invalid input naming the bad character and its position. Absolute values must stay inside JavaScript’s safe integer range so every digit remains exact when the result is serialized. If you only need ordinary positive-base binary or two’s-complement fixed-width encodings, those are separate search intents with their own landings rather than stretching this signed base-minus-two converter past its honest scope.

Where a negabinary converter earns its keep

Computer science courses introduce negative bases to show that sign bits and two’s complement are design choices rather than laws of nature: once students convert a few negative integers by hand, they see how alternating place values absorb the sign into the digits themselves. Exam-prep generators and LMS plugins need a deterministic oracle that marks wrong an ordinary binary decode of a string that was meant as negabinary, and that never invents a leading minus on a valid bit string.

Game designers invent alien numbering systems with no sign glyph and need deterministic conversion for lore tools and save formats. Treat ordinary binary or hexadecimal dumps, fixed-width two’s complement, Gray code, and multi-base radix tables that stop at thirty-six as separate search intents with their own landings rather than stretching this integer negabinary converter past its honest scope.