The division trick: divide by −2 but always keep the remainder 0 or 1 — never −1. So −13 ÷ −2 is 7 remainder 1 (because −2 × 7 + 1 = −13), not 6 remainder −1. That single rule is what lets negative numbers fall out of the algorithm with no sign attached. Fractions here are a half-and-half world too: 0.5 is written 1.1 — one, plus negative-a-half.
Base −2 uses only the digits 0 and 1, but its place values alternate: 1, −2, 4, −8, 16… The astonishing result: every number, positive or negative, gets its own bit pattern. No minus sign exists — or is needed.
Each digit's contribution — even places pull the value up, odd places drag it down. The final number is wherever the tug of war settles. Shown for up to 16 places.
A quick sign-reading trick: the leftmost 1 decides everything. If it sits in an even position (1, 4, 16, 64…), the number is positive; in an odd position (−2, −8, −32…), negative. The remaining digits can only tug the value around within that place's reach — never far enough to flip the sign.
Digits 0 and 1 with an optional fraction point. There's no minus sign to type — whether the value comes out negative is decided entirely by which places hold the 1s.
The full rulebook for a column sum s (digits plus incoming carry): if s is 0 or 1, write it, carry nothing. If s is 2 or 3, write s − 2 and carry −1. If s is −1, write 1 and carry +1. Subtraction never needs its own circuitry: since a left-shift multiplies by −2, the negation of B is simply B plus B-shifted — so A − B is just two additions. Hardware designers loved that.
Addition in negabinary is famous for its strange carries: 1 + 1 doesn't carry a 1 — it carries a −1, which can itself trigger a carry of +1 later. Watch it happen column by column.
Positive and negative numbers, shoulder to shoulder, not a sign among them.
Hover any entry to see how it's built.
The representable range for each digit count — it lurches alternately downward and upward as each new place opens either negative or positive territory. Binary with a sign bit would be symmetric; negabinary is deliberately lopsided.
Every integer from −16 to 16, each with its own unsigned bit pattern. Notice how counting upward makes the patterns jump around — that lurching walk is the price of banishing the minus sign.
Take the 6-digit bit patterns in ordinary counting order — 000000, 000001, 000010… — and plot the value each one means in base −2. Binary would draw a straight ramp; negabinary draws this sawtooth, visiting positives and negatives in leaps.
Negative bases aren't a thought experiment — real computers were designed and shipped around base −2. Pick a topic for the story.