Convert fractional decimal numbers to negative bases (like -2, -10) with detailed step-by-step derivation
Input Decimal Number
Negative Base (e.g., -2, -10)
Result
Negative Base Result
Step-by-Step Derivation
Negative Base Conversion Principle
① Integer part: divide by -b, adjust remainder to 0 ≤ r < |b|
② Fractional part: multiply by -b, extract integer part
③ If remainder negative: add |b| to remainder, add 1 to quotient
Negative bases use alternating sign positions. No minus sign needed—negative numbers are represented directly in the digit sequence!
⚠Negative base must be ≤ -2. Base -2 (negabinary) is most common. Fractional conversion may not terminate—limited to 10 digits for practicality.
What Are Negative Bases?
Negative base (negative radix) systems have a negative base like -2, -10. Positions alternate sign: rightmost = b^0 (positive), next = b^1 (negative), then b^2 (positive), etc.
Negabinary (Base -2)
Digits 0 and 1. Positions: ..., (-2)^3, (-2)^2, (-2)^1, (-2)^0. No sign bit needed: -1 = 11, -2 = 10, -3 = 1101, etc.
Integer Conversion
Divide by -b, get remainder. If remainder negative, add |b| and add 1 to quotient. Collect remainders in reverse order.
Fraction Conversion
Multiply fraction by -b, extract integer part (digit). Continue with new fraction. May be non-terminating!
Digit Values
Digits 0 to |b|-1. For base -10: digits 0-9. For base -2: digits 0-1. Remainder always non-negative!
💡 Teaching Example: Convert 0.5 to base -2. 0.5 × -2 = -1.0. Extract -1, but need non-negative digit: -1 + 2 = 1, adjust. So 0.5 in base -2 = 0.(10) repeating or 0.101010...
Applications
MathematicsComputer ScienceNumber TheoryEducationRecreational Math
Frequently Asked Questions
What is a negative base number system?▼
A negative base (or negative radix) is a numeral system with a negative base like -2, -10, etc. Unlike positive bases, digits alternate sign: rightmost is positive, next negative, then positive, etc. No minus sign needed for negative numbers.
How do you convert to negative base?▼
For integers: repeatedly divide by the negative base, make remainder positive (0 ≤ r < |base|) by adjusting quotient. For fractions: repeatedly multiply by negative base, extract integer part, continue until zero or limit reached.
What is negabinary (base -2)?▼
Negabinary uses base -2 with digits 0 and 1. It can represent any integer positive or negative without a sign bit. Example: decimal -1 = 11 in negabinary (1×(-2)^1 + 1×(-2)^0 = -2 + 1 = -1).
Can fractions be represented in negative bases?▼
Yes! Fractional negative bases use digits after a radix point with exponents -1, -2, etc. Note that negative base fractions may be non-terminating even when positive base representations terminate.
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