Negative radix bases have negative numbers as their base. They don't need a separate minus sign for negative numbers. Digits are always non-negative from 0 to |base|-1.
⚠For negative base conversion, ensure remainders stay non-negative. If remainder negative, add |base| to remainder and increment quotient by 1.
What Are Negative Radix Bases?
Negative radix (or negative base) is a positional numeral system with a negative base. It can represent both positive and negative numbers without a separate minus sign, having interesting mathematical properties.
No Minus Sign Needed
Negative bases represent negative numbers without a separate minus sign, using only non-negative digits.
Place Value Expansion
Negative base to decimal: digit × (negative base)^position. For example, base -2: 101 = 1×(-2)²+0×(-2)¹+1×(-2)⁰ = 5.
Non-Negative Remainders
Decimal to negative base: repeatedly divide by base, keep remainder non-negative by adding |base| and incrementing quotient.
Mathematical Interest
Negative bases are studied in computer science and mathematics, with interesting symmetry and representation properties.
💡 Teaching Example: In base -10, 190 = 1×(-10)²+9×(-10)¹+0×(-10)⁰ = 100 - 90 + 0 = 10. Conversely, 10 in base -10: divide 10 by -10, get 0 remainder 10 → add 10 to remainder, increment quotient → 1 with remainder 0. Then divide 1 by -10, get 0 remainder 1. So 10 in decimal = 190 in base -10.
A negative radix (or negative base) has a negative number as its base. For example, negabinary (base -2), negadecimal (base -10). They use digits 0 to |base|-1 and don't need a minus sign for negative numbers.
How do you convert negative base to decimal?▼
Place value expansion: each digit × (negative base)^position, then sum. The rightmost position is 0. For example, in base -2: 101 = 1×(-2)²+0×(-2)¹+1×(-2)⁰ = 5.
How do you convert decimal to negative base?▼
Repeated division: divide by base, keep remainder non-negative. If remainder negative, add |base| to remainder, increment quotient by 1. Collect remainders, read in reverse.
What are the advantages of negative radix?▼
Negative bases represent negative numbers without a separate minus sign. They have theoretical interest in computer science and mathematics, with interesting symmetry properties.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.