High-precision conversion between decimal fractions and binary, octal, hexadecimal, and more
Input Decimal Fraction
Target Base
Result
Conversion Result
Step-by-Step Derivation
Fractional Base Conversion Principle
Decimal fraction → Other base: Multiply by base, extract integer
Other base fraction → Decimal: Expand by place value and sum
Precision control: Default 10 fractional digits
The core method for fractional base conversion is "multiply by base, extract integer." For the integer part, use "repeated division." Together they handle full conversion of numbers with fractional parts.
⚠Not all decimal fractions can be exactly converted to binary. For example, 0.1 is a repeating fraction in binary and the result will be truncated to the specified precision.
What Is Fractional Base Conversion?
Fractional base conversion is the process of converting a number with a fractional part from one base representation to another. Unlike integer conversion, the fractional part uses the "multiply by base, extract integer" method for high-precision conversion.
Multiply-by-Base Method
Decimal fraction to other base: repeatedly multiply by the target base, extract the integer part as a digit, and continue with the remaining fraction until 0 or the desired precision.
Place Value Expansion
Other base fraction to decimal: similar to integers, the nth digit after the decimal point has weight base^(-n). Multiply each digit by its weight and sum.
Precision Issues
A finite decimal fraction may be a repeating fraction in binary, such as 0.1. Floating-point precision errors originate from this limitation.
Stepwise Conversion
Numbers with fractional parts are split into integer and fractional parts, converted separately, then combined into the final result.
💡 Teaching Example: Convert 0.625₁₀ to binary. 0.625×2=1.25→1, remainder 0.25; 0.25×2=0.5→0; 0.5×2=1.0→1. Read integer parts from top to bottom: 101. So 0.625₁₀ = 0.101₂.
Applications
Floating-Point ArithmeticScientific ComputingDigital Signal ProcessingComputer ArchitectureExam Preparation
Frequently Asked Questions
How do you convert a decimal fraction to binary?▼
Use the multiply-by-base method: repeatedly multiply the fractional part by 2, extract the integer part as a binary digit, and continue with the remaining fraction until 0 or desired precision. For example, 0.625×2=1.25→1, 0.25×2=0.5→0, 0.5×2=1.0→1, so 0.625₁₀=0.101₂.
Can all decimal fractions be converted to binary exactly?▼
Not always. Only fractions whose denominator is a power of 2 can be represented exactly in binary. For example, 0.1₁₀ is a repeating fraction in binary (0.000110011...), which can only be approximated. This is the root cause of floating-point precision issues.
How do you convert a binary fraction to decimal?▼
Similar to the integer part, use place value expansion. The first digit after the decimal point has weight 2⁻¹=0.5, the second 2⁻²=0.25, and so on. For example, 0.101₂ = 1×0.5+0×0.25+1×0.125 = 0.625₁₀.
Why is learning fractional base conversion important?▼
Fractional base conversion is key to understanding computer floating-point representation. The IEEE 754 standard uses binary scientific notation for floating-point numbers, involving concepts of precision, rounding, and overflow.
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