Binary Fraction Converter

Convert between binary and decimal fraction representations

This calculator converts binary fractions to decimal fractions and vice versa. Binary fractions use base-2 representation while decimal fractions use base-10.

Enter numbers separated by a decimal point. For binary: use only 0s and 1s

Example Conversions

Click on any example to load it into the calculator

Simple Binary Fraction

binaryToDecimal

Convert binary 101.101 to decimal

Type: binaryToDecimal

Value: 101.101

Binary with More Precision

binaryToDecimal

Convert binary 1010.1101 to decimal

Type: binaryToDecimal

Value: 1010.1101

Decimal to Binary

decimalToBinary

Convert decimal 5.625 to binary

Type: decimalToBinary

Value: 5.625

Precision: 8

Complex Decimal Fraction

decimalToBinary

Convert decimal 3.375 to binary

Type: decimalToBinary

Value: 3.375

Precision: 10

Other Titles
Understanding Binary Fraction Converter: A Comprehensive Guide
Master the art of converting between binary and decimal fraction representations

What is a Binary Fraction?

  • Understanding Binary Number System
  • Fractional Representation in Binary
  • Why Binary Fractions Matter
A binary fraction is a way to represent fractional numbers using the binary (base-2) number system, which uses only the digits 0 and 1. Just as decimal fractions use powers of 10, binary fractions use powers of 2.
Understanding Binary Number System
The binary number system is fundamental to computer science and digital electronics. In this system, each digit position represents a power of 2, starting from 2^0 for the rightmost digit before the decimal point.
Fractional Representation in Binary
In binary fractions, digits to the right of the decimal point represent negative powers of 2. The first position after the decimal represents 2^(-1) = 0.5, the second represents 2^(-2) = 0.25, and so on.
Why Binary Fractions Matter
Binary fractions are crucial in computer systems, digital signal processing, and embedded systems. Understanding them helps in optimizing algorithms and understanding how computers handle decimal numbers internally.

Basic Binary Fraction Examples

  • Binary 101.101 = 5.625 in decimal
  • Binary 11.011 = 3.375 in decimal

Step-by-Step Guide to Binary Fraction Conversion

  • Binary to Decimal Conversion
  • Decimal to Binary Conversion
  • Handling Precision and Rounding
Converting between binary and decimal fractions involves understanding positional notation and systematic calculation methods.
Binary to Decimal Conversion
To convert a binary fraction to decimal: 1) Identify each digit's position, 2) Multiply each digit by its corresponding power of 2, 3) Sum all the products. For example, 101.101₂ = 1×2² + 0×2¹ + 1×2⁰ + 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 4 + 0 + 1 + 0.5 + 0 + 0.125 = 5.625₁₀
Decimal to Binary Conversion
To convert decimal to binary: 1) Separate integer and fractional parts, 2) Convert integer part using repeated division by 2, 3) Convert fractional part using repeated multiplication by 2, 4) Combine the results.
Handling Precision and Rounding
Some decimal fractions cannot be exactly represented in binary, leading to repeating binary fractions. In such cases, we must round to a specified precision.

Conversion Process Examples

  • 5.625₁₀ → 101.101₂
  • 0.1₁₀ → 0.000110011...₂ (repeating)

Real-World Applications of Binary Fractions

  • Computer Graphics and Gaming
  • Digital Signal Processing
  • Embedded Systems and IoT
Binary fractions have numerous practical applications in modern technology and computer science.
Computer Graphics and Gaming
In 3D graphics, coordinates and transformations often use fixed-point arithmetic with binary fractions for performance reasons. Game engines use binary fractions for precise positioning and smooth animations.
Digital Signal Processing
DSP algorithms frequently use binary fractions to represent signal amplitudes and filter coefficients. This allows for efficient implementation in digital hardware while maintaining acceptable precision.
Embedded Systems and IoT
Microcontrollers and embedded systems often lack floating-point units, making binary fractions essential for representing fractional values in sensor readings and control algorithms.

Application Examples

  • RGB color values: 0.5 = 128/255
  • Audio sample values: -1.0 to +1.0 range

Common Misconceptions and Correct Methods

  • Precision Limitations
  • Repeating Binary Fractions
  • Conversion Accuracy
Understanding common pitfalls in binary fraction conversion helps avoid errors in practical applications.
Precision Limitations
Many decimal fractions cannot be exactly represented in binary. For example, 0.1 in decimal becomes an infinitely repeating binary fraction. This is why floating-point arithmetic can introduce small errors.
Repeating Binary Fractions
Just as 1/3 = 0.333... in decimal, some fractions repeat in binary. Understanding when this occurs helps in choosing appropriate precision levels for calculations.
Conversion Accuracy
Always verify conversions by converting back to the original base. This helps catch errors and understand the precision limitations of the conversion process.

Common Error Examples

  • 0.1₁₀ ≠ exact binary representation
  • 0.5₁₀ = 0.1₂ (exact)

Mathematical Derivation and Advanced Examples

  • Positional Notation Theory
  • Advanced Conversion Algorithms
  • Error Analysis and Optimization
The mathematical foundation of binary fractions lies in positional notation and power series representation.
Positional Notation Theory
Any number in base b can be expressed as: N = Σ(dᵢ × bⁱ) where dᵢ is the digit at position i and b is the base. For binary fractions, b = 2 and i can be negative for fractional parts.
Advanced Conversion Algorithms
Efficient algorithms for binary-decimal conversion include bit manipulation techniques for integer parts and iterative multiplication/division for fractional parts. These algorithms minimize rounding errors.
Error Analysis and Optimization
Understanding the error propagation in conversions helps in choosing optimal precision levels. IEEE 754 floating-point standard provides guidelines for handling such conversions in computer systems.

Advanced Mathematical Examples

  • π in binary: 11.001001000011111101...
  • e in binary: 10.101101111110000101...