Binary Fraction Converter

How Binary Fractions Work

In binary (base 2), the digit positions to the left of the binary point represent positive powers of 2 (1, 2, 4, 8, ...) and positions to the right represent negative powers of 2 (1/2, 1/4, 1/8, 1/16, ...). This is directly analogous to how decimal fractions work, except the base is 2 instead of 10.

For example, the binary number 110.101 means:

• 1 x 2² = 4
• 1 x 2¹ = 2
• 0 x 2⁰ = 0
• 1 x 2^-1 = 0.5
• 0 x 2^-2 = 0
• 1 x 2^-3 = 0.125

Total: 4 + 2 + 0 + 0.5 + 0 + 0.125 = 6.625

Converting Decimal Fractions to Binary

The integer part of a decimal number is converted to binary using the standard repeated division by 2 method. The fractional part is converted separately using repeated multiplication by 2:

1. Multiply the fractional part by 2.
2. The integer part of the result (0 or 1) becomes the next binary digit after the point.
3. Keep only the new fractional part and repeat.
4. Stop when the fractional part reaches 0 (exact result) or you have reached your required precision.

Worked Example: 6.625 to Binary

Integer part 6: 6 / 2 = 3 remainder 0, 3 / 2 = 1 remainder 1, 1 / 2 = 0 remainder 1. Reading remainders upward: 110.

Fractional part 0.625:

Binary fraction: .101. Combined result: 110.101

When Exact Conversion is Not Possible

Only decimal fractions whose denominator is a power of 2 convert exactly to binary. Common exact examples include 0.5 (= 1/2 = 0.1 in binary), 0.25 (= 1/4 = 0.01), 0.125 (= 1/8 = 0.001), and 0.75 (= 3/4 = 0.11). Fractions such as 0.1, 0.2, and 0.3 produce infinitely repeating binary sequences. This converter caps output at 32 fractional bits and notes when the result is approximate.

Common Reference Values

Related Calculators

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• Binary Converter: convert whole integers between binary, decimal, hex, and octal.
• Decimal to Binary Calculator: convert whole decimal integers to binary.
• Binary, Hex, Octal and Decimal Converter: multi-base integer converter.
• Decimal to Fraction Calculator: express a decimal as a simplified fraction.

Method: Decimal integer parts are converted to binary by repeated division by 2 (collecting remainders). Decimal fractional parts are converted by repeated multiplication by 2 (collecting integer parts). Binary numbers are converted to decimal using positional weights: each bit at position n after the binary point contributes bit x 2^-n. The converter applies up to 32 fractional bits and flags results that are truncated rather than exact.

This converter is provided for educational and practical use. Repeating binary fractions are shown to the selected precision and flagged as approximate. For integer-only binary conversions, use the Binary Converter or Decimal to Binary Calculator.