Binary Fraction Calculator

Convert decimal fractions to binary and binary fractions to decimal with bit positions and place values.

Select Conversion Mode

Result

Binary Representation
1101.101
binary
Integer Part1101
Fractional Part.101
Decimal Value13.625

Step-by-Step Conversion

13.625 (decimal) = 1101.101 (binary)

Understanding Binary Fractions

Binary fractions work similarly to decimal fractions, but use powers of 2 instead of powers of 10. In the decimal system, digits to the right of the decimal point represent 1/10, 1/100, 1/1000, etc. In binary, digits to the right of the binary point represent 1/2, 1/4, 1/8, 1/16, and so on.

Binary Place Values

Integer Positions

Positions to the left of the binary point represent increasing powers of 2.

2^0=1, 2^1=2, 2^2=4, 2^3=8 ...

Fractional Positions

Positions to the right represent negative powers of 2.

2^-1=0.5, 2^-2=0.25, 2^-3=0.125

Decimal to Binary (Integer)

Repeatedly divide by 2, collecting remainders from bottom to top.

13 / 2 = 6 R1, 6 / 2 = 3 R0 ...

Decimal to Binary (Fraction)

Repeatedly multiply by 2, collecting integer parts from top to bottom.

0.625 x 2 = 1.25, 0.25 x 2 = 0.5 ...

Converting Decimal Fractions to Binary

To convert a decimal fraction to binary, you need to handle the integer and fractional parts separately:

  1. Integer part: Divide by 2 repeatedly, recording the remainder at each step. Read the remainders from bottom to top.
  2. Fractional part: Multiply by 2 repeatedly. At each step, the integer part of the result becomes the next binary digit. Continue with the fractional part of the result.
  3. Combine the two parts with a binary point (period) separator.

Repeating Binary Fractions

Some decimal fractions that terminate (like 0.1 in decimal) become repeating fractions in binary. For example, decimal 0.1 in binary is 0.0001100110011... (repeating). This is why floating-point arithmetic in computers can have precision issues -- many common decimal fractions cannot be exactly represented in binary.

Converting Binary Fractions to Decimal

To convert a binary fraction to decimal, multiply each bit by its corresponding power of 2 and sum the results. For the integer part, use positive powers (2^0, 2^1, 2^2, ...) from right to left. For the fractional part, use negative powers (2^-1, 2^-2, 2^-3, ...) from left to right.

IEEE 754 Floating Point

Modern computers represent fractional numbers using the IEEE 754 standard, which stores a number as a sign bit, exponent, and mantissa (significand) in binary. Understanding binary fractions is essential for working with floating-point arithmetic, debugging precision issues, and understanding how numerical computation works at the hardware level.

Common Binary Fractions

  • 0.5 (decimal) = 0.1 (binary)
  • 0.25 (decimal) = 0.01 (binary)
  • 0.75 (decimal) = 0.11 (binary)
  • 0.125 (decimal) = 0.001 (binary)
  • 0.375 (decimal) = 0.011 (binary)
  • 0.1 (decimal) = 0.000110011... (binary, repeating)