Decimal to Binary Converter — Free Online Tool
Enter a decimal fraction below to instantly convert it to its binary representation with a detailed step-by-step breakdown.
Binary Result
Step-by-Step Solution
Visual Representation
How to Convert Decimal Fractions to Binary
To convert a decimal number with a fractional part to binary, you split the number into its integer and fractional parts and convert them separately.
Steps
- Separate the number into the integer part and the fractional part.
- Convert the integer part to binary by repeatedly dividing by 2 and recording the remainders.
- Convert the fractional part to binary by repeatedly multiplying by 2 and recording the integer part of the result.
- Combine the two binary parts with a decimal (binary) point.
Example
10.25 → 1010.01₂
integer: 10 = 2 × 5 + 0; 5 = 2 × 2 + 1; 2 = 2 × 1 + 0; 1 = 2 × 0 + 1 → 1010
Fractional Part (Binary):: 0.25 × 2 = 0.5 → 0; 0.5 × 2 = 1 → 1
1010 + .01 = 1010.01₂
Common Conversions
| Decimal | Binary |
|---|---|
| 0.125 | 0.001 |
| 0.25 | 0.01 |
| 0.5 | 0.1 |
| 0.75 | 0.11 |
| 1.5 | 1.1 |
Decimal to Hexadecimal
Convert decimal fractions to hex.
Decimal to Octal
Convert decimal fractions to octal.
Decimal to Fraction
Convert decimals to fractions.
Frequently Asked Questions
Common questions about binary number system, base-2, floating point representation
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01 What is the binary number system?
Binary (base-2) uses only the digits 0 and 1. Each position represents a power of 2 rather than a power of 10. For example, the binary number 1010 = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10 in decimal. Binary is the foundation of all digital computing because electronic circuits operate with two states: on (1) and off (0).
02 How do you represent a floating-point number in binary?
Floating-point numbers are represented in binary using the IEEE 754 standard, which allocates specific bits for the sign, exponent, and fraction (or mantissa).
03 Why do some decimal fractions have infinitely repeating binary representations?
A decimal fraction terminates in binary only if its denominator (in lowest terms) is a power of 2. For example, 0.25 = 1/4 = 1/2² converts exactly to 0.01 in binary. But 0.1 = 1/10, and since 10 has a factor of 5 (not a power of 2), it becomes 0.000110011… in binary — repeating forever. This is why 0.1 + 0.2 ≠ 0.3 in most programming languages.
04 What is the difference between binary integer conversion and binary fraction conversion?
For the integer part, repeatedly divide by 2 and collect remainders from bottom to top. For the fractional part, repeatedly multiply by 2 and collect the integer portion (0 or 1) of each result, reading from top to bottom. Example: 0.75 × 2 = 1.5 (take 1); 0.5 × 2 = 1.0 (take 1); stop → 0.11 in binary.
05 Can all decimal numbers be accurately converted to binary?
All decimal integers can be converted to binary exactly. Decimal fractions, however, can only be represented exactly if the denominator is a power of 2. Otherwise the binary expansion repeats indefinitely. In computing, these are stored as approximations using a fixed number of bits (e.g., 32 or 64 bits in IEEE 754 format), which introduces tiny rounding errors called floating-point errors.