Decimal Fraction to Binary
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
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How Decimal to Binary Conversion Works
Live Conversion Demo
Enter Your Decimal
Type any decimal number into the calculator. Supports standard decimals, repeating patterns, and measurement values.
Instant Conversion
Our algorithm converts your decimal to a fraction instantly, finding the greatest common divisor to simplify it to the lowest terms.
Get Results with Steps
See your fraction in simplified form, as a mixed number, and with a complete step-by-step breakdown of the conversion process.
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
Convert 10.25 to binary:
1. Integer part (10): 10 ÷ 2 = 5 rem 0, 5 ÷ 2 = 2 rem 1, 2 ÷ 2 = 1 rem 0, 1 ÷ 2 = 0 rem 1 → 1010
2. Fractional part (0.25): 0.25 × 2 = 0.50 (int 0), 0.50 × 2 = 1.00 (int 1) → .01
3. Combine: 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?
The binary number system, or base-2, uses only two digits: 0 and 1. It is the foundation of all modern computer systems and digital electronics.
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?
In base-2, only fractions whose denominator is a power of 2 can be expressed exactly. Others, like 0.1, result in repeating binary fractions, which leads to small rounding errors in computers.
04 What is the difference between binary integer conversion and binary fraction conversion?
Integers are converted by repeatedly dividing by 2 and taking remainders, while fractions are converted by repeatedly multiplying by 2 and taking the integer part of the result.
05 Can all decimal numbers be accurately converted to binary?
Yes, but some require an infinite number of bits to represent exactly. In computing, these are often truncated, causing minor precision issues known as floating-point errors.