Binary Fraction to Decimal Converter
Convert binary fractions to decimal instantly. Enter a binary value such as 101.101 or 0.011 to calculate its exact decimal value and see how every bit contributes through positive and negative powers of two.
What Is a Binary Fraction?
A binary fraction is a base-2 number containing digits on both sides of a binary point. The digits to the left represent positive powers of 2, while the digits to the right represent negative powers of 2.
Binary Fraction Place Values
Binary positions immediately to the right of the binary point represent successively smaller powers of two.
| Binary Position | Power of 2 | Decimal Value |
|---|---|---|
| First fractional bit | 2⁻¹ |
0.5 |
| Second fractional bit | 2⁻² |
0.25 |
| Third fractional bit | 2⁻³ |
0.125 |
| Fourth fractional bit | 2⁻⁴ |
0.0625 |
| Fifth fractional bit | 2⁻⁵ |
0.03125 |
| Sixth fractional bit | 2⁻⁶ |
0.015625 |
How to Convert Binary Fraction to Decimal
1. Separate the two parts
Split the binary value into the integer portion and fractional portion at the binary point.
2. Convert the integer portion
Use powers of 2 beginning with 2⁰ at the rightmost integer bit.
3. Start fractional powers at 2⁻¹
The first bit after the binary point has value one-half.
4. Continue with smaller powers
Each additional fractional position divides the previous place value by 2.
5. Add active place values
Only positions containing a binary 1 contribute to the total.
6. Combine both parts
Add the integer value and fractional value to get the final decimal number.
Example: Convert 0.101 Binary to Decimal
Example: Convert 101.101 Binary to Decimal
Example: Convert 1101.011 Binary to Decimal
Why Does the First Fraction Bit Equal 0.5?
The first position to the right of the binary point represents
2⁻¹.
Each movement one place to the right divides the positional value by 2.
Binary Fraction Conversion Formula
For a binary number with bits on both sides of the binary point, the decimal value can be described as the sum of each bit multiplied by its corresponding power of two.
Binary Fractions vs Decimal Fractions
| Feature | Binary | Decimal |
|---|---|---|
| Base | 2 | 10 |
| Digits | 0, 1 | 0–9 |
| First fractional place | 1/2 | 1/10 |
| Second fractional place | 1/4 | 1/100 |
| Third fractional place | 1/8 | 1/1000 |
Binary Fractions That Convert Exactly to Decimal
Every finite binary fraction has an exact terminating decimal representation because its denominator is a power of two.
Why Some Decimal Fractions Repeat in Binary
The reverse conversion is different. Some finite decimal fractions cannot be represented with a finite number of binary fractional bits.
For example, decimal 0.1 has a repeating binary expansion. This is
one reason floating-point arithmetic can contain small representation errors in
computer systems.
Can a Binary Fraction Have No Integer Part?
Yes. A value such as .101 can be interpreted as
0.101. For clarity, however, writing the leading zero is recommended.
Can a Binary Number Have No Fractional Part?
Yes. A regular binary integer such as 1011 is also valid input for
this converter.
When there is no binary point, the fractional part is simply zero.
Binary Fraction Examples
| Binary | Decimal |
|---|---|
0.1 |
0.5 |
0.01 |
0.25 |
0.001 |
0.125 |
0.11 |
0.75 |
0.101 |
0.625 |
1.1 |
1.5 |
10.01 |
2.25 |
101.101 |
5.625 |
1101.011 |
13.375 |
Binary Fractions in Computing
Binary fractions are fundamental to computer arithmetic. Fixed-point and floating-point systems both use binary positional values to represent numbers that are not whole integers.
Understanding powers such as 2⁻¹, 2⁻², and
2⁻³ helps explain how computers approximate and store real-world
numeric values.
Common Binary Fraction Conversion Mistakes
Using decimal place values
Binary fractional positions are halves, quarters, eighths and so on—not tenths and hundredths.
Starting at 2⁰ after the point
The first fractional position is 2⁻¹, not 2⁰.
Reading the fraction backward
Fractional powers become smaller as you move farther to the right.
Using invalid binary digits
Only 0 and 1 are allowed, along with at most one binary point.
Related BinaryCon Tools
Binary Fraction to Decimal Converter FAQs
How do you convert a binary fraction to decimal?
What is 0.1 in binary as decimal?
0.1₂ = 0.5₁₀.
What is 0.01 binary in decimal?
0.01₂ = 0.25₁₀.
What is 0.001 binary in decimal?
0.001₂ = 0.125₁₀.
What is 0.101 binary in decimal?
0.101₂ = 0.625₁₀.
What is 101.101 binary in decimal?
101.101₂ = 5.625₁₀.
What does the first bit after the binary point represent?
2⁻¹, which equals decimal 0.5.
What does the second fractional binary bit represent?
2⁻², equal to decimal 0.25.
What does the third fractional binary bit represent?
2⁻³, equal to decimal 0.125.
Can I enter a binary integer without a point?
Can a binary fraction start with a point?
.101 can be treated as 0.101, though
including the leading zero is clearer.