# Binary Fraction Calculator

URL: https://speedycalculator.com/calculators/math/binary-fraction/
Last updated: 2026-10-30

> The binary fraction calculator converts decimal fractions to binary (with the repeated-doubling steps) and binary fractions back to decimal, with a chosen number of fractional bits.

## Inputs

- Conversion
- Number
- Fractional bits (decimal → binary)

## Results

- Result
- Exactness
- Steps

## Formula

```
0.b₁b₂b₃…₂ = b₁·2⁻¹ + b₂·2⁻² + b₃·2⁻³ + …
Decimal → binary: multiply the fraction by 2, the integer part is the next bit
```

## Explanation

Many simple decimal fractions, such as 0.1, have no finite binary form – they repeat forever. That is why 0.1 + 0.2 ≠ 0.3 in most programming languages, and why money is stored in cents.

### How to use the Binary Fraction Calculator

- **Conversion**
- **Number**
- **Fractional bits (decimal → binary)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose the direction. Decimal to binary multiplies the fractional part by 2 repeatedly, taking the integer part as the next bit, up to the chosen number of bits.

## Worked example

Inputs — Conversion: Decimal → binary; Number: 0.1; Fractional bits (decimal → binary): 12.

Results — **Result = 0.000110011001**; Exactness = Truncated after 12 fractional bits (the exact expansion is longer); Steps = 0.1 × 2 = 0.2 → 0
0.2 × 2 = 0.4 → 0
0.4 × 2 = 0.8 → 0
0.8 × 2 = 1.6 → 1
0.6 × 2 = 1.2 → 1
0.2 × 2 = 0.4 → 0
0.4 × 2 = 0.8 → 0
0.8 × 2 = 1.6 → 1
0.6 × 2 = 1.2 → 1
0.2 × 2 = 0.4 → 0
0.4 × 2 = 0.8 → 0
0.8 × 2 = 1.6 → 1.

## Key terms

- **Binary point**: The binary equivalent of the decimal point.

## FAQ

**Why can't 0.1 be stored exactly in binary?**
Its binary expansion 0.000110011… repeats forever.

**Which decimal fractions are exact in binary?**
Those whose denominator is a power of 2, like 0.5, 0.25 and 0.375.

**What is 0.101₂ in decimal?**
0.625.

**How many bits does a double have for the fraction?**
52.


## Sources

- [Wolfram MathWorld – Binary](https://mathworld.wolfram.com/Binary.html)
- [Harvard CS50 – Binary and bitwise operators](https://cs50.harvard.edu/x/)