Binary Fractions: Converting Between Base-2 and Decimal Fractions
Convert between binary and decimal fractions with repeated multiplication, IEEE 754, and step-by-step examples.
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Binary fractions use base-2 place values after the binary point: 2⁻¹ (0.5), 2⁻² (0.25), 2⁻³ (0.125). Converting between binary and decimal fractions reveals why some decimal fractions cannot be represented exactly in binary.
Key Takeaway
Not all decimal fractions have exact binary representations. This is the root cause of floating-point rounding errors in computers.
1. Binary Fraction Place Values
| Position | Value |
|---|---|
| 2⁻¹ | 0.5 |
| 2⁻² | 0.25 |
| 2⁻³ | 0.125 |
| 2⁻⁴ | 0.0625 |
2. Why 0.1 is Not Exact
0.1 in binary is 0.0001100110011... (repeating). Since 10 is not a power of 2, 1/10 cannot be represented exactly. This causes 0.1 + 0.2 ≠ 0.3 in many programming languages.
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