Binary Decimal Representation
Binary fractions use a binary point. Each position to the right is 2โปยน, 2โปยฒ, 2โปยณ... = 1/2, 1/4, 1/8. Only fractions with denominator powers of 2 have exact finite binary.
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0.5 = 0.1โ, 0.25 = 0.01โ, 0.125 = 0.001โ โ powers of 2 have exact binary. 0.1 + 0.2 โ 0.3 in many languages due to binary approximation. IEEE 754 defines single (32-bit) and double (64-bit) precision formats.
Ready to run the numbers?
Why: 0.1 cannot be represented exactly in binary โ it's an infinite repeating pattern. This causes floating-point errors.
How: Binary to decimal: multiply each bit by its positional value and sum. Decimal to binary: repeatedly multiply by 2, take integer part.
Run the calculator when you are ready.
Binary Fractions โ Representing Decimals in Binary
Convert between binary fractions and decimals. See bit positions (2โปยน, 2โปยฒ...), precision, and integer vs fractional breakdown. Understand why 0.1 can't be exact.
๐ข Load Sample Example โ Click to Load
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
0.1 cannot be represented exactly in binary โ it's an infinite repeating pattern.
โ Floating-point arithmetic
IEEE 754 defines how computers store floating-point numbers.
โ IEEE 754
๐ Key Takeaways
- โข Binary point: The binary equivalent of a decimal point
- โข Positional values: 2โปยน, 2โปยฒ, 2โปยณ... = 1/2, 1/4, 1/8...
- โข Some decimals can't be exactly represented: 0.1 in binary is repeating (0.0(0011))
- โข Finite binary: Only fractions with denominator powers of 2 (e.g., 0.5, 0.25, 0.125) have exact finite binary
๐ก Did You Know?
๐ How Binary Fractions Work
Binary fractions use a binary point. Each position to the right represents a negative power of 2: 2โปยน (1/2), 2โปยฒ (1/4), 2โปยณ (1/8), etc.
Binary to Decimal
Multiply each bit by its positional value and sum. Example: 0.101โ = 1รยฝ + 0รยผ + 1รโ = 0.5 + 0 + 0.125 = 0.625โโ
Decimal to Binary (fractional)
Repeatedly multiply by 2; take the integer part as the next bit. Example: 0.625 ร 2 = 1.25 โ bit 1; 0.25 ร 2 = 0.5 โ bit 0; 0.5 ร 2 = 1.0 โ bit 1. Result: 0.101โ
Repeating decimals
Fractions with denominators not powers of 2 (e.g., 1/10, 1/3) produce infinitely repeating binary patterns, like 1/3 = 0.333... in decimal.
๐ฏ Expert Tips
๐ก Use Decimal for money
Financial apps use decimal types (e.g., Java BigDecimal) to avoid floating-point rounding errors.
๐ก Compare with tolerance
Never use == for floats. Use abs(a - b) < epsilon for comparisons.
๐ก Powers of 2 are exact
0.5, 0.25, 0.125, 0.0625... have exact finite binary representations.
๐ก IEEE 754 precision
Single precision โ 7 decimal digits; double precision โ 15โ17 digits.
โ๏ธ Decimal vs Binary Fraction Comparison
| Decimal | Binary | Exact? |
|---|---|---|
| 0.5 | 0.1 | Yes |
| 0.25 | 0.01 | Yes |
| 0.125 | 0.001 | Yes |
| 0.1 | 0.0(0011)... | No (repeating) |
| 0.333... | 0.0(01)... | No (repeating) |
| 0.75 | 0.11 | Yes |
โ Frequently Asked Questions
Why can't 0.1 be represented exactly in binary?
0.1 = 1/10. The denominator 10 has prime factor 5, which is not a factor of 2. So 1/10 has no finite binary expansion โ it repeats infinitely.
What is the binary point?
The binary point is the base-2 equivalent of the decimal point. Digits to the right represent negative powers of 2: 2โปยน, 2โปยฒ, 2โปยณ, etc.
What are the positional values in binary fractions?
First position after binary point = 2โปยน = 1/2; second = 2โปยฒ = 1/4; third = 2โปยณ = 1/8; etc.
Why does 0.1 + 0.2 โ 0.3 in programming?
Floating-point uses binary. 0.1 and 0.2 are approximated; their sum is slightly different from the approximation of 0.3.
What is IEEE 754?
The standard for floating-point arithmetic. Defines single (32-bit) and double (64-bit) precision formats used by most computers.
Which decimals have exact binary representation?
Only fractions whose denominator is a power of 2: 1/2, 1/4, 1/8, 1/16, etc. Examples: 0.5, 0.25, 0.125.
How do I convert 0.75 to binary?
0.75 = ยฝ + ยผ. So 0.75โ = 0.11โ (first bit is 1/2, second is 1/4).
What causes floating-point errors in finance?
Money amounts like $0.10 are stored as binary approximations. Adding many such values can accumulate rounding errors. Use decimal types for money.
๐ Binary Fractions by the Numbers
๐ Official Data Sources
โ ๏ธ Disclaimer: This calculator is for educational purposes. Results may be approximations for some decimals (e.g., 0.1, ฯ). Not a substitute for professional numerical analysis or programming advice. Always verify critical calculations.
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