Binary SI Prefixes: Extended to Fractional Values

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Introduction

\n The binary SI prefixes (kibi, mebi, gibi, etc.) were introduced to provide \n precise notation for binary multiples, distinguishing them from decimal SI prefixes. \n However, these prefixes only cover values ≥ 1. What if we extended this system \n to include fractional values?\n

\n Just as decimal SI prefixes have both large (kilo, mega, giga) and small \n (milli, micro, nano) variants, we can imagine a complete binary prefix system \n that includes fractional binary prefixes for values < 1.\n

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Current Binary Prefixes (≥ 1)

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Name	Symbol	Power of 2	Decimal Value	Common Usage
kibi	Ki	2¹⁰	1,024	KiB (kibibyte)
mebi	Mi	2²⁰	1,048,576	MiB (mebibyte)
gibi	Gi	2³⁰	1,073,741,824	GiB (gibibyte)
tebi	Ti	2⁴⁰	1,099,511,627,776	TiB (tebibyte)

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Extended Binary Prefixes (< 1)

\n Following the pattern of decimal SI prefixes, we can extend binary prefixes \n to fractional values using negative powers of 2:\n

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

Name	Symbol	Power of 2	Decimal Value	Fraction
mibi	mi	2⁻¹⁰	0.0009765625	1/1024
micbi	μbi	2⁻²⁰	0.00000095367	1/1,048,576
nibi	ni	2⁻³⁰	0.00000000093132	1/1,073,741,824
pibi	pi	2⁻⁴⁰	0.00000000000090949	1/1,099,511,627,776

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Interactive Calculator

\n \n Enter power of 2 (e.g., -10, 10, 20):\n \n \n

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Potential Applications

🔬 Scientific Computing

\n Precise representation of very small binary fractions in scientific \n calculations, quantum computing, or signal processing applications.\n

⚡ Hardware Design

\n Describing fractional clock cycles, timing intervals, or power \n consumption in binary-based hardware systems.\n

📊 Data Analysis

\n Representing probabilities or weights in binary tree structures, \n machine learning models, or statistical computations.\n

🎮 Game Development

\n Precise timing for frame rates, animation intervals, or probability \n calculations in procedural generation.\n

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Comparison: Binary vs Decimal Prefixes

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Scale	Decimal SI	Binary Extended	Ratio
Large	kilo (10³ = 1,000)	kibi (2¹⁰ = 1,024)	1.024×
Small	milli (10⁻³ = 0.001)	milli-binary (2⁻¹⁰ ≈ 0.000977)	0.977×

\n Note: Binary prefixes become increasingly different from their decimal counterparts \n as the scale increases, making the distinction more important for precision.\n

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Conclusion

\n While fractional binary prefixes aren't currently standardized, they represent \n a logical extension of the existing binary prefix system. As computing becomes \n more precise and we work with increasingly small binary fractions, such a system \n could provide valuable clarity and precision.\n

\n This extension maintains the mathematical elegance of powers of 2 while providing \n a complete prefix system that mirrors the comprehensiveness of decimal SI prefixes.\n