What is the difference between binary, decimal, octal, and hexadecimal?
A comparative study of the four core radices of digital computing, their power-of-two relationships, and architectural roles.
Binary, Decimal, Octal, and Hexadecimal form the quartet of number systems powering all computer hardware and software. Discover their technical differences, positional column weights, and direct bit-grouping shortcuts.
1. Understanding the Core Concept
Binary (Base 2), Octal (Base 8), Decimal (Base 10), and Hexadecimal (Base 16) are positional number systems differing primarily in their Radix (base), symbol sets, and computational roles. Decimal is the biological standard of human society. Binary is the physical language of silicon transistors. Octal and Hexadecimal serve as compact, human-readable representations of long binary strings because 8 and 16 are integer powers of 2 (2³ = 8, 2⁴ = 16), enabling lossless bit grouping without decimal math.
2. How Does It Work? Step-by-Step Methodology
The relationship between these systems stems from binary: because 2³ = 8, every octal digit corresponds to exactly 3 binary bits. Because 2⁴ = 16, every hexadecimal digit corresponds to exactly 4 binary bits (one nibble). Consequently, converting between binary and hex (or binary and octal) requires zero arithmetic—simply partition binary bits into groups of 3 or 4 and substitute the corresponding symbol. Converting to or from decimal, however, requires full polynomial evaluation or successive division by the target base.
3. Detailed Worked Example & Verification
4. Essential Rules & Edge Cases
- Binary uses base 2: allowed digits are {0, 1}.
- Octal uses base 8: allowed digits are {0, 1, 2, 3, 4, 5, 6, 7}. Digits 8 and 9 are strictly illegal.
- Decimal uses base 10: allowed digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Hexadecimal uses base 16: allowed digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}.
- Direct grouping shortcuts only work between bases that are powers of the same root (e.g., base 2, 4, 8, 16, 32).
5. Practical Engineering Applications
- Binary: Gate-level logic, FPGA bitstreams, flip-flops, CPU microcode.
- Hexadecimal: Memory dump inspection, color codes, cryptographic hashes (SHA-256, MD5), byte inspection.
- Octal: UNIX file permission masks (chmod 0777, 0755, 0644), legacy avionics and aviation transponder codes.
- Decimal: Human UI displays, financial transactions, metrics, measurements.
6. Common Mistakes to Avoid
- WarningAssuming hex numbers are larger in value just because they contain letters.
- WarningTreating 10_2, 10_8, 10_10, and 10_16 as equal (they equal 2, 8, 10, and 16 in decimal, respectively).
- WarningGrouping binary bits from left to right instead of right to left from the radix point.
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Frequently Asked Questions
Why is hexadecimal preferred over octal in modern computing?
Modern computer architectures are byte-oriented (8, 16, 32, 64 bits). Because one byte equals exactly two 4-bit hex nibbles, hexadecimal aligns seamlessly with hardware registers. Octal groups into 3 bits, which does not divide evenly into 8 or 16 bits.
How many bits does one hexadecimal character represent?
Exactly 4 bits, commonly referred to as one nibble.
How many bits does one octal character represent?
Exactly 3 bits.
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