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Number Systems Cheat Sheet

Quick reference for common number systems used in computing and mathematics.

Overview

System Base Digits Common Use
Binary 2 0, 1 Low-level computing, hardware
Octal 8 0–7 Unix permissions, legacy systems
Decimal 10 0–9 Human-readable numbers
Hexadecimal 16 0–9, A–F Memory, colors, debugging

Binary (Base 2)

  • Digits: 0, 1
  • Each position represents a power of 2
1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 11₁₀

Common prefixes: - 0b1010 (programming)

Octal (Base 8)

  • Digits: 0–7
  • Each position represents a power of 8
17₈ = 1×8¹ + 7×8⁰ = 15₁₀

Common prefixes: - 0755 (Unix file permissions)

Decimal (Base 10)

  • Digits: 0–9
  • Standard number system for humans
345₁₀ = 3×10² + 4×10¹ + 5×10⁰

Hexadecimal (Base 16)

  • Digits: 0–9, A–F
  • Each position represents a power of 16
2F₁₆ = 2×16¹ + 15×16⁰ = 47₁₀

Common prefixes: - 0xFF

Base Conversion Summary

From \ To Binary Octal Decimal Hex
Binary — Group by 3 Sum powers of 2 Group by 4
Octal Convert to binary — Sum powers of 8 Binary → Hex
Decimal Divide by 2 Divide by 8 — Divide by 16
Hex Convert to binary Binary → Octal Sum powers of 16 —

Binary ↔ Hex Mapping

Binary Hex
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

Common Ranges

Bits Unsigned Range Signed (Two's Complement)
4 0 – 15 −8 – 7
8 0 – 255 −128 – 127
16 0 – 65,535 −32,768 – 32,767
32 0 – 4,294,967,295 −2,147,483,648 – 2,147,483,647

Practical Examples

255₁₀ = 11111111₂ = FF₁₆
64₁₀  = 1000000₂  = 100₈
  • Binary → hardware & bitwise ops
  • Octal → permissions (rwx)
  • Hex → memory addresses, colors (#FFAA00)