Number Systems for Developers: Binary, Octal, Decimal and Hex Explained

Decimal is the number system you grew up with. Binary, octal, and hexadecimal are the number systems your computer thinks in. Understanding them makes you better at debugging, bit manipulation, memory analysis, and working with low-level APIs.

Why Multiple Number Systems?

Computers store and process everything as binary (1s and 0s). But binary is verbose for humans — 255 in binary is 11111111, requiring 8 characters for a number you could write as two hex digits (FF) or three decimal digits (255).

Different bases serve different purposes:

• Binary (base 2) — the actual hardware representation; essential for bit manipulation and understanding storage
• Octal (base 8) — 3 bits per digit; used in Linux file permissions, some C constants
• Decimal (base 10) — human-readable counting; what most application-level code outputs
• Hexadecimal (base 16) — 4 bits per digit; the standard for memory addresses, color codes, cryptographic hashes, and binary data

How Base Conversion Works

Every number is a sum of powers of its base. In decimal (base 10): 255 = (2 × 10²) + (5 × 10¹) + (5 × 10⁰) = 200 + 50 + 5.

In binary (base 2), 11111111 is:

1×2⁷ + 1×2⁶ + 1×2⁵ + 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰
= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1
= 255

In hex (base 16), digits go 0–9 then A=10, B=11, C=12, D=13, E=14, F=15. FF in hex:

F×16¹ + F×16⁰
= 15×16 + 15×1
= 240 + 15
= 255

The Hex ↔ Binary Shortcut

The most useful trick: each hex digit maps directly to exactly 4 binary bits (a nibble). You can convert between hex and binary without going through decimal:

Hex:    F    F    5    7    3    3
Binary: 1111 1111 0101 0111 0011 0011

So #FF5733 = 11111111 01010111 00110011 in binary

This is why hex is the standard for representing byte data — two hex digits = one byte.

Where Each Base Appears in Practice

Binary

// Bit flags in code (JavaScript)
const PERMISSIONS = {
  READ:    0b001,  // 1
  WRITE:   0b010,  // 2
  EXECUTE: 0b100,  // 4
};

// Check if a user has READ permission
const userPerms = 0b101; // read + execute
const canRead = (userPerms & PERMISSIONS.READ) !== 0; // true

// Add WRITE permission
const newPerms = userPerms | PERMISSIONS.WRITE; // 0b111 = 7

Octal

# Linux file permissions — chmod 755
# 7 = 111 in binary → rwx
# 5 = 101 in binary → r-x

# In C/C++ — octal literals start with 0
int permissions = 0755;  // NOT decimal 755!

# In Python — octal literals use 0o prefix
import os
os.chmod("deploy.sh", 0o755)

Hexadecimal

// Color values
const red   = 0xFF0000;  // R=255, G=0,   B=0
const white = 0xFFFFFF;  // R=255, G=255, B=255

// Extract RGB channels from a packed integer
const color = 0xFF5733;
const r = (color >> 16) & 0xFF;  // 255
const g = (color >> 8)  & 0xFF;  // 87
const b = color & 0xFF;           // 51

// Memory addresses in debuggers
// 0x7fff5fbff8a0  ← typical stack address

// File signatures (magic bytes) to identify file types
// JPEG: FF D8 FF
// PNG:  89 50 4E 47 (= 0x89PNG in ASCII)
// PDF:  25 50 44 46 (= %PDF in ASCII)

Number Representation in JavaScript

// Literals
const binary  = 0b11111111;  // 255
const octal   = 0o377;       // 255
const decimal = 255;
const hex     = 0xFF;        // 255

// All the same value
console.log(binary === octal && octal === hex); // true

// Converting to different base strings
(255).toString(2);   // "11111111"
(255).toString(8);   // "377"
(255).toString(16);  // "ff"

// Parsing from string
parseInt("11111111", 2);  // 255
parseInt("377", 8);       // 255
parseInt("FF", 16);       // 255

Use the Number Base Converter to convert between all four bases with a visual breakdown of the binary representation. For file permission octals specifically, see the Chmod Calculator.