Binary, Hex, Decimal: How to Convert Between Number Bases

Why Number Base Conversion Matters

Computers think in binary (base 2). Humans prefer decimal (base 10). Programmers use hexadecimal (base 16) as a compact shorthand. If you work with low-level data, you need all three.

Common Number Bases

How Number Bases Work

Every number is a sum of digits × base^position. The rightmost digit is position 0.

Decimal Example: 423₁₀

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4 × 10² = 4 × 100 = 400

2 × 10¹ = 2 × 10 = 20

3 × 10⁰ = 3 × 1 = 3

Sum: 423

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Binary Example: 11010101₂

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1 × 2⁷ = 128

1 × 2⁶ = 64

0 × 2⁵ = 0

1 × 2⁴ = 16

0 × 2³ = 0

1 × 2² = 4

0 × 2¹ = 0

1 × 2⁰ = 1

Sum: 213₁₀

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Conversion Methods

Decimal to Binary (Repeated Division)

Divide by 2 repeatedly, reading remainders from bottom to top:

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213 ÷ 2 = 106 remainder 1 ↑

106 ÷ 2 = 53 remainder 0 │

53 ÷ 2 = 26 remainder 1 │

26 ÷ 2 = 13 remainder 0 │

13 ÷ 2 = 6 remainder 1 │

6 ÷ 2 = 3 remainder 0 │

3 ÷ 2 = 1 remainder 1 │

1 ÷ 2 = 0 remainder 1 │

Result: 11010101₂

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Binary to Hexadecimal (Grouping)

Group binary digits into sets of 4 (from right), then convert each group:

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Binary: 1101 0101

Hex: D 5

Result: 0xD5

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Hexadecimal to Decimal

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0xD5 = 13 × 16¹ + 5 × 16⁰

= 208 + 5

= 213₁₀

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Using the ToolboxPro Converter

Visit our Number Base Converter and:

1. Type any number — it auto-detects the base

2. See all bases simultaneously — binary, octal, decimal, hex side by side

3. Copy any format with one click

4. Works with very large numbers — up to 64-bit values

Practical Use Cases

1. RGB Color Values

\\\`css

/* Hex is shorthand for RGB in base-10 */

#FF5733

/* FF = 255 red, 57 = 87 green, 33 = 51 blue */

background-color: rgb(255, 87, 51);

\\\`

2. Unix File Permissions

\\\`bash

# chmod uses octal

chmod 755 script.sh

# 7 = rwx (owner), 5 = r-x (group), 5 = r-x (others)

# 7 in octal = 111 in binary = read + write + execute

\\\`

3. Bitwise Flags

\\\`javascript

// Each bit is a flag

const READ = 0b0001; // 1

const WRITE = 0b0010; // 2

const EXECUTE = 0b0100; // 4

const permissions = READ | WRITE; // 0b0011 = 3

const canRead = permissions & READ; // 0b0001 = true

\\\`

4. Memory Addresses

\\\`c

// Debuggers show addresses in hex

int *ptr = malloc(64);

printf("%p", ptr); // 0x7ffeefbff5e0

\\\`

5. Network MAC Addresses

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MAC: 00:1A:2B:3C:4D:5E

Each pair is one byte (0-255 in decimal, 00-FF in hex)

First 3 bytes: vendor ID, Last 3 bytes: device ID

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Common Conversion Table

FAQ

What base do computers actually use? Binary (base 2). Every value in memory — numbers, text, images — is ultimately stored as sequences of 0s and 1s.

Why do programmers use hex? Hex is a human-readable shorthand for binary. One hex digit = 4 binary digits. It's much easier to read \0xFF\ than \0b11111111\.

What about base64? Base64 uses 64 characters (A-Z, a-z, 0-9, +, /) and is used for encoding binary data as text — see our Base64 Encoder/Decoder.

Is there a base higher than hex? Yes — base32, base36, base58 (Bitcoin addresses), and base64 are common. Our tool handles bases 2 through 36.

Bitwise Operations: The Foundation

Number base conversion is tightly connected to bitwise operations — the manipulation of individual bits within a binary number. Understanding bit manipulation makes you a more effective programmer, especially in systems programming, embedded development, and performance-critical code.

Essential Bitwise Operators

Practical Bit Tricks

// Check if a number is even or odd (faster than modulo)
const isEven = (n) => (n & 1) === 0;

// Swap two values without a temp variable
a ^= b; b ^= a; a ^= b;

// Multiply/divide by powers of 2 (faster on some architectures)
const doubled = value << 1;    // value * 2
const halved  = value >> 1;    // value / 2 (integer division)

// Set, clear, toggle, and check a specific bit
const setBit    = (n, pos) => n | (1 << pos);
const clearBit  = (n, pos) => n & ~(1 << pos);
const toggleBit = (n, pos) => n ^ (1 << pos);
const checkBit  = (n, pos) => (n >> pos) & 1;

Number Base Conversion in Programming

Every programming language provides built-in functions for base conversion. Knowing the idiomatic approach saves time and prevents subtle bugs.

JavaScript

// Decimal to other bases
(255).toString(2);    // "11111111" (binary)
(255).toString(16);   // "ff" (hex)
(255).toString(8);    // "377" (octal)

// Other bases to decimal
parseInt("ff", 16);    // 255
parseInt("11111111", 2); // 255
parseInt("377", 8);    // 255

// Literal prefixes
0xFF   // 255 (hex literal)
0b1111 // 15 (binary literal, ES6+)
0o17   // 15 (octal literal, ES6+)

Python

# Built-in conversion functions
bin(255)   # '0b11111111'
hex(255)   # '0xff'
oct(255)   # '0o377'

# int() with base argument
int("ff", 16)   # 255
int("11111111", 2)  # 255

# Format strings (Python 3.6+)
f"{255:#x}"  # '0xff'
f"{255:#b}"  # '0b11111111'

Color Code Parsing

Hexadecimal color codes are the most common real-world application of base conversion for web developers. Understanding how they work lets you manipulate colors programmatically.

How Hex Colors Work

A hex color like #FF5733 is actually three two-digit hex numbers representing red, green, and blue:

#FF5733
  │ │ │
  │ │ └─ Blue:  0x33 = 51
  │ └─── Green: 0x57 = 87
  └───── Red:   0xFF = 255

Converting Between Hex and RGB in Code

// Hex to RGB
function hexToRgb(hex) {
  const r = parseInt(hex.slice(1, 3), 16);
  const g = parseInt(hex.slice(3, 5), 16);
  const b = parseInt(hex.slice(5, 7), 16);
  return { r, g, b };
}
// hexToRgb('#FF5733') → { r: 255, g: 87, b: 51 }

// RGB to Hex
function rgbToHex(r, g, b) {
  return '#' + [r, g, b]
    .map(x => x.toString(16).padStart(2, '0'))
    .join('');
}
// rgbToHex(255, 87, 51) → '#ff5733'

8-Digit Hex Colors (with Alpha)

Modern CSS supports 8-digit hex: #FF573380. The last two digits (80) represent opacity: 0x80 = 128, which is 128/255 ≈ 50% opacity.

Common Mistakes to Avoid

1. Confusing 0x prefix with the value — 0x10 is 16 in decimal, not 10. The 0x is just a prefix indicating hex; the actual digits are 1 and 0.

2. Forgetting hex letters are case-insensitive — 0xFF and 0xff are identical. Don't write code that treats them differently.

3. Overflowing integer sizes — A 32-bit unsigned integer maxes out at 0xFFFFFFFF (4,294,967,295). Converting a 40-bit hex value to a 32-bit integer produces incorrect results silently in some languages.

Real-World Examples

Debugging Network Packets

When analyzing a network capture (Wireshark, tcpdump), packet data is displayed in hex. Understanding hex lets you read MAC addresses, port numbers, and protocol flags directly from the raw bytes without reaching for a calculator.

Configuring Microcontroller Registers

Embedded developers regularly set hardware registers using hex values. To enable UART with specific baud rate and parity settings, you might write USART1->CR1 = 0x200C; — knowing what each hex digit maps to in binary is essential for getting the configuration right.

Comparison: Manual vs. Tool-Assisted Conversion

Recommendation: Learn manual conversion to build fundamental understanding, then use a converter tool or programming function for all real work. Reach for mental conversion only for values under 256 (one byte) — beyond that, tools are faster and more reliable.