Number System Converter - Binary, Decimal, Octal, Hexadecimal
Convert numbers between different number systems instantly. Enter a value in any field and see automatic conversions to Binary (Base 2), Octal (Base 8), Decimal (Base 10), and Hexadecimal (Base 16).
How to use: Type or paste a number into any field below. All other number systems will be calculated automatically. Use the copy buttons to copy results.
Decimal (Base 10)
Standard number system (0-9)
Length: 3 digits
Binary (Base 2)
Only 0 and 1 (computers use this)
Length: 7 bits
Octal (Base 8)
Digits 0-7 (used in Unix permissions)
Length: 3 digits
Hexadecimal (Base 16)
Digits 0-9 and A-F (used in colors)
Length: 2 characters
Quick Reference: Common Conversions
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 8 | 1000 | 10 | 8 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 100 | 1100100 | 144 | 64 |
| 255 | 11111111 | 377 | FF |
| 256 | 100000000 | 400 | 100 |
| 1024 | 10000000000 | 2000 | 400 |
Understanding Number Systems
Decimal (Base 10)
The standard number system we use daily. It has 10 digits (0-9) and each position represents a power of 10.
Example: 125 = (1×10²) + (2×10¹) + (5×10⁰)
= 100 + 20 + 5
Binary (Base 2)
Used by computers. Only uses two digits (0 and 1). Each position represents a power of 2.
Example: 101 = (1×2²) + (0×2¹) + (1×2⁰)
= 4 + 0 + 1 = 5 in decimal
Octal (Base 8)
Uses 8 digits (0-7). Commonly used in Unix file permissions. Each position represents a power of 8.
Example: 17 = (1×8¹) + (7×8⁰)
= 8 + 7 = 15 in decimal
Hexadecimal (Base 16)
Uses 16 digits (0-9, A-F). Common in web colors and memory addresses. Each position represents a power of 16.
Example: 2F = (2×16¹) + (15×16⁰)
= 32 + 15 = 47 in decimal
Real-World Applications
Computer Science
Binary is fundamental to how computers process and store information. Understanding binary helps programmers work with bits, bytes, and low-level operations.
Web Design
Hexadecimal is used for color codes in CSS and HTML. For example, #FF0000 represents red, where FF is the maximum red value (255 in decimal).
File Permissions
Unix/Linux systems use octal notation for file permissions. For example, chmod 755 sets read-write-execute permissions using octal values.
Network Addresses
IP addresses and MAC addresses are often represented in hexadecimal or binary. Understanding conversions helps with network configuration and troubleshooting.
Digital Electronics
Binary logic gates (AND, OR, NOT) form the basis of all digital circuits. Understanding binary is essential for hardware design and debugging.
Data Storage
Understanding how numbers are stored in binary helps optimize data structures, compression algorithms, and memory usage in applications.
How to Convert Between Number Systems
Decimal to Binary
Divide the decimal number by 2 repeatedly and record the remainders from bottom to top.
Example: Convert 13 to binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Result: 1101 (read remainders from bottom to top)
Binary to Decimal
Multiply each digit by 2 raised to its position (counting from right, starting at 0) and sum the results.
Example: Convert 1011 to decimal
(1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
= 8 + 0 + 2 + 1
Result: 11
Hexadecimal to Decimal
Multiply each digit by 16 raised to its position and sum the results. Remember A=10, B=11, C=12, D=13, E=14, F=15.
Example: Convert 2A to decimal
(2 × 16¹) + (10 × 16⁰)
= 32 + 10
Result: 42
Binary to Hexadecimal (Shortcut)
Group binary digits into sets of 4 (from right to left) and convert each group to its hex equivalent.
Example: Convert 11010110 to hex
Group: 1101 0110
1101 = D (13 in decimal)
0110 = 6
Result: D6
Frequently Asked Questions
Why do computers use binary?
Computers use binary because digital circuits can easily represent two states: on (1) and off (0). These correspond to high and low voltage levels in transistors. It's much more reliable and simpler to distinguish between two states than multiple voltage levels.
What's the difference between bits and bytes?
A bit is a single binary digit (0 or 1). A byte is a group of 8 bits. For example, the binary number 11111111 is one byte (8 bits) and equals 255 in decimal. Bytes are the standard unit for measuring computer memory and file sizes.
Why is hexadecimal used in programming?
Hexadecimal is more compact than binary while being easy to convert. One hex digit represents exactly 4 binary digits. This makes it perfect for representing large binary numbers concisely, especially for memory addresses, color codes, and data dumps.
What is the largest number I can convert?
This converter supports numbers up to JavaScript's maximum safe integer (2⁵³ - 1, or 9,007,199,254,740,991 in decimal). Beyond this limit, precision may be lost. For extremely large numbers, specialized arbitrary-precision libraries are needed.
Can I convert negative numbers?
This tool currently supports positive integers only. Negative numbers in binary use different representations like two's complement, which requires specifying a bit width (8-bit, 16-bit, etc.). For negative number conversions, specialized tools are more appropriate.
What are common prefix notations?
In programming, prefixes indicate number bases: 0b or 0B for binary (0b1010), 0o or 0O for octal (0o755), 0x or 0X for hexadecimal (0xFF). These help prevent confusion when working with different number systems in code.
How accurate is this converter?
The converter uses JavaScript's built-in parseInt() and toString() methods with specified radix values. It's accurate for all integers within JavaScript's safe integer range. All conversions are performed client-side in your browser.
Can I convert decimal numbers (fractions)?
This tool handles whole numbers (integers) only. Converting fractional numbers to binary, octal, or hexadecimal requires different algorithms as the fractional part is converted separately from the integer part.
Tips for Working with Number Systems
- • Memorize Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024... These are essential for quick binary conversions
- • Group Binary Digits: For readability, group binary in sets of 4 or 8. Example: 1010 1100 instead of 10101100
- • Use Prefixes in Code: Always use 0x for hex, 0b for binary, and 0o for octal in your code to avoid confusion
- • Practice Hex-to-Binary: Learn to quickly convert hex to binary by memorizing the 16 patterns (0=0000, 1=0001, ... F=1111)
- • Check Your Work: Always verify conversions, especially when working with critical systems or security-related code
- • Understand Context: Different fields use different systems. Web developers use hex for colors, system admins use octal for permissions
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