Binary Calculator
Convert between binary, decimal, octal, and hexadecimal
How to Use
- Select the base of your input number (binary, decimal, octal, or hexadecimal)
- Enter the number you want to convert
- Click calculate to see conversions to all other bases
- View all conversions displayed simultaneously
Understanding Number Systems
Number systems are methods of representing numbers using different bases. While we commonly use the decimal system (base 10) in everyday life, computers internally use binary (base 2), and programmers often work with octal (base 8) and hexadecimal (base 16) systems.
Each number system uses a specific set of digits and a base value that determines place values. Understanding these systems is essential for computer science, programming, and digital electronics.
Types of Number Systems
| System | Base | Digits Used | Common Use |
|---|---|---|---|
| Binary | 2 | 0, 1 | Computer internal representation |
| Octal | 8 | 0-7 | Unix file permissions, legacy systems |
| Decimal | 10 | 0-9 | Everyday mathematics |
| Hexadecimal | 16 | 0-9, A-F | Colors, memory addresses, debugging |
How Number System Conversion Works
Converting between number systems involves understanding place values. In binary (base 2), each position represents a power of 2. In hexadecimal (base 16), each position represents a power of 16.
For example, the binary number 1011 equals: (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11 in decimal.
Practical Applications
- Programming: Understanding memory addresses and bitwise operations
- Web Development: Working with color codes (#FF0000 for red)
- Network Administration: IP address calculations and subnet masks
- Digital Electronics: Circuit design and logic gates
- Data Encoding: Understanding how computers store information
- Debugging: Reading memory dumps and register values
Frequently Asked Questions
- What is binary and why do computers use it?
- Binary is a base-2 number system using only 0 and 1. Computers use binary because electronic circuits have two stable states (on/off, high voltage/low voltage), making binary the natural choice for digital systems.
- How do I convert decimal to binary manually?
- Divide the decimal number by 2 repeatedly, recording the remainder each time. Read the remainders from bottom to top to get the binary equivalent. For example, 13 ÷ 2 = 6 R1, 6 ÷ 2 = 3 R0, 3 ÷ 2 = 1 R1, 1 ÷ 2 = 0 R1, giving 1101 in binary.
- Why is hexadecimal useful in programming?
- Hexadecimal is compact and easier to read than binary while maintaining a direct relationship with binary (each hex digit represents exactly 4 binary digits). It's commonly used for memory addresses, color codes, and debugging.
- What's the difference between octal and hexadecimal?
- Octal uses base 8 (digits 0-7) where each digit represents 3 binary digits, while hexadecimal uses base 16 (digits 0-9, A-F) where each digit represents 4 binary digits. Hexadecimal is more commonly used today.