Binary Calculator
Convert between binary and decimal numbers, and perform binary arithmetic operations with this free calculator.
About Binary Calculator
This Binary Calculator allows you to convert between binary (base-2) and decimal (base-10) number systems, as well as perform various binary arithmetic and logical operations. Whether you're a student learning computer science, a programmer, or simply curious about binary mathematics, this tool provides a simple way to work with binary numbers.
Binary Number System:
The binary number system is a base-2 numeral system that uses only two symbols: 0 and 1. It forms the foundation of digital computing and electronics. Each digit in a binary number is called a "bit" (binary digit), and each bit represents a power of 2.
For example, the binary number 101101 represents:
1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20
= 32 + 0 + 8 + 4 + 0 + 1
= 45 in decimal
Calculator Features:
- Conversion: Convert numbers between binary and decimal formats.
- Basic Arithmetic: Perform addition, subtraction, multiplication, and division operations on binary or decimal inputs.
- Logical Operations: Execute bitwise operations including AND, OR, XOR, and NOT (complement).
- Bit Shifting: Perform binary shift operations (left and right shifts).
- Additional Information: View equivalent hexadecimal and octal values, as well as bit length information.
Binary Operations Explained:
| Operation | Description | Example |
|---|---|---|
| Addition | Adds two binary numbers | 1011 + 101 = 10000 |
| Subtraction | Subtracts the second binary number from the first | 1011 - 101 = 110 |
| Multiplication | Multiplies two binary numbers | 101 × 11 = 1111 |
| Division | Divides the first binary number by the second (integer division) | 1010 ÷ 10 = 101 |
| AND | Returns 1 if both corresponding bits are 1 | 1010 AND 1100 = 1000 |
| OR | Returns 1 if at least one corresponding bit is 1 | 1010 OR 1100 = 1110 |
| XOR | Returns 1 if exactly one corresponding bit is 1 | 1010 XOR 1100 = 0110 |
| NOT | Inverts all bits (0 becomes 1, 1 becomes 0) | NOT 1010 = 0101 |
| Shift Left | Shifts all bits to the left by a specified number, adding 0s on the right | 1010 << 2 = 101000 |
| Shift Right | Shifts all bits to the right by a specified number, removing rightmost bits | 1010 >> 2 = 10 |
Applications of Binary Numbers:
- Computer Systems: All data in computers is ultimately represented in binary form.
- Digital Electronics: Binary is used to represent the two states in digital electronics: on/off, high/low, true/false.
- Programming: Understanding binary operations is essential for low-level programming and optimization.
- Networking: IP addresses and subnet masks use binary representation.
- Data Storage: File sizes, memory allocation, and storage capacities are measured in binary units.
- Digital Logic: Boolean logic and digital circuits operate using binary principles.
Binary Number System in Computing:
Binary is the foundation of computing because electronic components can easily represent two states: on (1) or off (0). These states can be represented by voltage levels, magnetic polarities, or optical signals.
In computing, binary numbers are grouped in different ways:
- 4 bits = 1 nibble (ranges from 0000 to 1111, equivalent to a single hexadecimal digit)
- 8 bits = 1 byte (ranges from 00000000 to 11111111, can represent values from 0 to 255)
- 16 bits = 2 bytes = 1 word (on some systems)
- 32 bits = 4 bytes = 1 double word
- 64 bits = 8 bytes = 1 quad word
Disclaimer
This calculator is designed for educational and practical purposes. While it handles most common binary operations, it may have limitations for very large numbers or special cases. For professional or critical applications, please verify results with specialized tools or professional software.