Perform binary arithmetic operations (add, subtract, multiply, divide) and convert between binary and decimal values. Perfect for computer science, programming, and digital electronics.
The binary number system is a base-2 numeral system that uses only two digits: 0 and 1. It's the fundamental number system used by computers and digital electronics because it directly corresponds to the two states of electronic switches: off (0) and on (1).
Understanding binary is essential for computer science, programming, digital electronics, and understanding how computers process information at the most fundamental level.
| Decimal | Binary | Calculation | Powers of 2 |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 2⁰ |
| 2 | 10 | 1×2¹ + 0×2⁰ | 2¹ |
| 3 | 11 | 1×2¹ + 1×2⁰ | 2¹ + 2⁰ |
| 4 | 100 | 1×2² + 0×2¹ + 0×2⁰ | 2² |
| 7 | 111 | 1×2² + 1×2¹ + 1×2⁰ | 2² + 2¹ + 2⁰ |
| 8 | 1000 | 1×2³ + 0×2² + 0×2¹ + 0×2⁰ | 2³ |
| 15 | 1111 | 1×2³ + 1×2² + 1×2¹ + 1×2⁰ | 2³ + 2² + 2¹ + 2⁰ |
| 16 | 10000 | 1×2⁴ + 0×2³ + 0×2² + 0×2¹ + 0×2⁰ | 2⁴ |
| 255 | 11111111 | 8 ones in binary | 2⁷ + 2⁶ + ... + 2⁰ |
Where n is the position from right (starting at 0)
Computer Science: All data in computers is stored and processed in binary format. Every program, image, video, and document is ultimately a series of 0s and 1s.
Digital Electronics: Binary represents the two voltage states in digital circuits - typically 0V (off/0) and 5V (on/1) in many systems.
Programming: Understanding binary is crucial for low-level programming, bit manipulation, memory management, and optimizing code performance.
Data Storage: File sizes, memory capacity, and data transfer rates are all based on binary units (bits, bytes, kilobytes, etc.).
| Unit | Binary Value | Decimal Value | Common Usage |
|---|---|---|---|
| Bit | 1 binary digit | 0 or 1 | Smallest unit of data |
| Byte | 8 bits | 256 possible values | Character storage |
| Kilobyte (KB) | 2¹⁰ bytes | 1,024 bytes | Small files |
| Megabyte (MB) | 2²⁰ bytes | 1,048,576 bytes | Images, documents |
| Gigabyte (GB) | 2³⁰ bytes | 1,073,741,824 bytes | Videos, large files |
| Terabyte (TB) | 2⁴⁰ bytes | 1,099,511,627,776 bytes | Hard drives |
Bit Manipulation: Operations like AND, OR, XOR, and bit shifting are fundamental in optimizing algorithms and working with flags.
Boolean Logic: Binary directly represents true/false values in programming logic and conditional statements.
Memory Addressing: Computer memory addresses are binary numbers, determining where data is stored and retrieved.
Network Protocols: Internet protocols, IP addresses, and data transmission all rely on binary representation.
Powers of 2: Memorize common powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) for quick mental conversions.
Grouping: Group binary digits in sets of 4 (nibbles) or 8 (bytes) for easier reading and conversion to hexadecimal.
Validation: Always verify that binary numbers contain only 0s and 1s - any other digit indicates an error.