Perform binary arithmetic operations, conversions, and bitwise calculations with real-time results, step-by-step solutions, and comprehensive validation. Supports multiple number systems and formats.
Enter binary numbers (0s and 1s only)
Enter two numbers and select an operation to see real-time results.
Master binary mathematics, conversions, and computer science applications with our comprehensive guide.
The binary number system, also known as base-2, is the fundamental language of computers and digital systems. Unlike our familiar decimal system that uses ten digits (0-9), binary uses only two digits: 0 and 1. This simplicity makes it perfect for representing the on/off states of electronic switches in computer circuits.
Every digital device you use—from smartphones to supercomputers—processes information in binary. Understanding binary is essential for computer science, programming, digital electronics, and cybersecurity. It's the foundation that enables everything from simple calculations to complex artificial intelligence algorithms.
Each position in a binary number represents a power of 2:
While binary mathematics has ancient roots in Indian mathematics (Pingala, 3rd century BCE), modern binary was formalized by Gottfried Wilhelm Leibniz in 1679. The system gained practical importance with the development of electronic computers in the 1940s, when engineers realized that binary perfectly matched the on/off nature of electronic switches.
A binary digit, or "bit," is the smallest unit of data in computing. Each bit can hold one of two values: 0 (representing "off," "false," or "low") or 1 (representing "on," "true," or "high"). Multiple bits combine to represent larger numbers and more complex information.
Binary numbers are read from right to left, with each position representing an increasing power of 2. The rightmost bit (position 0) represents 2⁰ = 1, the next bit represents 2¹ = 2, then 2² = 4, and so on.
Binary addition follows simple rules similar to decimal addition, but uses only two digits. The key is understanding when to carry over to the next position.
Binary subtraction requires borrowing from the next higher position when subtracting 1 from 0. This is similar to decimal subtraction but uses powers of 2 instead of 10.
Binary multiplication is simpler than decimal because you only multiply by 0 or 1. Division uses repeated subtraction or the same long division process as decimal.
There are multiple methods to convert decimal numbers to binary. The division method is most common and intuitive, while the subtraction method helps understand the positional value system.
Converting binary to other number systems leverages the relationships between bases. Since 8 = 2³ and 16 = 2⁴, conversions to octal and hexadecimal are particularly efficient.
Group binary digits in sets of 3 (from right):
Group binary digits in sets of 4 (from right):
Bitwise operations perform logical operations on individual bits of binary numbers. These operations are fundamental to computer processors, digital logic circuits, and programming optimizations. For hexadecimal conversions, these principles remain essential.
Returns 1 only when both bits are 1:
Used for masking, clearing bits, and filtering operations.
Returns 1 when at least one bit is 1:
Used for setting bits, combining flags, and merging operations.
Returns 1 when bits are different:
Used for toggling bits, encryption, and error detection.
Flips all bits (1s complement):
Used for bit inversion and creating bit masks.
Bit shifting operations move bits left or right, effectively multiplying or dividing by powers of 2. These operations are much faster than traditional multiplication/division in computer processors.
| A | B | A&B |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
| A | B | A|B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
| A | B | A^B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Binary mathematics forms the foundation of all modern computing. From simple calculations to complex artificial intelligence algorithms, everything ultimately relies on binary operations.
Binary is essential in networking, data transmission, and communication protocols. Understanding binary helps in network administration, cybersecurity, and protocol design.
IPv4 Address: 192.168.1.1
Subnet Mask: /24 = 255.255.255.0
Binary operations are fundamental to modern cryptography and cybersecurity. Many encryption algorithms rely on bitwise operations for their security properties.
Two's complement is the standard method for representing negative numbers in computer systems. It allows for efficient arithmetic operations without separate addition and subtraction circuits.
To represent -13 in 8-bit two's complement:
Range for 8-bit: -128 to +127
IEEE 754 floating-point standard uses binary to represent real numbers with fractional parts. This is crucial for scientific computing, graphics, and any application requiring decimal precision.
Value = (-1)^S × (1.M) × 2^(E-127)
Binary mathematics continues to evolve with new computing paradigms and technologies. Understanding these applications helps prepare for future developments in computing.
Given IP 192.168.10.45/28, find the network address and broadcast address.
Convert Unix permission 755 (octal) to binary and explain the permissions.
Write expressions to set the 4th bit, clear the 2nd bit, and toggle the 6th bit of a number.
Implement a simple XOR cipher and decrypt: 01001000 01100101 01101100 01101100 01101111
Convert RGB color (255, 128, 64) to a 24-bit binary representation.
Replace multiplication by 8 using bit shift operations. Explain the efficiency gain.
Common questions about binary numbers and calculations
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