Binary Calculator
Convert between binary, decimal, octal, and hexadecimal number systems. Perform arithmetic operations on binary numbers and see results in multiple formats instantly. Perfect for students, programmers, and computer science professionals.
What Is a Binary Calculator?
A binary calculator is a tool that performs calculations using the binary number system (base-2). Unlike the decimal system which uses ten digits (0-9), binary uses only two digits: 0 and 1. Computers and digital electronics operate on binary because it directly represents the on-and-off states of electrical signals. For students, programmers, and engineers, a binary calculator simplifies working with binary numbers by automatically performing conversions, arithmetic, and logic operations that would otherwise be tedious by hand.
This calculator supports both conversion between number systems (binary, octal, decimal, hexadecimal) and arithmetic operations (addition, subtraction, multiplication, division) on binary numbers.
How Does the Binary Calculator Work?
The calculator uses standard mathematical methods for number system conversions and binary arithmetic:
Binary to Decimal: Each binary digit represents a power of 2. For example, 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀.
Binary Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (carry 1).
Binary Subtraction: Borrow when needed, similar to decimal subtraction.
Binary Multiplication: 0×0=0, 0×1=0, 1×0=0, 1×1=1.
Binary Division: Mirrors decimal long division but with simpler rules.
Why Use This Binary Calculator?
- Two Modes: Converter mode for number system conversion, Arithmetic mode for binary calculations.
- Instant Results: See results in binary, octal, decimal, and hexadecimal simultaneously.
- Accurate: Prevents human error when handling long binary sequences.
- Learning Aid: Step-by-step calculation display helps students practice binary arithmetic.
- Free & Private: No registration, no data storage.
Common Uses for Binary Calculations
- Computer Science: Understanding how computers represent and process data.
- Programming: Bitwise operations, data encoding, and low-level programming.
- Networking: IP addressing, subnet masks, and network calculations.
- Digital Electronics: Circuit design and logic gate operations.
- Education: Learning number systems and binary arithmetic.
❓ Binary Calculator FAQ
What is a binary number?
A binary number is a number expressed in the base-2 numeral system, using only two digits: 0 and 1. Each digit represents a power of 2. For example, 1011₂ = 11₁₀.
How do I convert binary to decimal?
Multiply each binary digit by its corresponding power of 2 and sum the results. For example, 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀.
How do I convert decimal to binary?
Divide the decimal number by 2 repeatedly and record the remainders. The binary number is the remainders read in reverse order.
How do I add two binary numbers?
Binary addition follows the same rules as decimal addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1). For example, 1011 + 1101 = 11000.
How do I subtract binary numbers?
Binary subtraction uses borrowing when needed: 0-0=0, 1-0=1, 1-1=0, 0-1=1 (borrow 1). For example, 1101 - 1011 = 0010.
How do I multiply binary numbers?
Binary multiplication uses simple rules: 0×0=0, 0×1=0, 1×0=0, 1×1=1. Multiply each digit and add the results. For example, 101 × 11 = 1111.
How do I divide binary numbers?
Binary division mirrors decimal long division. For example, 1010 ÷ 10 = 101.
What is the difference between binary and decimal?
Binary uses base-2 with digits 0 and 1. Decimal uses base-10 with digits 0-9. Computers use binary because it aligns with two-state logic (on/off).
What is octal?
Octal is a base-8 number system using digits 0-7. It is often used as a shorthand for binary because each octal digit represents exactly three binary bits.
What is hexadecimal?
Hexadecimal is a base-16 number system using digits 0-9 and letters A-F. It is commonly used in programming and computing because each hex digit represents exactly four binary bits.
How do I convert binary to hexadecimal?
Group the binary digits into sets of four (starting from the right). Convert each group to its hexadecimal equivalent. For example, 11010111₂ = D7₁₆.
How do I convert hexadecimal to binary?
Convert each hexadecimal digit to its 4-bit binary equivalent. For example, D7₁₆ = 11010111₂.
What are bitwise operations?
Bitwise operations (AND, OR, XOR, NOT, shifts) work directly on the binary representation of numbers. They are fundamental in low-level programming and digital logic design.
Is this calculator accurate for large binary numbers?
Yes, JavaScript's BigInt is used internally to handle arbitrarily large integers without losing precision.
Can I use this calculator for negative binary numbers?
This calculator works with positive binary numbers (unsigned). For signed binary operations, you would need to specify the bit width and two's complement representation.
What is the binary representation of 255?
255₁₀ = 11111111₂ (eight ones).
What is the binary representation of 1024?
1024₁₀ = 10000000000₂ (1 followed by 10 zeros).
How do I use this calculator for my programming work?
Use the Converter mode to quickly convert between number systems for debugging or data representation. Use the Arithmetic mode to verify binary calculations for low-level programming tasks.