Binary Calculator

What is Binary Arithmetic?

Binary arithmetic is the foundation of all computer operations. It uses only two digits (0 and 1) to represent numbers and perform calculations. Understanding binary arithmetic is essential for computer science, digital electronics, and low-level programming.

How to Use This Binary Calculator

1. Enter the first binary number using only 0s and 1s
2. Enter the second binary number in the same format
3. Select the operation from arithmetic or bitwise options
4. Click "Calculate" to see the result with step-by-step explanation

Supported Operations

This calculator supports both arithmetic operations (addition, subtraction, multiplication, division) and bitwise operations (AND, OR, XOR) commonly used in digital logic and computer programming.

Binary Arithmetic Operations

Binary Addition

Binary addition follows these simple rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (0 with carry 1)

Binary Subtraction

Binary subtraction rules:

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1 = 1 (with borrow from next digit)

Binary Multiplication

Binary multiplication is similar to decimal multiplication but simpler since you only multiply by 0 or 1. Multiplying by 0 gives 0, and multiplying by 1 gives the original number.

Binary Division

Binary division follows the same long division process as decimal division, but with binary arithmetic rules. The result is the integer quotient.

Bitwise Operations

Bitwise AND (&)

Returns 1 only when both bits are 1:

• 0 & 0 = 0
• 0 & 1 = 0
• 1 & 0 = 0
• 1 & 1 = 1

Bitwise OR (|)

Returns 1 when at least one bit is 1:

• 0 | 0 = 0
• 0 | 1 = 1
• 1 | 0 = 1
• 1 | 1 = 1

Bitwise XOR (^)

Returns 1 when bits are different:

• 0 ^ 0 = 0
• 0 ^ 1 = 1
• 1 ^ 0 = 1
• 1 ^ 1 = 0

Examples & Worked Problems

Example 1: Binary Addition

Problem: 1010 + 0110

Solution:

1. Convert to decimal: 1010₂ = 10₁₀, 0110₂ = 6₁₀
2. Add: 10 + 6 = 16
3. Convert result: 16₁₀ = 10000₂

Example 2: Bitwise AND

Problem: 1100 AND 1010

Solution:

  1100
& 1010
------
  1000

Result: 1000₂ = 8₁₀

Applications of Binary Arithmetic

Computer Programming

• Bit manipulation and optimization
• Cryptography and hashing algorithms
• Graphics and image processing
• Network protocols and data compression

Digital Electronics

• Logic circuit design
• Processor arithmetic units
• Memory addressing
• Signal processing

Tips for Binary Calculations

Best Practices

• Always align digits properly for manual calculations
• Double-check your binary-to-decimal conversions
• Use the step-by-step approach for complex operations
• Verify results by converting back to decimal

Common Mistakes

• Forgetting to handle carries in addition
• Misaligning digits in multi-bit operations
• Confusing bitwise operations with logical operations
• Not accounting for sign bits in signed arithmetic

Frequently Asked Questions

What's the difference between arithmetic and bitwise operations?

Arithmetic operations (addition, subtraction, etc.) treat binary numbers as mathematical values, while bitwise operations work on individual bits without considering the overall numeric value.

How do I convert between binary and decimal?

To convert binary to decimal, multiply each digit by its corresponding power of 2 and sum the results. To convert decimal to binary, repeatedly divide by 2 and collect the remainders.

Why are bitwise operations important in programming?

Bitwise operations are extremely fast and memory-efficient. They're used for flags, masks, optimization, and low-level system programming where performance is critical.

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