Binary Calculator

About Binary Calculator

Welcome to the Binary Calculator, a comprehensive free online tool for performing binary arithmetic and bitwise operations. Whether you are a student learning computer science, a programmer debugging low-level code, or an electronics enthusiast working with digital systems, this calculator provides detailed step-by-step solutions with results displayed in multiple number bases.

What is Binary?

Binary is a base-2 numeral system that uses only two digits: 0 and 1. Each digit in a binary number is called a bit (binary digit). Binary is the foundation of all modern computing because electronic circuits can easily represent two states: on (1) and off (0).

Understanding Binary Place Values

Each position in a binary number represents a power of 2, starting from 2⁰ (which equals 1) on the right:

Binary Arithmetic Operations

Binary Addition

Binary addition follows these simple rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (write 0, carry 1 to the next column)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Binary Subtraction

Binary subtraction uses borrowing when needed:

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1: Borrow 1 from the next column, making it 10 - 1 = 1

Binary Multiplication

Binary multiplication is simpler than decimal because you only multiply by 0 or 1:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

Binary Division

Binary division works like long division in decimal but uses binary subtraction. This calculator handles both integer division and fractional results.

Bitwise Operations

AND Operation (&)

Returns 1 only when both corresponding bits are 1. Used for masking specific bits.

OR Operation (|)

Returns 1 when either corresponding bit is 1. Used for setting specific bits.

XOR Operation (^)

Returns 1 when corresponding bits are different. Used for toggling bits and simple encryption.

NOT Operation (~)

Flips all bits: 0 becomes 1, and 1 becomes 0.

Left Shift (<<) and Right Shift (>>)

Shifts move all bits left or right by the specified number of positions:

• Left shift multiplies by 2 for each position shifted
• Right shift divides by 2 for each position shifted

How to Use This Calculator

1. Enter first binary number: Type your first binary number using only 0s and 1s. You can use the quick example buttons to try common calculations.
2. Select operation: Choose your operation: arithmetic (+, -, *, /, %) or bitwise (AND, OR, XOR, NOT, <<, >>). Click operation buttons or use the dropdown.
3. Enter second binary number: Input your second binary number. For NOT operation, only the first number is needed. For shifts, this is the number of positions to shift.
4. View results: Click Calculate to see results in binary, decimal, and hexadecimal formats with step-by-step solution breakdown and interactive diagrams.

Frequently Asked Questions

What is binary and how does it work?

Binary is a base-2 numeral system that uses only two digits: 0 and 1. Each digit represents a power of 2, from right to left. For example, 1010 in binary equals 1×8 + 0×4 + 1×2 + 0×1 = 10 in decimal. Computers use binary because electronic circuits can easily represent two states: on (1) and off (0).

How do you add binary numbers?

Binary addition follows simple rules: 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (write 0, carry 1). Add column by column from right to left, carrying when the sum is 2 or more. For example, 1011 + 1101: starting from right, 1+1=10 (write 0, carry 1), 1+0+1=10 (write 0, carry 1), 0+1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1). Result: 11000.

What are bitwise operations in binary?

Bitwise operations manipulate individual bits in binary numbers. AND (&) returns 1 only if both bits are 1. OR (|) returns 1 if either bit is 1. XOR (^) returns 1 if bits are different. NOT (~) flips all bits. Left shift (<<) moves bits left, multiplying by 2. Right shift (>>) moves bits right, dividing by 2. These are essential for low-level programming and optimization.

How do you multiply binary numbers?

Binary multiplication works like decimal multiplication but simpler: 0×0=0, 0×1=0, 1×0=0, 1×1=1. Multiply the multiplicand by each bit of the multiplier, shift left for each position, then add all partial products. For example, 101 × 11: 101×1=101, 101×1 shifted left = 1010, then add: 101 + 1010 = 1111 (15 in decimal, which is 5×3).

What is the difference between signed and unsigned binary?

Unsigned binary represents only positive numbers (0 and above). Signed binary uses the leftmost bit as a sign bit: 0 for positive, 1 for negative. The most common signed representation is two's complement, where negative numbers are represented by inverting all bits and adding 1. This calculator supports both positive and negative binary numbers for arithmetic operations.

Related Resources

• Binary Number - Wikipedia
• Bits and Binary - Khan Academy
• Binary Converter - Convert between binary, decimal, hex, and octal
• Hex Calculator - Perform hexadecimal arithmetic and bitwise operations
• Hex Converter - Convert hexadecimal numbers

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