Binary Calculator

Enter binary numbers and select operations

First Binary Number

Please enter only 0s and 1s

Second Binary Number

Please enter only 0s and 1s

Bit Length

Results

View calculations in different bases

Binary Result

0111

Decimal

Hexadecimal

0x7

Operation

ADD

Operation: ADD

Input 1: 1010

Input 2: 1101

Result: 0111

How It Works

Select operation type, enter binary numbers, choose bit length, and get instant results. Our calculator uses standard binary arithmetic and logic operations for accuracy.

Common Uses

Debug low-level code, design digital circuits, calculate network addresses, learn computer science fundamentals, and prepare for technical interviews.

Always Accessible

Works completely in your browser - no data sent to servers. Use it anytime, anywhere with full privacy protection for your calculations.

How the Binary Calculator Works

Follow these steps to understand binary calculations:

Input Validation: The calculator first validates that all inputs contain only 0s and 1s.
Bit Length Application: Numbers are padded or truncated to the selected bit length (8, 16, 32, or 64 bits).
Arithmetic Operations: For addition/subtraction, calculations proceed bit-by-bit from right to left using binary addition rules, with carries handled automatically.
Two's Complement: Subtraction uses two's complement representation for negative numbers.
Bitwise Operations: AND, OR, and XOR compare corresponding bits according to their truth tables. NOT inverts each bit.
Bit Shifts: Left shift moves bits left, filling with 0s. Right shifts move bits right, filling based on logical or arithmetic rules.
Base Conversion: Results are converted between binary, decimal, and hexadecimal representations.
Display: All results update in real-time with appropriate visualizations.

Binary Calculation Logic

Binary Arithmetic Rules

Addition: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0 with carry 1

Subtraction: Uses two's complement: invert all bits and add 1 to get negative representation

Binary arithmetic follows these fundamental rules for all calculations.

Bitwise Operation Truth Tables

AND (&): 1 only if both bits are 1. Used for masking.

OR (|): 1 if either bit is 1. Used for setting bits.

XOR (^): 1 if bits are different. Used for toggling.

NOT (~): Inverts each bit (0 becomes 1, 1 becomes 0).

These operations work on corresponding bits position by position.

Bit Shift Operations

Left Shift (<<): Move bits left, fill with 0s. Equivalent to multiplication by 2ⁿ.

Logical Right Shift (>>): Move bits right, fill with 0s. For unsigned numbers.

Arithmetic Right Shift (>>>): Move bits right, fill with sign bit. Preserves sign for signed numbers.

Shifts are fundamental operations in low-level programming and digital design.

Base Conversion Formulas

Binary to Decimal: Sum of (bit × 2^position) where position starts at 0 from right

Decimal to Binary: Repeated division by 2, collecting remainders in reverse order

P = bit position from right (0-indexed), D = decimal value, B = binary digits

Step-by-Step Examples

Example 1: Binary Addition

Inputs: Binary1: 11001010, Binary2: 00110101, Operation: Addition, Bit Length: 8-bit Steps: 1. Align numbers: 11001010 + 00110101 2. Add from right with carries 3. Result: 11111111

Binary: 11111111, Decimal: 255, Hexadecimal: FF

Example 2: Bitwise AND

Inputs: IP: 11000000.10101000.00000001.00000101 (192.168.1.5), Mask: 11111111.11111111.11111111.00000000 (255.255.255.0), Operation: AND Steps: 1. Apply AND to each bit position 2. 1 AND 1 = 1, 1 AND 1 = 1, 0 AND 1 = 0, etc. 3. Result: 11000000.10101000.00000001.00000000

Network Address: 192.168.1.0

Example 3: Left Shift

Inputs: Binary: 00000101 (5 decimal), Shift: Left by 2 positions Steps: 1. Original: 00000101 2. Shift left 2: move all bits left two positions 3. Fill right with 0s: 00010100

5 × 2² = 20 (confirmed)

Use Cases & Applications

Binary calculations are fundamental to computing and digital systems. Here are practical applications:

Debugging Low-Level Code

Examine bit flags, masks, and register values in embedded systems or driver development. When working with hardware registers, you often need to set, clear, or test specific bits without affecting others.

Computer Science Education

Learn fundamental concepts of binary arithmetic, logic gates, and number representation. Understanding binary is essential for anyone studying computer architecture, digital logic, or low-level programming.

Digital Circuit Design

Verify logic gate combinations and arithmetic logic unit (ALU) operations before hardware implementation. Designers use binary calculations to test their circuits' behavior with various inputs.

Network Configuration

Calculate network addresses, subnet masks, and CIDR notations for IP addressing. Network administrators use binary operations daily when designing and troubleshooting networks.

Technical Interview Preparation

Practice common bit manipulation problems asked in software engineering interviews. Companies like Google, Facebook, and Amazon frequently test candidates' understanding of binary operations.

Practical Example: Subnet Calculation

Problem: Find network address for IP 192.168.1.5 with mask 255.255.255.0 Solution: 1. Convert to binary: 11000000.10101000.00000001.00000101 2. Mask: 11111111.11111111.11111111.00000000 3. Apply AND operation 4. Result: 11000000.10101000.00000001.00000000 5. Convert back: 192.168.1.0

Network Address: 192.168.1.0

How to Read the Results

Understanding the output formats is crucial for interpreting binary calculations correctly:

Binary Result

The direct output of your operation in base-2. This is how computers store the result. Pay attention to bit length - results may be truncated or sign-extended based on your selection. For example, in 8-bit arithmetic, 255 + 1 = 0 (overflow).

Decimal Equivalent

The human-readable integer value. For signed numbers using two's complement, negative values will display correctly. This helps verify the calculation makes mathematical sense. Remember that decimal representation depends on whether you're treating the number as signed or unsigned.

Hexadecimal Representation

A compact format where each hex digit represents 4 bits. Widely used in programming (e.g., memory addresses, color codes) as it's more readable than binary while maintaining bit-level visibility. The "0x" prefix indicates hexadecimal notation.

Bit Visualization

Shows how each bit was affected by the operation. Essential for understanding bitwise operations and verifying specific bit manipulations. This visualization helps debug complex bit masks and understand the step-by-step transformation of your binary data.

Interpreting Negative Numbers

In two's complement representation (used by most modern systems), negative numbers have their most significant bit set to 1. For example, -5 in 8-bit is 11111011 (two's complement of 00000101). The calculator handles this conversion automatically.

Frequently Asked Questions

What's the difference between logical and arithmetic right shift?

Logical right shift (>>) always fills the leftmost bits with zeros, treating the number as unsigned. Arithmetic right shift (>>>) fills with the sign bit (0 for positive, 1 for negative), preserving the number's sign for signed integer representation.

Why does binary subtraction use two's complement?

Two's complement allows subtraction to be performed using the same circuitry as addition, simplifying computer hardware. To subtract, we take the two's complement of the subtrahend (invert bits, add 1) and then add it to the minuend.

How do I handle overflow in binary addition?

Overflow occurs when a result exceeds the bit length. In 8-bit arithmetic, 255 + 1 = 0 with overflow. Our calculator shows the truncated result but consider the bit length limit. In programming, overflow flags indicate this condition.

When would I use XOR in real programming?

XOR (exclusive OR) is useful for toggling bits, simple encryption (XOR cipher), checking parity, and swapping values without temporary variables. For example, a ^= b; b ^= a; a ^= b; swaps two integers.

What does 8-bit, 16-bit, 32-bit mean in this context?

These refer to the number of bits used to represent each number. 8-bit means numbers are treated as having 8 binary digits. Choosing the correct bit length matches how your target system (microcontroller, processor, protocol) handles data.

How are negative numbers represented in binary?

Most modern systems use two's complement: invert all bits of the positive number and add 1. The leftmost bit becomes the sign bit (1 for negative). For example, -5 in 8-bit is 11111011 (two's complement of 00000101).

Binary Calculator