Binary Calculator

Free online Binary Calculator with step-by-step solutions. 100% client-side, no data upload.

Binary InputEnter binary numbers (0s and 1s)
💡 Tip: Only digits 0 and 1 are allowed. Leading zeros are ignored.
ResultBinary Arithmetic
Click "Calculate" to see the detailed steps.

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Binary Calculator

This free online Binary Calculator performs arithmetic operations (addition, subtraction, multiplication, division) and bitwise operations (AND, OR, XOR, NOT, NAND, NOR, XNOR, shift left, shift right) on binary numbers instantly in your browser, with no signup, no ads, and no server uploads.

What is the Binary Number System? Binary is a base-2 number system that uses only two symbols: 0 and 1. It is the fundamental language of all modern digital computers and electronic devices. Every piece of data in a computer — numbers, text, images, videos, and programs — is ultimately represented as a sequence of binary digits (bits).

The binary system was first described by Gottfried Leibniz in 1679. However, it was the development of digital computers in the 20th century that made binary arithmetic essential. Claude Shannon's 1948 work on information theory cemented binary as the language of computing.

Bitwise Operations - AND: Both bits must be 1 for the result to be 1. - OR: At least one bit must be 1 for the result to be 1. - XOR: Bits must differ (one is 0, the other is 1) for the result to be 1. - NOT: Inverts every bit (1 becomes 0, 0 becomes 1). - NAND: Inverse of AND (result is 1 unless both bits are 1). - NOR: Inverse of OR (result is 1 only when both bits are 0). - XNOR: Inverse of XOR (result is 1 when both bits are equal).
Binary Arithmetic - Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=0 with carry 1. - Subtraction: Uses borrowing when subtracting 1 from 0. - Multiplication: Based on shift-and-add, similar to decimal multiplication but simpler. - Division: Uses repeated subtraction or shift-subtract algorithm.
Applications Binary numbers are used everywhere in computing: digital logic circuits (AND, OR, XOR gates), CPU instructions and memory addressing, data representation, networking (IP addresses, subnet masks), cryptography, and error detection/correction codes.
Privacy & Performance All calculations run locally in your browser. No data is transmitted, logged, or stored on any server. Results appear instantly with no page reloads.
🔒 Client-side only⚡ Web Worker processing🌐 Works offline🛡️ Zero server logs

Frequently Asked Questions

Got questions about binary numbers, bitwise operations, or how to use this tool? We've got answers.

What is the binary number system?

Binary is a base-2 number system that uses only two digits: 0 and 1. Each position in a binary number represents a power of 2, just as each position in a decimal number represents a power of 10. For example, 1010₂ = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 10 in decimal.

What are bitwise operations?

Bitwise operations work on binary numbers at the level of individual bits. Common operations include AND (both bits must be 1), OR (at least one bit is 1), XOR (bits must differ), and NOT (invert all bits). These are fundamental in digital electronics and computer programming.

How does binary addition work?

Binary addition follows the same principle as decimal addition but with base 2: 0+0=0, 0+1=1, 1+0=1, 1+1=0 with a carry of 1. When a carry propagates, it's added to the next higher bit.

What is the difference between AND, OR, and XOR?

AND: 1 AND 1 = 1, all other combinations = 0. OR: 0 OR 0 = 0, all other combinations = 1. XOR (exclusive OR): 1 XOR 1 = 0, 1 XOR 0 = 1, 0 XOR 1 = 1, 0 XOR 0 = 0. XOR is particularly useful in cryptography and parity checking.

What does shift left / shift right do?

Shifting left by n bits multiplies the number by 2ⁿ, filling the rightmost bits with zeros. Shifting right by n bits divides by 2ⁿ (integer division), discarding the rightmost bits. These operations are extremely efficient in digital circuits.

How do I convert binary to decimal?

Multiply each binary digit (bit) by 2 raised to its position power (starting from 0 on the right), then sum all results. For example: 1010₂ = 1×8 + 0×4 + 1×2 + 0×1 = 10.

Is this calculator private?

Yes. All calculations run locally in your browser. No data is transmitted, logged, or stored on any server.