Binary Calculator

Convert between binary, octal, decimal, and hexadecimal. This calculator is useful for programming, networking, and digital electronics when you need to translate values across number systems.

Operation Manual

  1. Choose the base of your input value (binary, octal, decimal, or hexadecimal).
  2. Enter the number (use a leading minus sign for negative values).
  3. Click Convert to generate equivalent values in the other bases.
  4. Copy the output for use in code, debugging, or documentation.

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Binary Number System Basics

Binary is a base-2 number system using only 0 and 1:

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

Example: 1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 13₁₀

Base Prefixes and Quick Conversions

In many programming languages, values are written with prefixes to indicate the base: 0b for binary, 0o for octal, and 0x for hexadecimal.

A practical shortcut: group binary into 4-bit chunks (nibbles) to convert to hex. Example: 11110000₂ becomes F0₁₆.

Common Binary Operations

AND (∧): 1 if both bits are 1

OR (∨): 1 if at least one bit is 1

XOR (⊕): 1 if bits are different

NOT (¬): Inverts each bit

Shift Left (≪): Multiply by 2

Shift Right (≫): Divide by 2

These operations are commonly used in programming, bit masks, permissions, and low level protocols. Even if you only convert bases, understanding these symbols helps you read documentation and code examples.

Signed Integers and Two's Complement

Computers typically store negative integers using two's complement, which depends on a fixed bit width (8-bit, 16-bit, 32-bit, and so on). In that representation, the most significant bit acts as a sign bit, and a negative value is encoded by inverting bits and adding 1.

This converter supports negative values by using a leading minus sign in the output (for example, -1010). That is a mathematical representation, not a fixed-width two's complement encoding. To work with two's complement, you must choose a bit width and pad the binary string accordingly.

Common Mistakes

  • Mixing up uppercase and lowercase hex digits. Both are valid, but output is often uppercase.
  • Dropping leading zeros when a fixed width is required (for example, bytes or 32-bit values).
  • Confusing decimal 10 with hex A or binary 1010.
  • For signed values, forgetting that two's complement interpretation requires a chosen bit width.

Applications in Computing

Binary is fundamental to digital systems:

  • Data Storage: All data stored as binary
  • Digital Logic: CPU operations use binary
  • Networking: Data transmission in binary
  • Error Detection: Parity bits and checksums

Understanding binary is crucial for computer science and digital electronics.

Binary Patterns

Common binary patterns and their uses:

  • 00000000: Null byte, string terminator
  • 11111111: All bits set, broadcast address
  • 10000000: Sign bit in signed integers
  • 01111111: Maximum positive signed byte

These patterns are frequently used in programming and networking protocols.

Related calculators: big number calculator and character counter.