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2s Complement Calculator

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Convert between decimal and 2’s complement binary with our free calculator. Supports 4-64 bit widths with step-by-step explanations and bit visualization.

2’s Complement Calculator: Complete Reference Guide

Description

Easily convert between decimal and binary using 2’s complement representation. Ideal for computer science students and engineers working with signed binary numbers and bitwise operations.

What Is It?

This calculator helps you:

  • Convert decimal numbers to their 2’s complement binary representation
  • Convert 2’s complement binary back to decimal
  • Understand the bit-level representation of signed numbers
  • Perform bit negation and value sign flipping
  • Visualize how numbers are stored in computer memory

Calculation Formula

2’s Complement = (Invert Bits of Absolute Value) + 1
For negative numbers:

  1. Convert absolute value to binary
  2. Invert all bits (1’s complement)
  3. Add 1 to the result

How to Use

  1. Select bit width (4, 8, 16, 32, or 64 bits)
  2. Choose conversion direction:
    • Decimal → Binary: Enter any integer within bit range
    • Binary → Decimal: Enter binary digits (e.g., “1101”)
  3. View results:
    • Binary representation with highlighted sign bit
    • Decimal equivalent
    • Hexadecimal conversion
    • Step-by-step explanation for negative numbers

Key Terms

  • Sign Bit: Leftmost bit (1 = negative, 0 = positive)
  • Overflow: When result exceeds bit width capacity
  • Endianness: Byte ordering (calculator assumes big-endian)
  • Bit Flipping: Inverting all bits (equivalent to 1’s complement)

FAQs

Q: Why does 8-bit 2’s complement range from -128 to 127?
A: The sign bit uses one bit position, leaving 7 bits for magnitude (2^7 = 128 possible values per sign).

Q: How is -1 represented in 8-bit 2’s complement?
A: As 11111111 (all bits set) because:

  1. +1 = 00000001
  2. Inverted = 11111110
  3. Add 1 = 11111111

Q: Can I convert floating-point numbers?
A: No - this calculator only handles integers. Floating-point uses different representation.

Formula Source

The 2’s complement method is derived from:

  • IEEE 754 standards for binary arithmetic
  • Computer architecture principles (e.g., Von Neumann architecture)
  • Digital logic design fundamentals

Important Notes

⚠️ Overflow Handling: Results exceeding bit width will show warning
⚠️ Sign Extension: When increasing bit width, sign bit replicates leftward
⚠️ Zero Representation: There’s only one zero (000…0), unlike 1’s complement
⚠️ Minimum Value: -2^(n-1) has no positive counterpart (e.g., -128 in 8-bit)

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