2 Complement Calculator To Decimal

2's Complement Conversion:

\[ Decimal = (-b_{n-1} \times 2^{n-1}) + \sum_{i=0}^{n-2} b_i \times 2^i \]

Decimal Result:

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1. What Is Two's Complement?

Two's complement is a mathematical operation on binary numbers, as well as a binary signed number representation. It is the most common method of representing signed integers on computers.

2. How Two's Complement Conversion Works

The conversion from two's complement binary to decimal follows this formula:

\[ Decimal = (-b_{n-1} \times 2^{n-1}) + \sum_{i=0}^{n-2} b_i \times 2^i \]

Where:

\( b_{n-1} \) — The most significant bit (sign bit)
\( b_i \) — The i-th bit of the binary number
\( n \) — The total number of bits

Explanation: If the sign bit is 1, the number is negative and we subtract 2^(n-1) from the positive value of the remaining bits.

3. Applications of Two's Complement

Details: Two's complement is widely used in computer systems for arithmetic operations because it simplifies the hardware design for addition and subtraction of signed numbers.

4. Using This Calculator

Tips: Enter a binary string consisting of only 0s and 1s. The calculator will automatically detect if it's a positive or negative number based on the most significant bit.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of numbers representable in n-bit two's complement?
A: The range is from -2^(n-1) to 2^(n-1)-1. For example, 8-bit two's complement can represent numbers from -128 to 127.

Q2: How do I convert a negative decimal to two's complement?
A: To convert a negative number: 1) Take the absolute value, 2) Convert to binary, 3) Invert all bits, 4) Add 1 to the result.

Q3: Why is two's complement used instead of sign-magnitude?
A: Two's complement has several advantages: only one representation for zero, simpler arithmetic operations, and no need for special handling of negative numbers.

Q4: What happens if I enter more bits than typical computer word sizes?
A: The calculator will still work correctly regardless of the number of bits, though practical computer systems have fixed word sizes (8, 16, 32, or 64 bits).

Q5: Can two's complement represent fractions?
A: No, two's complement is specifically for integers. Fractional numbers are typically represented using floating-point formats.