What is 11111111 in 2s complement?
The definitive breakdown of 0xFF in signed 8-bit registers, negative weights, and sign extension principles.
In an 8-bit Two’s Complement signed binary system, the bit pattern 11111111 represents decimal -1. Explore the mathematical proofs, MSB negative weighting, and modular arithmetic behind this fundamental value.
1. Understanding the Core Concept
In an 8-bit signed Two’s Complement system, the binary string 11111111 represents the decimal value -1. In contrast, in an 8-bit unsigned integer system, 11111111 represents the maximum possible value: +255. The interpretation of the exact same 8 bits depends entirely on whether the microprocessor register is treated as signed or unsigned.
2. How Does It Work? Step-by-Step Methodology
There are three mathematically rigorous ways to prove why 11111111_2 equals -1: Method 1: Negative MSB Weighting In an 8-bit Two’s Complement system, the most significant bit (bit 7) carries a negative weight of -2⁷ = -128. All other bits carry positive powers of 2. Summing all weights: (-128) + 64 + 32 + 16 + 8 + 4 + 2 + 1 = -128 + 127 = -1. Method 2: Inversion and Increment Algorithm To find the decimal value of a negative Two’s complement number (where MSB = 1): 1. Invert all bits (One’s complement): ~11111111 = 00000000. 2. Add 1 to the result: 00000000 + 1 = 00000001 (Decimal magnitude 1). 3. Apply the negative sign: -1. Method 3: Modular Arithmetic An 8-bit register operates modulo 256 (2⁸). The unsigned value is 255. In modular arithmetic: 255 - 256 = -1.
3. Detailed Worked Example & Verification
4. Essential Rules & Edge Cases
- In 8-bit signed Two’s complement, 11111111 is ALWAYS -1.
- In 8-bit unsigned integer arithmetic, 11111111 is ALWAYS +255.
- To represent -1 in wider bit widths (16-bit, 32-bit, 64-bit), you must sign-extend the MSB (replicate 1s all the way to the new register width).
- A 16-bit value of 00000000 11111111 is NOT -1; it is positive +255 because its sign bit (bit 15) is 0.
5. Practical Engineering Applications
- Return Codes: C/C++ functions returning -1 (EOF or error status) encoded in memory as 0xFF or 0xFFFFFFFF.
- Bitwise Masks: Bitwise NOT of 0 (~0) produces all 1s (0xFF or -1), commonly used as an all-ones bitmask.
- ALU Comparisons: Evaluating negative flags in conditional branch instructions (B.LT, JL).
6. Common Mistakes to Avoid
- WarningConfusing signed Two’s complement with unsigned binary and assuming 11111111 is always 255.
- WarningThinking that 11111111 in signed magnitude is -1 (in signed magnitude, -1 is 10000001, where MSB is sign and remaining 7 bits are magnitude).
- WarningForgetting to sign-extend when casting an 8-bit signed byte to a 16-bit signed integer in C/C++.
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Frequently Asked Questions
Why is 11111111 not -127 in Two’s Complement?
In One’s Complement, 11111111 is negative zero (-0), and in Signed Magnitude, 11111111 is -127. But Two’s Complement adds 1 to eliminate dual zeros, shifting values so that all ones represents -1.
What is 10000000 in 8-bit Two’s Complement?
10000000 represents -128, which is the most negative number representable in an 8-bit signed register.
How does hexadecimal 0xFF relate to 11111111?
0xFF is the exact hexadecimal representation of the binary byte 11111111. When interpreted as a signed 8-bit char/int8, 0xFF equals -1.
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