Below is a 2's complement binary number:

10101101

Please write its base 10 value in the space provided.

Question

Below is a 2's complement binary number:

10101101

Please write its base 10 value in the space provided.

BlackTom AI · Accepted Answer

Because the leftmost bit is 1 in an 8-bit two's-complement number, this represents a negative integer.

1) Relevant concepts
- Two's-complement representation uses the most significant bit (MSB) as the sign bit. For n bits, values range from -2^(n-1) to 2^(n-1)-1 (for 8 bits: -128 to +127).
- To find the magnitude of a negative two's-complement number, invert all bits and add 1, or compute the unsigned value and subtract 2^n (256 for 8 bits).

2) Step-by-step reasoning (method A: invert-and-add-1)
Original 8-bit pattern: 10101101

Invert bits:
10101101 -> 01010010

Add 1 to the inverted result:
01010010
+       1
---------
01010011

3) Intermediate calculations (convert 01010011 to decimal)
Bit weights (from MSB to LSB): 128 64 32 16 8 4 2 1
01010011 = 0*128 + 1*64 + 0*32 + 1*16 + 0*8 + 0*4 + 1*2 + 1*1
= 64 + 16 + 2 + 1
= 83

Since the original number was negative, the value is -83.

Alternate check (method B: unsigned then subtract 256)
Unsigned interpretation of 10101101 = 1*128 +0*64 +1*32 +0*16 +1*8 +1*4 +0*2 +1*1
= 128 + 32 + 8 + 4 + 1 = 173
Two's-complement value = 173 - 256 = -83 (same result).

4) Why this is correct
Both accepted two's-complement conversion methods lead to 83 as the magnitude and a negative sign because the MSB is 1. The arithmetic check (unsigned minus 256) confirms the result.

5) Common incorrect answers and why they are wrong
- 173: This treats the bits as an unsigned byte; invalid for two's-complement signed interpretation.
- Any other positive number: ignores the sign bit or misapplies the invert/add-1 procedure.

6) Conclusion
The correct two's-complement interpretation of 10101101 (8-bit) is -83, which matches the provided answer: -83.

Below is a 2's complement binary number: 10101101 Please write its base 10 value in the space provided.简答题

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