if a 16 bit machine adds 0101 0101 0101 0101 and 0011 1001 1100 1111, and both numbers are 2’s complement what is the answer

To solve this, we will add the two binary numbers directly and take into account that these are 16-bit, 2's complement numbers.

### Step-by-step solution:

1. **Write the numbers in binary:**
   - \( 0101\ 0101\ 0101\ 0101_2 \)
   - \( 0011\ 1001\ 1100\ 1111_2 \)

2. **Add the two binary numbers:**

```
  0101 0101 0101 0101
+ 0011 1001 1100 1111
----------------------
  1000 1111 0001 0100
```

3. **Check for overflow:**
   Since both operands are positive (their most significant bits, MSB, are 0), and the result's MSB is 1, this suggests an overflow has occurred. In a 16-bit 2's complement system, the result would actually represent a negative number.

4. **Interpret the result:**
   The binary result is \( 1000\ 1111\ 0001\ 0100_2 \).

   To interpret this in 2's complement, we need to find its decimal value:
   - The MSB is 1, so it's a negative number.
   - To find the magnitude, take the 2's complement (invert all bits and add 1):

     ```
     Invert: 0111 0000 1110 1011
     Add 1:  0111 0000 1110 1100
     ```

     \( 0111\ 0000\ 1110\ 1100_2 \) is \( 28908_{10} \).

   - Since the original number was negative, the final answer is \( -28908 \).

### Final Answer: 
The sum of the two 16-bit 2's complement numbers is \( -28908 \).

Would you like more details or explanations on any step?

Here are 5 related questions you might find interesting:
1. How do you detect overflow in binary addition?
2. What are the advantages of 2's complement over 1's complement?
3. How does a 16-bit machine handle overflow?
4. How would you subtract binary numbers using 2's complement?
5. How would the addition differ if the numbers were unsigned?

**Tip:** When adding 2's complement numbers, always check the sign bits before and after the addition to detect overflow.

If a 16-bit machine adds 0101 0101 0101 0101 and 0011 1001 1100 1111, and both numbers are 2’s complement, what is the answer?

Learn how to add two 16-bit 2's complement numbers (0101 0101 0101 0101 and 0011 1001 1100 1111) and handle overflow detection. This detailed step-by-step solution explains binary addition, overflow detection, and how to interpret negative results in 2's complement arithmetic. Suitable for Grades 11-12 or introductory college level.

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