A computer uses 32-bit as a Word for storing data and instruction. In terms of  the 2's complement binary number system, what is the result of conducting the arithmetic for the following two binary numbers (the result is to be expressed in decimal number):

1111 0000 0001 1100 1110 0001 0011  1101  

plus

1100 1110 1000 0011 0010 0010 0001  0111 

Please note that there is no space in the computer memory for storing the binary numbers; a space is included here for the "convenience of our human eyes".

To solve this problem, we need to add the two given 32-bit binary numbers using the two's complement system and then express the result in decimal. Here are the steps:

### Step 1: Analyze the Binary Numbers
The given binary numbers are:

- **Binary 1:**  
  \(1111\ 0000\ 0001\ 1100\ 1110\ 0001\ 0011\ 1101\)
- **Binary 2:**  
  \(1100\ 1110\ 1000\ 0011\ 0010\ 0010\ 0001\ 0111\)

### Step 2: Determine if the Numbers are Negative or Positive
In two's complement, the most significant bit (MSB) indicates the sign of the number:
- If the MSB is **1**, the number is negative.
- If the MSB is **0**, the number is positive.

- **Binary 1** has an MSB of **1**, so it is negative.
- **Binary 2** also has an MSB of **1**, so it is also negative.

### Step 3: Convert the Numbers to Decimal
First, we convert each binary number to its absolute value in decimal, then apply the sign.

#### **Binary 1:**
1. **Invert the bits** (flip 0s to 1s and 1s to 0s):
   \[
   0000\ 1111\ 1110\ 0011\ 0001\ 1110\ 1100\ 0010
   \]
2. **Add 1**:
   \[
   0000\ 1111\ 1110\ 0011\ 0001\ 1110\ 1100\ 0011
   \]
3. **Convert to decimal**:
   - \(0 \times 2^{31} + 0 \times 2^{30} + \cdots + 1 \times 2^4 + 1 \times 2^1 + 1 \times 2^0\)
   - Decimal value: \( 267401411 \)
   - Since the original number was negative, Binary 1 is \( -267401411 \).

#### **Binary 2:**
1. **Invert the bits**:
   \[
   0011\ 0001\ 0111\ 1100\ 1101\ 1101\ 1110\ 1000
   \]
2. **Add 1**:
   \[
   0011\ 0001\ 0111\ 1100\ 1101\ 1101\ 1110\ 1001
   \]
3. **Convert to decimal**:
   - Decimal value: \( 833731305 \)
   - Since the original number was negative, Binary 2 is \( -833731305 \).

### Step 4: Add the Two Numbers
- **Binary 1**: \( -267401411 \)
- **Binary 2**: \( -833731305 \)

Adding these:
\[
-267401411 + (-833731305) = -1101132716
\]

### Step 5: Express the Result in Decimal
The final result of adding the two binary numbers is \( \boxed{-1101132716} \) in decimal.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. What happens if you add two negative numbers in two's complement?
2. How do you determine if an overflow occurs in two's complement addition?
3. Can you explain how to subtract two binary numbers directly using two's complement?
4. How does two's complement handle large negative numbers?
5. How does the process change if the computer uses 64-bit words instead of 32-bit?

**Tip:** When adding two negative numbers in two's complement, the result will also be negative unless an overflow occurs, in which case it might wrap around to a positive number.

Explore how to add two 32-bit binary numbers using the two's complement system and convert the result to decimal. This problem covers binary arithmetic concepts, including binary addition and two's complement representation. Suitable for advanced high school students and beyond.

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