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 1111 1111 1111 1111 1111 1111  1001

minus

0000 0000 0000 0000 0000 0000 0001  0101

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 the problem, we'll follow these steps:

1. **Understand the binary representation:**
   - The two binary numbers given are:
     - \( \text{Binary 1: } 1111\ 1111\ 1111\ 1111\ 1111\ 1111\ 1111\ 1001 \)
     - \( \text{Binary 2: } 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0001\ 0101 \)
   - These are 32-bit binary numbers using two's complement representation.

2. **Convert the binary numbers to decimal:**
   - In two's complement, the most significant bit (MSB) determines the sign. If the MSB is 1, the number is negative; if it is 0, the number is positive.
   
   - **For Binary 1:**
     - MSB is 1, so the number is negative.
     - To find its decimal value:
       - Invert all bits and add 1 to the result.
       - Binary 1: \( 1111\ 1111\ 1111\ 1111\ 1111\ 1111\ 1111\ 1001 \)
       - Inverted: \( 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0110 \)
       - Add 1: \( 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0111 \)
       - Decimal value: \( 7 \)
       - So, Binary 1 is \( -7 \).

   - **For Binary 2:**
     - MSB is 0, so the number is positive.
     - Decimal value: \( 1 \times 2^4 + 1 \times 2^2 + 1 \times 2^0 = 16 + 4 + 1 = 21 \)
       - So, Binary 2 is \( 21 \).

3. **Subtract the binary numbers:**
   - The problem asks to subtract Binary 2 from Binary 1:
     \[
     -7 - 21 = -28
     \]

4. **Express the result in decimal:**
   - The result of the subtraction is \( -28 \) in decimal.

So, the final answer is:

\(\boxed{-28}\)

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you convert a binary number to its two's complement?
2. How would you subtract two binary numbers directly without converting them to decimal first?
3. What is the significance of the most significant bit (MSB) in two's complement numbers?
4. How does the two's complement system differ from the sign-magnitude representation?
5. How would the operation change if the numbers were 16-bit instead of 32-bit?

**Tip:** When working with binary subtraction in two's complement, remember that subtraction can be done by adding the negative of the second number to the first.

Learn how to subtract binary numbers using the two's complement representation. This problem involves converting binary numbers to decimal, understanding two's complement arithmetic, and performing subtraction. Suitable for advanced high school students or anyone interested in computer arithmetic.

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