To solve the problem of adding two 32-bit binary numbers in a 2's complement system and expressing the result in decimal, we can follow these steps:

1. Interpret the Binary Numbers: Identify whether each binary number is positive or negative based on the most significant bit (MSB).

   • If the MSB (leftmost bit) is 0, the number is positive.
   • If the MSB is 1, the number is negative.

2. Convert the Binary Numbers to Decimal: Convert both numbers to their decimal equivalents.

3. Add the Two Numbers: Perform the arithmetic addition.

4. Express the Result: Convert the final binary sum into its decimal equivalent.

Step 1: Interpret the Binary Numbers

Given binary numbers:

• First number: 0000 0000 0000 0000 0000 0000 0011 1101

• Second number: 1111 1111 1100 1010 1111 0001 0001 0111

• The first number has MSB 0, so it's positive.

• The second number has MSB 1, so it's negative.

Step 2: Convert the Binary Numbers to Decimal

First Number (Positive):

The binary 0000 0000 0000 0000 0000 0000 0011 1101 in decimal:

• Directly convert it as a positive binary number.

$2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 = 61$

Second Number (Negative):

The binary 1111 1111 1100 1010 1111 0001 0001 0117 is in 2's complement form.

To find its decimal equivalent:

1. Invert the bits: 0000 0000 0011 0101 0000 1110 1110 1000.
2. Add 1 to the inverted bits to get the positive version:
   • 0000 0000 0011 0101 0000 1110 1110 1001
3. Convert the result to decimal:
   • This equals 14,472,441
   • Then, negate the result to represent the original negative number:
   • Decimal Value = $$-14,472,441]

Step 3: Add the Two Numbers

Now add the decimal equivalents:

[ 61 + (-14,472,441) = -14,472,380$$

Step 4: Result

The final result in decimal is $-14,472,380$.

Do you want further clarification or details?

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Here are some follow-up questions:

1. How does the 2's complement system work with signed integers?
2. How do you convert a negative 2's complement binary number to decimal?
3. What happens if the sum of two 32-bit numbers exceeds the 32-bit range?
4. What is the role of the most significant bit (MSB) in 2's complement?
5. How can you detect overflow in binary arithmetic?

Tip: Understanding how to convert 2's complement binary to decimal will help in interpreting both positive and negative binary numbers correctly.