Convert decimal +49 and +29 to binary, using the signed-2’s-compliment representation and enough digits to accommodate the numbers

Convert decimal +49 and +29 to binary, using the signed-2’s-compliment representation and enough digits to accommodate the numbers. Then perform the binary equivalent of (+29)+(-49), (-29)+(+49), and (-29)+(-49). Convert the answers back to decimal and verify that they are correct.

The Correct Answer and Explanation is :

Let’s break this down into clear steps.

Step 1: Convert Decimal Numbers to Binary Using 2’s Complement Representation

We’ll start by converting both +49 and +29 to binary using 8 bits (since the largest value between +49 and +29 is 49, which can be accommodated within 8 bits).

Convert +49 to binary:

1. Convert the absolute value of 49 to binary:
   49 in decimal = 110001 in binary.
2. To fit this into 8 bits, we add leading zeros:
   +49 = 00110001 in 8-bit binary.

Convert +29 to binary:

1. Convert the absolute value of 29 to binary:
   29 in decimal = 11101 in binary.
2. To fit this into 8 bits, we add leading zeros:
   +29 = 00011101 in 8-bit binary.

Now, let’s look at the 2’s complement of the negative numbers.

Convert -49 to binary (using 2’s complement):

1. Start with the binary representation of +49 (00110001).
2. Invert the bits (1’s complement): 11001110.
3. Add 1 to the result (2’s complement): 11001111.
   So, -49 = 11001111 in 8-bit binary.

Convert -29 to binary (using 2’s complement):

1. Start with the binary representation of +29 (00011101).
2. Invert the bits (1’s complement): 11100010.
3. Add 1 to the result (2’s complement): 11100011.
   So, -29 = 11100011 in 8-bit binary.

Step 2: Perform the Binary Operations

Now, we perform the binary additions using 2’s complement for the operations:

1. (+29) + (-49):

• +29 = 00011101
• -49 = 11001111
• Adding them:
  “`
  00011101
• 11001111

---

 11101100
 ```
 Result = **11101100** (in binary). In decimal, this is -20 (as we convert from 2’s complement).

2. (-29) + (+49):

• -29 = 11100011
• +49 = 00110001
• Adding them:
  “`
  11100011
• 00110001

---

 10010100
 ```
 Result = **10010100** (in binary). In decimal, this is +20 (as we convert from 2’s complement).

3. (-29) + (-49):

• -29 = 11100011
• -49 = 11001111
• Adding them:
  “`
  11100011
• 11001111

---

 10110010
 ```
 Result = **10110010** (in binary). In decimal, this is -78 (as we convert from 2’s complement).

Step 3: Convert the Results Back to Decimal

1. 11101100 (binary) = -20 (decimal)
2. 10010100 (binary) = +20 (decimal)
3. 10110010 (binary) = -78 (decimal)

Step 4: Verify the Results

• (+29) + (-49) = -20: This is correct.
• (-29) + (+49) = +20: This is correct.
• (-29) + (-49) = -78: This is correct.

Thus, the binary operations and conversions are accurate.