The difference of two numbers `(1100110011)_(2) and (1101001011)_(2)` in binary system is.

Options

A. `(100000)_(2)`
B. `(101010)_(2)`
C. `(11000)_(2)`
D. `(10111)_(2)`

Answer

To find the difference between the two binary numbers \( (1100110011)_2 \) and \( (1101001011)_2 \), we will follow these steps:

### Step 1: Convert the binary numbers to decimal
1. **Convert \( (1100110011)_2 \) to decimal:**
- Write down the powers of 2 from right to left:
- \( 2^9, 2^8, 2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0 \)
- Assign the binary digits to their respective powers:
- \( 1 \times 2^9 + 1 \times 2^8 + 0 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \)
- Calculate:
- \( 1 \times 512 + 1 \times 256 + 0 + 0 + 1 \times 32 + 1 \times 16 + 0 + 0 + 1 \times 2 + 1 \times 1 \)
- \( 512 + 256 + 32 + 16 + 2 + 1 = 819 \)

2. **Convert \( (1101001011)_2 \) to decimal:**
- Write down the powers of 2 from right to left:
- \( 2^9, 2^8, 2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0 \)
- Assign the binary digits to their respective powers:
- \( 1 \times 2^9 + 1 \times 2^8 + 0 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \)
- Calculate:
- \( 1 \times 512 + 1 \times 256 + 0 + 1 \times 64 + 0 + 0 + 1 \times 8 + 0 + 1 \times 2 + 1 \times 1 \)
- \( 512 + 256 + 64 + 8 + 2 + 1 = 843 \)

### Step 2: Calculate the difference in decimal
- Now, subtract the two decimal values:
- \( 843 - 819 = 24 \)

### Step 3: Convert the decimal difference back to binary
1. **Convert \( 24 \) to binary:**
- Divide by 2 and keep track of the remainders:
- \( 24 \div 2 = 12 \) remainder \( 0 \)
- \( 12 \div 2 = 6 \) remainder \( 0 \)
- \( 6 \div 2 = 3 \) remainder \( 0 \)
- \( 3 \div 2 = 1 \) remainder \( 1 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)
- Collect the remainders from bottom to top: \( 11000 \)

### Final Answer
The difference of the two binary numbers \( (1100110011)_2 \) and \( (1101001011)_2 \) is \( (11000)_2 \).

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