What is the sum of two numbers `(11110)_(2) and (1010)_(2)` ?

Options

A. `(101000)_(2)`
B. `(110000)_(2)`
C. `(100100)_(2)`
D. `(101100)_(2)`

Answer

To find the sum of the two binary numbers \( (11110)_2 \) and \( (1010)_2 \), we can follow these steps:

### Step 1: Convert the binary numbers to decimal
1. **Convert \( (11110)_2 \)**:
- Each digit represents a power of 2, starting from the right (0th power).
- \( 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \)
- \( = 1 \times 16 + 1 \times 8 + 1 \times 4 + 1 \times 2 + 0 \times 1 \)
- \( = 16 + 8 + 4 + 2 + 0 = 30 \)

2. **Convert \( (1010)_2 \)**:
- Similarly, calculate the decimal value.
- \( 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \)
- \( = 1 \times 8 + 0 \times 4 + 1 \times 2 + 0 \times 1 \)
- \( = 8 + 0 + 2 + 0 = 10 \)

### Step 2: Add the decimal values
- Now that we have both numbers in decimal:
- \( 30 + 10 = 40 \)

### Step 3: Convert the sum back to binary
1. **Convert \( 40 \) to binary**:
- Use the division method:
- \( 40 \div 2 = 20 \) remainder \( 0 \)
- \( 20 \div 2 = 10 \) remainder \( 0 \)
- \( 10 \div 2 = 5 \) remainder \( 0 \)
- \( 5 \div 2 = 2 \) remainder \( 1 \)
- \( 2 \div 2 = 1 \) remainder \( 0 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)
- Reading the remainders from bottom to top gives us \( (101000)_2 \).

### Final Answer
The sum of \( (11110)_2 \) and \( (1010)_2 \) is \( (101000)_2 \).

---