Sum of the two binary numbers `(1000010)_(2)` and `(110011)_(2)` is

Options

A. `(111101)_(2)`
B. `(111111)_(2)`
C. `(101111)_(2)`
D. `(111001)_(2)`

Explanation

To solve the problem of adding the two binary numbers \( (1000010)_2 \) and \( (110011)_2 \), we can follow these steps:

### Step 1: Align the Binary Numbers
Align the two binary numbers for addition, ensuring that they are properly lined up by their least significant bits (rightmost side).

```
1000010
+ 110011
```

### Step 2: Add the Binary Numbers
We will add the numbers column by column from right to left, just like in decimal addition. Remember that in binary:
- \( 0 + 0 = 0 \)
- \( 0 + 1 = 1 \)
- \( 1 + 0 = 1 \)
- \( 1 + 1 = 10 \) (which means write down 0 and carry over 1)
- \( 1 + 1 + 1 = 11 \) (which means write down 1 and carry over 1)

Now let's perform the addition:

```
Carry: 1 1 1 1
1000010
+ 110011
__________
```

Starting from the rightmost column:
1. **Rightmost column**: \( 0 + 1 = 1 \) (write down 1)
2. **Second column**: \( 1 + 1 = 10 \) (write down 0 and carry 1)
3. **Third column**: \( 0 + 0 + 1 = 1 \) (write down 1)
4. **Fourth column**: \( 0 + 1 = 1 \) (write down 1)
5. **Fifth column**: \( 0 + 1 = 1 \) (write down 1)
6. **Sixth column**: \( 1 + 0 = 1 \) (write down 1)
7. **Seventh column**: \( 1 + 0 = 1 \) (write down 1)

Putting it all together, we get:

```
Carry: 1 1 1 1
1000010
+ 110011
__________
1111001
```

### Step 3: Write the Result
The sum of the two binary numbers \( (1000010)_2 \) and \( (110011)_2 \) is \( (1111001)_2 \).

### Final Answer
Thus, the sum of the two binary numbers is:
\[ (1111001)_2 \]

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