The Binary Number `(110111111011)_2` when written in base 10 equals :

Options

A. 1985
B. 3357
C. 2797
D. 3579

Answer

To convert the binary number `(110111111011)_2` to its decimal (base 10) equivalent, we will follow these steps:

### Step 1: Write down the binary number and assign powers of 2
The binary number is `110111111011`. We will assign powers of 2 starting from the rightmost bit (which is 0) to the leftmost bit (which is 11).

```
Binary: 1 1 0 1 1 1 1 1 0 1 1
Powers: 11 10 9 8 7 6 5 4 3 2 1 0
```

### Step 2: Calculate the value of each bit
Now we will calculate the value of each bit by multiplying it by 2 raised to the power of its position.

- \(1 \times 2^{11} = 1 \times 2048 = 2048\)
- \(1 \times 2^{10} = 1 \times 1024 = 1024\)
- \(0 \times 2^{9} = 0 \times 512 = 0\)
- \(1 \times 2^{8} = 1 \times 256 = 256\)
- \(1 \times 2^{7} = 1 \times 128 = 128\)
- \(1 \times 2^{6} = 1 \times 64 = 64\)
- \(1 \times 2^{5} = 1 \times 32 = 32\)
- \(1 \times 2^{4} = 1 \times 16 = 16\)
- \(0 \times 2^{3} = 0 \times 8 = 0\)
- \(1 \times 2^{2} = 1 \times 4 = 4\)
- \(1 \times 2^{1} = 1 \times 2 = 2\)
- \(1 \times 2^{0} = 1 \times 1 = 1\)

### Step 3: Add all the values together
Now we will sum all the calculated values:

\[ 2048 + 1024 + 0 + 256 + 128 + 64 + 32 + 16 + 0 + 4 + 2 + 1 = 3565 \]

### Final Answer
The decimal (base 10) equivalent of the binary number `(110111111011)_2` is **3565**.

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