To convert the binary number $(1\ 1011)_2$ to its decimal expansion, let's break it down step by step:

Step 1: Identify the place values

The binary number $(1\ 1011)_2$ is equivalent to:

$1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0$

Step 2: Calculate the powers of 2

• $2^4 = 16$
• $2^3 = 8$
• $2^2 = 4$
• $2^1 = 2$
• $2^0 = 1$

Step 3: Multiply each bit by its corresponding power of 2

$1 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1 = 16 + 8 + 0 + 2 + 1$

Step 4: Add the results

$16 + 8 + 2 + 1 = 27$

Conclusion:

The decimal expansion of $(1\ 1011)_2$ is $27$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you convert a fractional binary number to decimal?
2. How can you convert decimal numbers back to binary?
3. What is the significance of each bit in a binary number?
4. How do you convert binary numbers larger than 8 bits into decimal?
5. What is the binary equivalent of decimal fractions?

Tip: When converting binary to decimal, remember that the rightmost bit represents $2^0$, and each bit to the left increases the power of 2.