To find the decimal equivalent of the binary number 1011011, we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understanding Binary Numbers Binary numbers are base-2 numbers, which means they only use two digits: 0 and 1. Each digit in a binary number represents a power of 2, starting from the rightmost digit, which represents \(2^0\). ### Step 2: Assigning Powers of 2 For the binary number 1011011, we will assign powers of 2 to each digit from right to left: - The rightmost digit (1) is in the \(2^0\) place. - The next digit (1) is in the \(2^1\) place. - The next digit (0) is in the \(2^2\) place. - The next digit (1) is in the \(2^3\) place. - The next digit (1) is in the \(2^4\) place. - The next digit (0) is in the \(2^5\) place. - The leftmost digit (1) is in the \(2^6\) place. So, we can write it out as follows: \[ \begin{align*} 1 & \times 2^6 \\ 0 & \times 2^5 \\ 1 & \times 2^4 \\ 1 & \times 2^3 \\ 0 & \times 2^2 \\ 1 & \times 2^1 \\ 1 & \times 2^0 \\ \end{align*} \] ### Step 3: Calculating Each Term Now, we will calculate the value of each term: - \(1 \times 2^6 = 1 \times 64 = 64\) - \(0 \times 2^5 = 0 \times 32 = 0\) - \(1 \times 2^4 = 1 \times 16 = 16\) - \(1 \times 2^3 = 1 \times 8 = 8\) - \(0 \times 2^2 = 0 \times 4 = 0\) - \(1 \times 2^1 = 1 \times 2 = 2\) - \(1 \times 2^0 = 1 \times 1 = 1\) ### Step 4: Summing the Values Next, we will sum all these values together: \[ 64 + 0 + 16 + 8 + 0 + 2 + 1 = 91 \] ### Conclusion Thus, the decimal equivalent of the binary number 1011011 is **91**. ### Explanation of Options - **Option A (91)**: This is the correct answer, as we have calculated. - **Option B (75)**: This is incorrect. It does not match our calculated sum. - **Option C (63)**: This is also incorrect. It is less than our calculated value. - **Option D (45)**: This is incorrect as well. It is significantly lower than the correct value. ### Common Pitfalls - **Miscounting Powers**: Ensure you start counting powers from \(2^0\) on the right. - **Forgetting to Add**: Always remember to sum all the calculated values to get the final decimal equivalent. - **Binary to Decimal Confusion**: Be careful not to confuse binary with decimal; they are fundamentally different number systems. ### Revision Summary - Binary numbers are base-2, using only 0s and 1s. - Each binary digit represents a power of 2, starting from the right. - To convert binary to decimal, calculate the value of each digit and sum them. - The binary number 1011011 equals 91 in decimal.