To find the decimal equivalent of the binary number 101101, we will follow a step-by-step process to convert it from binary (base 2) to decimal (base 10). ### Step-by-Step Explanation 1. **Understanding Binary Numbers**: - Binary numbers are composed of only two digits: 0 and 1. Each digit represents a power of 2, starting from the rightmost digit, which represents \(2^0\), the next one to the left represents \(2^1\), and so on. 2. **Identifying the Binary Number**: - The binary number given is **101101**. We will break it down from right to left: - The rightmost digit (1) is in the \(2^0\) place. - The next digit (0) is in the \(2^1\) place. - The next digit (1) is in the \(2^2\) place. - The next digit (1) is in the \(2^3\) place. - The next digit (0) is in the \(2^4\) place. - The leftmost digit (1) is in the \(2^5\) place. 3. **Calculating the Decimal Value**: - Now, we will calculate the decimal value by multiplying each binary digit by its corresponding power of 2: - \(1 \times 2^5 = 1 \times 32 = 32\) - \(0 \times 2^4 = 0 \times 16 = 0\) - \(1 \times 2^3 = 1 \times 8 = 8\) - \(1 \times 2^2 = 1 \times 4 = 4\) - \(0 \times 2^1 = 0 \times 2 = 0\) - \(1 \times 2^0 = 1 \times 1 = 1\) 4. **Adding the Values Together**: - Now, we sum all these values: \[ 32 + 0 + 8 + 4 + 0 + 1 = 45 \] 5. **Final Result**: - Therefore, the decimal equivalent of the binary number 101101 is **45**. ### Explanation of Other Options - **Option B: 22**: - This is incorrect because the calculations do not support this value. The binary number 101101 does not add up to 22 when converted properly. - **Option C: 37**: - This is also incorrect. The binary number does not yield 37 when calculated using the powers of 2 as shown above. - **Option D: 53**: - This option is incorrect as well. The sum of the powers of 2 from the binary number does not equal 53. ### Common Pitfalls - **Miscounting Powers**: Ensure you start counting powers of 2 from the rightmost digit (starting at \(2^0\)). - **Forgetting to Add All Values**: It’s crucial to add all the calculated values together to get the final decimal equivalent. - **Binary to Decimal Confusion**: Remember that binary is base 2, and each digit represents a power of 2, not 10. ### Revision Summary - The binary number 101101 converts to decimal by calculating the sum of powers of 2. - Each binary digit is multiplied by \(2^n\) where \(n\) is the position from the right (starting at 0). - The correct decimal equivalent of 101101 is **45**. - Always double-check calculations and ensure all binary digits are accounted for in the final sum.