To find the binary representation of the decimal number 13, we will convert the decimal number into binary step by step. ### Step-by-Step Explanation 1. **Understanding Binary Representation**: - Binary is a base-2 numeral system that uses only two digits: 0 and 1. Each digit represents a power of 2, starting from the rightmost digit, which represents \(2^0\), the next \(2^1\), then \(2^2\), and so on. 2. **Finding the Largest Power of 2**: - First, we need to find the largest power of 2 that is less than or equal to 13. The powers of 2 are: - \(2^0 = 1\) - \(2^1 = 2\) - \(2^2 = 4\) - \(2^3 = 8\) - \(2^4 = 16\) (this is too large) - The largest power of 2 less than or equal to 13 is \(2^3 = 8\). 3. **Subtracting the Largest Power of 2**: - We subtract 8 from 13: \[ 13 - 8 = 5 \] 4. **Finding the Next Largest Power of 2**: - Now we need to find the largest power of 2 that is less than or equal to 5. The powers of 2 are: - \(2^0 = 1\) - \(2^1 = 2\) - \(2^2 = 4\) - The largest power of 2 less than or equal to 5 is \(2^2 = 4\). 5. **Subtracting Again**: - We subtract 4 from 5: \[ 5 - 4 = 1 \] 6. **Finding the Next Largest Power of 2**: - Now we need to find the largest power of 2 that is less than or equal to 1. The only power of 2 that fits is: - \(2^0 = 1\) - So we take \(2^0\). 7. **Final Subtraction**: - We subtract 1 from 1: \[ 1 - 1 = 0 \] - We have now expressed 13 as the sum of powers of 2: \[ 13 = 2^3 + 2^2 + 2^0 \] 8. **Writing the Binary Representation**: - In binary, we represent the presence of each power of 2 with a 1 and the absence with a 0. The powers of 2 we used are \(2^3\), \(2^2\), and \(2^0\). Therefore, we write: - \(2^3\) (8) is present → 1 - \(2^2\) (4) is present → 1 - \(2^1\) (2) is absent → 0 - \(2^0\) (1) is present → 1 - This gives us the binary representation: \[ 1101 \] ### Conclusion Thus, the binary representation of the decimal number 13 is **B. 1101**. ### Explanation of Other Options - **A. 1100**: This represents \(8 + 4 = 12\), which is not equal to 13. - **C. 1011**: This represents \(8 + 2 + 1 = 11\), which is also not equal to 13. - **D. 1110**: This represents \(8 + 4 + 2 = 14\), which exceeds 13. ### Revision Summary - The binary system uses only 0s and 1s, representing powers of 2. - To convert a decimal number to binary, find the largest powers of 2 that sum to the number. - The binary representation of 13 is 1101, which corresponds to \(2^3 + 2^2 + 2^0\). - Always check each option against the decimal value to confirm correctness.