In the binary system the number 13 is written as _______.

A. 1011

B. 1101

C. 1010

D. 1110

To convert the decimal number 13 into its binary equivalent, we can break it down step by step. The binary system is base-2, which means it uses only two digits: 0 and 1. Each digit represents a power of 2, starting from the rightmost digit, which represents 2^0, then 2^1, 2^2, and so on.

Step-by-Step Conversion

Let’s convert 13 to binary:

• First, identify the largest power of 2 that fits into 13. The powers of 2 are: 1 (2^0), 2 (2^1), 4 (2^2), 8 (2^3), and 16 (2^4). The largest power of 2 less than or equal to 13 is 8 (2^3).
• Next, subtract 8 from 13, which leaves us with 5. Now we look for the largest power of 2 that fits into 5. The largest is 4 (2^2).
• Subtract 4 from 5, leaving us with 1. The largest power of 2 that fits into 1 is 1 (2^0).
• Now we have used 8, 4, and 1, which correspond to the binary digits. We can represent 13 as follows:

Binary Representation

In binary, we write a '1' for each power of 2 we have used and a '0' for those we haven’t:

• 2^3 (8) → 1
• 2^2 (4) → 1
• 2^1 (2) → 0
• 2^0 (1) → 1

Putting it all together, we get:

1 (for 8) + 1 (for 4) + 0 (for 2) + 1 (for 1) = 1101

Final Answer

So, the binary representation of the decimal number 13 is 1101. Therefore, the correct answer is B. 1101.