Hexadecimal to Binary Conversion

Converting a hexadecimal number to a binary number is a fundamental concept in computer science and digital electronics. Each hexadecimal digit corresponds to exactly four binary digits (bits).

Understanding Hexadecimal Digits

The hexadecimal number system (base 16) uses 16 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

• Digits 0 through 9 have their usual values.
• Letters A through F represent values 10 through 15, respectively.

The question asks for the conversion of the hexadecimal digit "D" into binary.

Value of Hexadecimal "D"

The hexadecimal digit "D" corresponds to the decimal value 13.

Converting Decimal 13 to Binary

To convert the decimal number 13 to a 4-bit binary number, we can use the division-by-2 method or consider the place values in binary (8s, 4s, 2s, 1s).

Using place values:

• We need to find which combination of \(2^3=8\), \(2^2=4\), \(2^1=2\), and \(2^0=1\) adds up to 13.
• Start with the largest place value, 8. Is 8 needed? Yes, \(13 \ge 8\). So the \(2^3\) bit is 1. Remaining value: \(13 - 8 = 5\).
• Next place value, 4. Is 4 needed? Yes, \(5 \ge 4\). So the \(2^2\) bit is 1. Remaining value: \(5 - 4 = 1\).
• Next place value, 2. Is 2 needed? No, \(1 < 2\). So the \(2^1\) bit is 0. Remaining value: \(1 - 0 = 1\).
• Next place value, 1. Is 1 needed? Yes, \(1 \ge 1\). So the \(2^0\) bit is 1. Remaining value: \(1 - 1 = 0\).

Reading the bits from left to right (8s, 4s, 2s, 1s), we get 1101.

Alternatively, using division by 2:

13 ÷ 2 = 6  Remainder: 1 (Least Significant Bit)
 6 ÷ 2 = 3  Remainder: 0
 3 ÷ 2 = 1  Remainder: 1
 1 ÷ 2 = 0  Remainder: 1 (Most Significant Bit)

Reading the remainders from bottom up gives the binary number 1101.

Summary of Hexadecimal to Binary Conversion

Here is a table showing the conversion for some hexadecimal digits:

From the table and the conversion steps, the binary representation of hexadecimal "D" is 1101.