Converting Between Decimal & Hexadecimal

• Decimal (Base 10): The standard number system used by humans, consisting of ten unique digits from $0$ to $9$. Each position in a decimal number represents a power of $10$.

• Hexadecimal (Base 16): A positional number system used extensively in computing that utilizes sixteen unique symbols. It includes the digits $0-9$ and the letters $A-F$ to represent values from $10$ to $15$.

• Alphanumeric Mapping: In hexadecimal, the letters $A, B, C, D, E,$ and $F$ correspond to the decimal values $10, 11, 12, 13, 14,$ and $15$ respectively. This allows a single character to represent a value that would require two digits in decimal.

• The Radix: The base (or radix) of a system determines how many digits are available and the multiplier for each column. Hexadecimal has a radix of $16$, meaning each column is $16$ times greater than the column to its right.

• Positional Notation: Like decimal, hexadecimal is a weighted system where the value of a digit depends on its position. The positions are weighted by powers of $16$, starting from $16^0$ (the units column) on the far right.

• Power Series Expansion: Any hexadecimal number can be expressed as a sum of its digits multiplied by their respective positional weights. For a two-digit hex number $d_1d_0$, the value is $(d_1 \times 16^1) + (d_0 \times 16^0)$.

• Relationship to Binary: Hexadecimal is highly efficient for computing because $16$ is a power of $2$ ($2^4$). This means exactly one hexadecimal digit can represent a $4$-bit binary sequence (a nibble), making it a compact shorthand for binary data.

Decimal to Hexadecimal (Successive Division)

• Step 1: Divide the decimal number by $16$. Record the whole number quotient and the remainder.
• Step 2: Convert the remainder (which will be between $0$ and $15$) into its corresponding hexadecimal digit ($0-9, A-F$).
• Step 3: Use the quotient from the previous step as the new number to divide. Repeat the process until the quotient reaches $0$.
• Step 4: The hexadecimal result is formed by reading the remainders in reverse order (the last remainder found is the most significant digit).

Hexadecimal to Decimal (Positional Expansion)

• Step 1: Identify the hexadecimal digits and their positions, starting from position $0$ on the right.
• Step 2: Convert any letter digits ($A-F$) back into their decimal equivalents ($10-15$).
• Step 3: Multiply each digit by $16$ raised to the power of its position ($16^{position}$).
• Step 4: Sum all the resulting values to find the final decimal total.

• Base Comparison: Decimal uses a base of $10$, while Hexadecimal uses a base of $16$. This means Hexadecimal can represent larger numbers using fewer digits.

• Conversion Direction: Converting to Hexadecimal typically involves division, whereas converting from Hexadecimal typically involves multiplication. Both rely on the same underlying positional logic.

• Letter Mapping Errors: A common mistake is misidentifying the letters. Always double-check that $A=10$ and $F=15$. Some students mistakenly start $A$ at $11$.

• Ignoring the Zero Quotient: The division process must continue until the quotient is $0$, not just until the quotient is less than $16$. The final quotient (if less than $16$) becomes the final remainder.

• Confusing Hex with Binary: While they are related, ensure you do not accidentally use base $2$ or base $8$ logic when performing base $16$ calculations.