Converting Between Denary & Hexadecimal

Like denary, hexadecimal is a positional system. The value of a digit depends on its position relative to the radix point, with each place value being a power of $16$.

The weights for hexadecimal positions starting from the right are $16^0 = 1$, $16^1 = 16$, $16^2 = 256$, $16^3 = 4096$, and so on.

Mathematically, a hexadecimal number $d_n...d_1d_0$ can be expressed in denary as: $\sum_{i=0}^{n} d_i \times 16^i$ where $d_i$ is the denary value of the hex digit.

Successive Division by 16: Divide the denary number by $16$ and record the remainder. Repeat the process with the quotient until the quotient reaches zero.

Remainder Mapping: If a remainder is between $10$ and $15$, it must be converted to its corresponding letter ($A=10, B=11, C=12, D=13, E=14, F=15$).

Ordering: The first remainder calculated represents the Least Significant Digit (LSD), and the final remainder represents the Most Significant Digit (MSD). Read the remainders from bottom to top.

Expansion Method: Multiply each hexadecimal digit by its corresponding power of $16$ based on its position.

Step-by-Step Calculation: Convert any letter digits ($A-F$) back to their denary equivalents ($10-15$) before performing the multiplication.

Summation: Add the results of all multiplications together to find the final denary value.

The Binary Bridge: For complex conversions, it is often faster to convert to binary first. Since $16 = 2^4$, you can convert each hex digit into a 4-bit binary group and then convert the binary string to denary.

Verification: Always perform a 'sanity check'. If you convert a large denary number to hex, the resulting hex string should have fewer digits than the denary version because the base is larger.

Common Values: Memorize the first few powers of $16$ ($16, 256, 4096$) and the $A-F$ mappings to save time during exams.

The 'Off-by-One' Error in A-F: A common mistake is starting the alphabet at $11$ instead of $10$. Remember that $9$ is the last number, so $10$ must be $A$.

Reversing the Order: Students often write the remainders in the order they were calculated (top-to-bottom). Always write them from the last remainder to the first (bottom-to-top).

Ignoring the Zero Power: Forgetting that the first column on the right is $16^0$ (which equals $1$) rather than $16^1$ leads to significant calculation errors.