Comparing Hexadecimal to Denary Methods

There are two ways to turn a two-digit hexadecimal number into denary. The place-value method works on the digits themselves: the left digit counts 16s and the right digit counts 1s. The binary method changes each digit into a 4-bit nibble, joins them into 8 bits, then adds the binary place values. They look different but do exactly the same job, so both must reach the same denary answer — which makes one a ready-made check on the other.

One number, two routes

Take 4D, remembering that D is denary 13.

Both routes give 77 because they are the same sum written differently. The place-value method counts 16s with the left digit and 1s with the right digit. In the binary method each nibble uses place values 8, 4, 2 and 1, and every bit of the left nibble is worth 16 times the matching bit of the right nibble, so the left nibble ends up counting 16s in just the same way.

What a sensible answer looks like

Two hexadecimal digits make up one byte, so the denary result must be a positive whole number from 0 to 255. An answer of 300, or a negative answer, proves something has gone wrong: digits swapped, a letter read with the wrong value, or the right digit multiplied by 16 instead of added. Adding the two digits together is another slip, because 4 + 13 = 17 ignores the 16s completely. Once your answer is inside that range, convert the hexadecimal number again with the other method and check that both results match.

Finally, check which direction the question wants. 4D converts to 77, but starting from 77 and producing 4D is a different procedure. Read carefully and label your final value as denary.