Checking Binary to Denary Answers

Once you have converted a binary number to denary, the two end bits let you catch a slip in seconds: the most significant bit on the left and the least significant bit on the right.

What the end bits tell you

In an 8-bit number the most significant bit sits in the 128 column. When it is 1, that column adds 128 to the total, so the result cannot be below 128. When it is 0, the column contributes nothing and the rest add up to 127 at most. The least significant bit is worth 1, and every other place value is even, so the total is odd exactly when that bit is 1 and even when it is 0.

Testing an answer

Take 01010110. Its MSB is 0, so the answer must be 127 or less, and its LSB is 0, so it must be even. Adding the columns holding a 1 gives 64 + 16 + 4 + 2 = 86, which fits both. An answer of 186 fails at once, because a number starting with 0 can never reach 128, and 91 fails because a number ending in 0 is never odd. Either mistake sends you back to the place values. A range check catches the rest: an 8-bit pattern holds values from 0 to 255 only, so an answer outside that range is impossible too.

Converting back

The strongest check is to rebuild the pattern from your denary answer. Take 10101101: MSB 1 means 128 or more, LSB 1 means odd, and 128 + 32 + 8 + 4 + 1 = 173 fits both. Now build 173 from the place values: 173 − 128 = 45, 45 − 32 = 13, 13 − 8 = 5, 5 − 4 = 1, 1 − 1 = 0. That puts 1s in the 128, 32, 8, 4 and 1 columns and 0s in the rest, giving 10101101 — the pattern you started with. If the rebuilt pattern differs, the original conversion was wrong.