Positional weight in hexadecimal: For the hexadecimal number 1AB16, what is the column weight (positional value) of the leftmost digit “1”?

Introduction / Context:
Understanding positional weights in different bases is crucial for converting and analyzing numbers in digital systems. Hexadecimal uses base 16, so each position moving left multiplies the weight by 16. This question asks for the positional value (column weight) of the leftmost digit “1” in the hex number 1AB16.

Given Data / Assumptions:

• Number is 1AB16 (three hex digits).
• Positions from rightmost: 16^0, 16^1, 16^2.
• We need the weight of the leftmost “1”.

Concept / Approach:
In base 16, the rightmost digit has weight 16^0 = 1. The next digit to the left has weight 16^1 = 16. The next one (leftmost in this 3-digit number) has weight 16^2 = 256. Therefore, the column weight of the leftmost “1” is 256, independent of the actual digit values in the other positions.

Step-by-Step Solution:

Verification / Alternative check:
Expand 1AB16 to decimal: 1 * 256 + A * 16 + B * 1 = 256 + 10 * 16 + 11 = 256 + 160 + 11 = 427. The component for the leftmost digit is indeed 256.

Why Other Options Are Wrong:
64 is 16^1 for a different position count; 512 and 1024 correspond to 16^2 * 2 and 16^2 * 4 and are not the direct column weight for the leftmost digit in a 3-digit hex number.

Common Pitfalls:
Confusing value contribution (digit * weight) with weight itself; miscounting positions from the rightmost digit; assuming decimal place values.

Final Answer:
256