91Ó°ÊÓ

The hexadecimal number system, commonly used in computing and digital electronics, is a base-16 numbering system. It employs 16 distinct symbols: 0-9 to represent values zero to nine, and A-F (or a-f) to represent values ten to fifteen. Therefore, 'A' in hexadecimal is equivalent to the decimal value 10.

In the given exercise, the hexadecimal number 'A03' consists of A which is 10 in decimal, followed by two other digits: 0 and 3, which are the same in both the decimal and hexadecimal systems. Understanding each symbol's decimal equivalent is crucial for converting any hexadecimal number to decimal.

Base conversion is the process of changing a number from one base to another. For hexadecimal to decimal conversion, each digit of the hexadecimal number is taken into account along with its position in the number. Each digit is then multiplied by its base, which is 16 for hexadecimal, raised to the power corresponding to its position. Positions are counted from right to left, starting with zero.

For example, in 'A03', '3' is in position 0, '0' in position 1, and 'A' in position 2. These positions influence how each digit contributes to the final decimal number. This procedure ensures that the same numeric value is represented accurately across different number systems. Simplification of this process can be achieved using a calculator or conversion chart for those less familiar with powers of 16.

Positional notation is a method of using the same set of symbols to represent different values based on the symbols' positions in the number. This method applies to any base system. In base-16 as with hexadecimal, the far right position (least significant digit) is the 'ones' place, and each position to the left is 16 times greater than the one to its right. This ensures that each digit’s place value is 16 times that of a digit to its immediate right.

When converting 'A03' to decimal, you would calculate as follows: for 'A' in the leftmost position (also called the most significant digit), you would calculate its contribution as 'A' times 16 squared, because it is in the second position from the right. Likewise, each digit's contribution is calculated based on its positional value and then these values are summed to find the final decimal number. Clear comprehension of positional notation is fundamental to mastering base conversions.