Add the following hex values: \(\begin{array}{ll}\text { (a) } 2 \mathrm{CH}+3 \mathrm{FH} & \text { (b) } \mathrm{F} 34 \mathrm{H}+5 \mathrm{D} 6 \mathrm{H}\end{array}\) (c) \(20000 \mathrm{H}+12 \mathrm{FFH}\) (d) \(\mathrm{FFFFH}+2222 \mathrm{H}\)

Understanding how to convert hexadecimal (hex) numbers to decimal form is crucial when working with digital systems that represent values in hex format. Hexadecimal is a base-16 number system, which means it has 16 distinct symbols ranging from 0 to 9 and the letters A to F, with A representing 10 and F representing 15 in decimal.
Each digit in a hex number represents a power of 16, with the rightmost digit reflecting 16 to the power of 0 (which is 1), and each digit to the left increasing the exponent by one. To convert hex to decimal, you multiply each digit by its corresponding power of 16 and then sum all those products.
For example, converting the hex number 2CH to decimal involves calculating \(2 \times 16^1) + (12 \times 16^0) = 32 + 12\), which equals 44 in decimal.

Once you've converted hex numbers to decimal, adding them is as simple as summing up ordinary numbers. Addition involves lining up the digits by their place value and summing each column, carrying over any excess to the next column whenever a sum exceeds the base, which in decimal is 10.
For instance, adding the decimal values 44 (from 2CH) and 63 (from 3FH) would go as follows: you align the units digits (4 and 3) and the tens digits (4 and 6) respectively, add each pair of digits, and combine the results to get \(44 + 63 = 107\) in decimal. Adding decimal values is a universal skill, but it's particularly important here as it bridges understanding between different numerical systems.

After performing addition in the decimal system, you may need to convert the sum back to hex to maintain consistency with the system you're working with. Converting from decimal to hex involves dividing the decimal number by 16 and keeping track of the remainders.
As an example, converting the decimal number 107 back to hex works like this: \(107 \div 16 = 6\) with a remainder of 11. The quotient is part of the hex value, and the remainder represents the last hex digit. Since 11 corresponds to the letter B in hex, the full conversion results in the hex number 6B.

When adding hex numbers, you may encounter an 'overflow,' which occurs when the result exceeds the maximum value that can be displayed with a fixed number of digits. In computers and digital systems, handling overflow properly is essential to maintain data integrity.
If we consider adding FFFFFH and 2222H, the calculation leads to an overflow beyond the 4-digit limit for hex numbers. Normally, you would acknowledge the overflow and consider it when interpreting the result, especially if the data represents something like an address or a specific instruction. Depending on context, you might also truncate or wrap around the result; for example, a five-digit overflow in a system that handles only four digits would have the overflow 'disappear,' and the system would register only the last four digits of the result.