write a c perform A-B and B-A where A=F A C 9) base 16 and B=( C A D) base 16 using 16's compliment

To perform the operations $A - B$ and $B - A$ using 16's complement in C, we will follow these steps:

1. Break Down the Problem

1. Identify the hexadecimal values of $A$ and $B$.
2. Convert these hexadecimal values to their decimal equivalents.
3. Compute $A - B$ and $B - A$ using 16's complement.
4. Convert the results back to hexadecimal.

2. Relevant Concepts

• Hexadecimal Representation: Each digit represents a value from 0 to 15.
• 16's Complement: For a number $x$ in base 16, the 16's complement is calculated as $16^n - x$, where $n$ is the number of digits in the hexadecimal representation.

3. Analysis and Detail

Step 1: Identify the Values

• $A = \text{F A C 9}_{16}$
• $B = \text{C A D}_{16}$

Step 2: Convert to Decimal

• Calculate the decimal equivalent of $A$: $A_{10} = F \cdot 16^3 + A \cdot 16^2 + C \cdot 16^1 + 9 \cdot 16^0 = 15 \cdot 4096 + 10 \cdot 256 + 12 \cdot 16 + 9 \times 1$ $A_{10} = 61440 + 2560 + 192 + 9 = 64001$

• Calculate the decimal equivalent of $B$: $B_{10} = C \cdot 16^2 + A \cdot 16^1 + D \cdot 16^0 = 12 \cdot 256 + 10 \cdot 16 + 13 \cdot 1$ $B_{10} = 3072 + 160 + 13 = 3245$

Step 3: Perform the Operations

• Calculate $A - B$:

  To find $A - B$ using 16's complement, find the 16's complement of $B$:

  1. $B$ in 4 digits: $0000 C A D$
  2. Find the 16's complement of $B$:

  $16^3 - B = 16^3 - 3245 = 4096 - 3245 = 851$

  Convert $851$ to hexadecimal: $851 \div 16 = 53 \quad \text{remainder} = 3$ $53 \div 16 = 3 \quad \text{remainder} = 5$ So, $B_{16's\ complement} = 0000 0353_{16}$

  Now perform $A + B_{16's\ complement}$: $A_{16} = \text{F A C 9}_{16} = 64001_{10}$ $64001 + 851 = 64852$

  Convert $64852$ to hexadecimal: $64852 \div 16 = 4053 \quad \text{remainder} = 4$ $4053 \div 16 = 253 \quad \text{remainder} = 5$ $253 \div 16 = 15 \quad \text{remainder} = D$ $15 \div 16 = 0 \quad \text{remainder} = F$

  So $A - B = F D 5 4_{16}$.

• Calculate $B - A$:

  To find $B - A$, calculate the 16's complement of $A$:

  1. $A$ in 4 digits: $0000 F A C 9$
  2. Calculate the 16's complement of $A$:

  $16^4 - A = 65536 - 64001 = 1535$

  Convert $1535$ to hexadecimal: $1535 \div 16 = 95 \quad \text{remainder} = 15$ $95 \div 16 = 5 \quad \text{remainder} = 15$ $5 \div 16 = 0 \quad \text{remainder} = 5$

  So $A_{16's\ complement} = 0005 FF_{16}$

  Now perform $B + A_{16's\ complement}$: $B_{16} = \text{C A D}_{16} = 3245_{10}$ $3245 + 1535 = 4780$

  Convert $4780$ to hexadecimal: $4780 \div 16 = 298 \quad \text{remainder} = 12$ $298 \div 16 = 18 \quad \text{remainder} = 10$ $18 \div 16 = 1 \quad \text{remainder} = 2$ $1 \div 16 = 0 \quad \text{remainder} = 1$

  So $B - A = 12 A C 4_{16}$.

4. Verify and Summarize

• $A - B = F D 5 4_{16}$
• $B - A = 12 A C 4_{16}$

Final Answer

1. $A - B = F D 5 4_{16}$
2. $B - A = 12 A C 4_{16}$