data_representation.md

Exercise 1

Given the binary number ((-1101.1101)_2), calculate its floating-point representation, and then determine the values of (S), (M), and the exponent (esp), following the convention (1.M). Consider 1 bit for the sign (S), 5 bits for the exponent (esp) (with a bias of 15), and 10 bits for the mantissa (M). Indicate the sequence of bits for (S), (esp), and (M) separated by a comma (0.0000000000).

Solution

1. Sign Bit (S): 1 (since the number is negative)
2. Normalized Form: (-1.1011101 \times 2^3)
3. Exponent Calculation:
   • Actual exponent: 3
   • Biased exponent: (3 + 15 = 18)
   • Binary representation of 18: (10010)
4. Mantissa (M):
   • From the normalized form: (1011101000) (10 bits)

Final Answer:
1,10010,1011101000

Exercise 2

Given the decimal number (16) and its corresponding binary number in 8 bits ( (16)_{10} = 0001\ 0000_2 ), determine the negative corresponding number using 1's complement and 2's complement representation. Write the values of both representations separated by a comma (for example, 00000000, 00000000).

Solution

1. Binary Representation of (16):

   • (0001\ 0000)

2. 1's Complement:

   • Invert all bits: (1110\ 1111)

3. 2's Complement:

   • Add 1 to the 1's complement:
     • (1110\ 1111 + 1 = 1111\ 0000)

Final Answer:
11101111,11110000