Question:
Question 1
- a) A Computer uses IEEE-754 single precision format to represent floating points. What value (in decimal) the computer represents if the floating point is represented using the following binary digits: [6 marks]
0 01111110 10100000000000000000000
- b) Using a “word” of 5 bits, list all of the possible signed binary numbers and their decimal equivalents that are represent able in: [3+3+3 = 9 marks]
- Signed magnitude
- One’s complement
iii. Two’s complement
Question 2
- a) Write a Boolean function and construct a logic diagram of a circuit which use of basic logic gates to activate CSU main entrance door during 9:00 am to 12:00 pm and after lunch time during 1:00 pm – 4:00 pm. You need to use 24 hour clock timing when designing this circuit. [9 marks]
Please insert the Boolean expression, minimisation, and the diagram
- b) Using basic Boolean algebra identities for Boolean variables A, B and C, prove that X’Y + XYZ’ + Y’ + XZ (Y+Y’) = 1. Please show all steps and mention the identities used. [6 marks]
Answer:
Question 1
Section a
The binary digits in the IEEE 754 number system is as shown below,
0 01111110 10100000000000000000000
The IEE 754 is a floating point standard that was defined in 1985. It was developed in response to various representation and the rising need for portability for the scientific code. There are two representations:
- Single precision IEEE 754 format with 32-bit representation
- Double precision IEEE 754 format with 64-bit representation
The IEEE 754 format is converted using the following formula,
When S, denotes a sign bit, is 0 – shows that a number is a non-negative and when s=1, it is a negative number. The significand is the fraction that is restored with a single addition. For a single precision, the bias is 127.
| 0 | Positive number |
| 01111110 | Exponent |
| 10100000000000000000000 | Fraction |
Converting the three sections to decimals,
Exponent – 126
Fraction = 0.625
Section b
- Signed magnitude
| First Binary | First decimal | Last binary | Last decimal |
| +1510 | -1510 |
- One’s complement
| First Binary | First decimal | Last binary | Last decimal |
| -1510 | +1510 |
- Two’s complement
| First Binary | First decimal | Last binary | Last decimal |
| -1610 | +1510 |
Question 2
Section a
The truth table showing the circuit input and outputs for a 4 input circuit is as shown below,
The truth table is simplified further to obtain the circuit diagram that is implemented in the door system,
Section b
The variables A, B, & C are denoted as X, Y, Z respectively.
The solution is simplified further using the exclusive OR gate as denoted below,
