Problem 1:

Use the repeated subtraction method to convert the number 91 to binary (show your conversion steps clearly). What would be the representation of (-91)₁₀ in an 8-bit binary signed magnitude form?

Problem 2:

Convert the following binary numbers to the destination base (show your conversion steps clearly):

a) (1011.11000)₂ = ( )₁₀

b) (111110110001)₂ = ( )₁₆

Problem 3:

A computer system uses a simplified 14-bit floating point with excess-16 bias to represent binary numbers. What are the least and the biggest binary number handled by this computer? (Give the numbers in a scientific binary representation)

Problem 4:

Use only two input gates to draw the logical diagram of the following Boolean expression: F(x,y,z) = x ¯y+x ¯yz+((y+z)) ¯. Simplification is not allowed.

Problem 5: Binary representation systems:

Consider the binary number 10110100. What does this binary number represents in decimal (base 10) when it is used in: 

a) An 8-bit unsigned whole numbers system?

b) An 8-bit signed magnitude system?

c) An 8-bit one’s complement system?

d) An 8-bit two’s complement system?

Problem 6: Explaining an advertisement

The detailed specifications of a laptop are: 

- Intel Core i3 / 2.2 GHz

- 4 GB ( 2 x 2 GB) DDR2 SDRAM

- 500 GB HDD / 5400 rpm

- 15.6-inch, 1366 x 768 ( HD )

Monitor resolution is 1366x768 pixels

- VGA ¦ 2 x headphone output ¦ Microphone input ¦ 2 x USB 2.0 ¦ 2 x USB 3.0 ¦ LAN ¦ HDMI

External ports

Problem 7: Combinational and sequential logic

Consider the Boolean expression F(a,b,c) = a ¯bc+((b ¯+c)) ¯+((b+c ¯)) ¯+a((b ¯+c ¯)) ¯.

a) Use DeMorgan’s law to convert F(a,b,c) to a sum of products form.

b) Draw the truth table of F(a,b,c).

c) Use Boolean identities and Boolean algebra to simplify F. Show your simplification steps clearly.

Hint: Your answer should contain only two terms.

Problem 8:

The capacity of a recent external hard disk drive is 2 Terabytes (TB). How many Gigabytes (GB) and Kilobytes (KB) are there in this hard disk drive?

(The student can represent his answer is a power of 2 or a power of ten. Both can be considered as correct.)

Problem 9:

Express the number -33 in an 8-bit Complement one and complement two binary forms.

Problem 10: Signed Magnitude representation and arithmetic

A computer uses an 8-bit Signed Magnitude system to represent numbers.

a) What is the decimal equivalent of the following binary numbers: 00001001₂ and 00011100₂?

b) Use signed magnitude arithmetic to add 00001001₂ to 00011100₂.

c) Does the system overflow in the previous question (Problem 1-b)? Why/Why not?

Problem 11: Boolean Algebra

Consider the following Boolean function: F(a,b,c)=a ¯b ¯c ¯+ab ¯.

a) Use DeMorgan’s law, other Boolean identities and Boolean algebra to find the complement F ¯ of the function F. You should also represent F ¯ in its simplest sum of products form.

b) Draw the truth table of the function F and deduce the truth table of its complement function F ¯.

Hint: Including the terms a ¯b ¯c ¯  ;ab ¯c ; F and F ¯ is quite enough.

c) Using the pre-drawn truth table give another sum of products form for the complement function F ¯.

Hint: One of the terms is a ¯b ¯c and still four other terms: F ¯=a ¯b ¯c+xxx+xxx+xxx+xxx

Problem 12:

Using a "word" of 3 bits, list all of the possible signed binary numbers and their decimal equivalents that are representable in a Signed magnitude system.

Problem 13:

Draw the truth table of  (xy) ¯ and x ¯+y ¯ to prove that the following DeMorgan's Laws is valid: (xy) ¯ = x ¯+y ¯.

Problem 14: Boolean Algebra and digital logic

The following table is the truth table of a function F.

a) Give the Boolean expression of the function F(a,b,c) in a sum of products form.

b) Draw the truth table of the complement F ¯ of the function F. Deduce F ¯ in a sum of products form.

c) Use Boolean algebra and Boolean identities to simplify F ¯ to its simplest form. 

Problem 15: Binary representations

Show how the number -127 is represented in binary using:

a) 8-bit Signed magnitude system

b) 8-bit two’s complement system

c) 14-bit floating point representation with excess 16 bias (1 sign bit, 5 bits exponent, 8 bits significand)

Problem 16:

Convert the following decimal fraction to binary with a maximum of four places to the right of the binary point: 57.55. Show your conversion steps clearly.

Problem 17:

Construct the XOR operator using only NAND gates.

Hint: x XOR y = ((x ' y)' (xy' )' )'

Problem 18: Boolean Algebra and digital logic

Consider the Boolean expression F= a(b+c ¯) + a ¯((bc ¯ ) ¯ )+ac+b

a) Use the distributive law and DeMorgan’s law to represent F in a sum of product form.

b) Draw the truth table of F.

c) Analyze your truth table, what can you deduce?

d) What would be the most simplified version of F and its complement function F ¯?

Hint: No need to do any simplification. Recall what you deduced in part c.

Problem 19: Binary representations and arithmetic

A computer system has two different ways to represent signed numbers: Signed magnitude and two’s complement.

a) Show how this computer represents the number 7710in these two different binary systems (an 8-bit signed magnitude and 8-bit two’s complement).

b) Show how this computer represents the number (-42)10 in these two different binary systems (an 8-bit signed magnitude and 8-bit two’s complement).  

c) Show how this computer uses two’s complement arithmetic to add these numbers in binary (77 + (-42)). Is there an overflow? Why or why not?

Problem 20: 

Convert the following binary numbers to the destination base.

a. (A4D.F)₁₆ = (……………)₂ ? 

b. (732)₈ = (……………)₂ ? 

Problem 21: Binary representation systems

Consider the following decimal numbers: + 17 and -26.

A. Signed magnitude system:

a. How does an 8-bit signed-magnitude computer system represent these numbers in binary?

In an 8 bit signed magnitude system:

b. Add these binary numbers in binary (in a signed magnitude system). Show your calculations clearly.

B. Two’s complement system:

a. How does an 8-bit two’s complement computer system represent these numbers in binary?

b. Add these binary numbers in binary (in a two’s complement system). Show your calculations clearly.

Problem 22: A simple floating point system

A computer system uses a simplified 14-bit floating point with excess-16 bias to represent binary numbers.

a) How does this system represent the number -33 in binary? +33?

b) How does this system represent the number 64 in binary?

c) Show how can this system add +33 and 64 in binary? Represent your answer in a 14-bit floating point with excess-16 bias form.

Problem 23: Boolean algebra

Consider the following Boolean function:

F(x,y,z) = x‾y‾z‾ + xyz + xy‾z + x‾yz + x‾y‾z + x‾yz‾

a) Draw the truth table of F(x,y,z). 

Hint: No need for too many details in the truth table (x,y,z and F are enough)

b) Use Boolean identities and Boolean algebra to simplify F to its simplest form. Show your simplification steps clearly.

c) Verify your answer by drawing the truth table of your simplified F.

d) Draw the logical diagram of your simplified F.

Problem 24:

Convert the following binary numbers to the destination base. Show your conversion steps clearly. 

a. (1001100)2 = ()8

b. (011010101)2 = ()16 or ()16

Problem 25:

Use the division remainder method to convert the number 115 to binary. Show your conversion steps clearly. 

Problem 26:

Convert 0.0625 from base 10 to base 2. Show your conversion steps clearly.

Problem 27:

Construct an AND gate using only NAND gates. Put labels on your logical circuit to explain how you have obtained the output “xy” from the inputs “x” and “y”.

Problem 28: Binary representations and arithmetic 

Consider the decimal numbers -93.

a) Represent this number in an 8-bit binary signed magnitude form.

b) Represent this number in an 8-bit binary two complement’s form.

c) Use 8-bit two complement’s arithmetic to subtract 93 from 01100111 (+103). Show your calculations clearly. Show the result in two complement and decimal form. 

Hint: you should calculate 103 - 93.

Problem 29: Boolean expressions and digital logic

The truth table of the Boolean expression F(a,b,c) is given below:

a) Give the Boolean expression F(a,b,c) in a sum of products form.

b) Use Boolean identities and Boolean algebra to simplify F. Show your simplification steps clearly.

Hint: Your answer should contain only two terms.

Problem 30: A simple floating point system 

A computer system uses a simplified 14-bit floating point with excess-16 bias to represent binary numbers.

a) How does this system represent the number -0.062510 in binary?

Hint: You have already converted 0.062510 to binary in Part 2 - question 5.

b) What is the decimal version of: 01010010110000? 

Problem 31:

Assuming 2’s complement 8-bit representation, consider the following:

Is the result correct? Why or why not? What can you deduce?

Problem 32:

Draw the combinational circuit that directly implements the Boolean expression:

F(x,y,z)= xz + (xy + z')

Problem 33: Boolean Algebra and Digital Logic 

Consider the following logical diagram of a logical function F:

a) Give the Boolean expression of the above logical diagram’s output function F(a,b,c).

b) Draw the detailed truth table of F.

c) If F1 is the simplified version of F and   what would be the Boolean expression of F1?

d) Use Boolean algebra and Boolean identities to prove that F1 is really a simplified version of F (found in part a).

Problem 34: Binary representations 

Consider the binary number 10011100. What does this binary number represent in decimal (base 10) when it is used in:

a) An 8-bit unsigned whole numbers system?

b) An 8-bit signed magnitude system? 

c) An 8-bit one’s complement system? 

d) An 8-bit two’s complement system? 

Problem 35:

Convert each of the following decimal numbers to the destination radix (unsigned 8 bits representation).

a) (102)10 = (……………)2 

b) (168)10=(………….)16

c) (01001010)2 = (………)10

d) (01001010)2= (……….)16

e) 61₈ = (……………)2

Problem 36:

A) How many nano seconds (ns) are in 12 microseconds (?s)?  

B) How many bits do we have in 6 MB file?  

Problem 37:

Draw the truth table and rewrite the expression below as the products of sum: f=xz’+y’z+x’y

Problem 38:

Given  the following expression F=(x+y)(x+z)(x’+z)

a) Use Boolean algebra and Boolean identities to simplify F to its simplest form. 

b) Draw the truth table of the function F.

c) Draw the logic diagram for simplified F.

Problem 39: The Von-Neumann Model

The Von-Neumann model given below contains three main components: CPU, Main Memory and Input/Output system. It also contains the following sub-components: Control Unit (CU), Arithmetic Logic Unit (ALU), Program Counter Register (PC), Instruction Register (IR), data registers (also called general purpose registers). Use the two lists given above to answer the following questions.

1) During the fetch process: 

a. In which component does the system find the needed instruction?

b. In which sub-component does the system find the address of the needed instruction?

c. Which sub-component is responsible for fetching the instruction?

d. In which sub-component does the system store the fetched instruction? 

In the following, consider that the fetched instruction is an “Add” instruction (add two numbers).

2) During the decode process, in which sub-components does the system store the needed operands? 

3) During the execute process: 

a. Which sub-component is responsible for executing the instruction?

b. Where does this sub-component store the result?