Computer Architecture Assignment: Data Representation & Logic

This document presents a solved homework assignment focusing on computer organization and architecture, specifically addressing data representation and digital logic. The assignment includes problems related to base conversion, including converting between decimal, binary, octal, hexadecimal, and other bases. It also covers number representation methods such as one's complement, two's complement, and signed magnitude, determining the highest and lowest values representable with a given number of bits. Furthermore, the assignment explores digital logic circuits, including simplifying Boolean expressions and proving the equivalence of combinational circuits using truth tables and De-Morgan's Law. Desklib offers a wide array of solved assignments and past papers for students seeking assistance with their studies.

Running head: COMPUTER ORGANIZATION AND ARCHITECTURE
Computer organization and Architecture
Name of the Student
Name of the University
Author’s Note

1COMPUTER ORGANIZATION AND ARCHITECTURE
Contents
Question1:............................................................................................................................2
Answer to question number a).........................................................................................2
Answer to question number b).........................................................................................2
Answer to question c)......................................................................................................5
Question 2............................................................................................................................6
Answer to question a)......................................................................................................6
Answer to question b)......................................................................................................7
Answer to question c)......................................................................................................8

2COMPUTER ORGANIZATION AND ARCHITECTURE
Question1:
Answer to question number a)
By considering the base as X,
Given,
(152) x = (6A) 16
Or, X^2 + 5X + 2*X = 6*16 + A
Or, X^2 + 5X + 2*X = 6*16 + 10
X2 + 5X + 2 = 106
X^2 + 5X- 104 = 0
X^2 + 13X- 8X – 104 = 0
X (X + 13) – 8(X + 13) = 0
(X - 8) (X + 13) = 0
X = 8 and X = -13
Hence, X is 8.
The value of the base is 8.
Answer to question number b)
Conversions:

3COMPUTER ORGANIZATION AND ARCHITECTURE
i) BED16 converting to base-3
Solution:
= B * 16*16 + E * 16 + D
= 2816 + 224 + 13
= (3053)10
(3053)10 =
3 3053
3 1017 2
3 339 0
3 113 0
3 37 2
3 12 1
3 4 0
3 1 1
So, (BED)16 = (11012002)3
ii) 3217 into 2-base (binary) representation
Solution:
(321)7 = (3 * 72) + (2 * 71) + (1 * 70)
= (162)10

4COMPUTER ORGANIZATION AND ARCHITECTURE
Again, (162)10 =
2 162
2 81 0
2 40 1
2 20 0
2 10 0
2 5 0
2 2 1
2 1 0
Hence, (162)10 = (10100010)2
iii) (1235)10 conversion to octal representation
Solution:
8 1235
8 154 3
8 19 2
8 2 3
Hence, (1235)10 = (2323)8

5COMPUTER ORGANIZATION AND ARCHITECTURE
iv) 21.218 conversion to decimal representation
Solution:
21.218 = (2 * 81) + (1 * 80). (2 * 8-1) + (1 * 7=8-2)
= 17 + 0.25 + 0.015625
= 17.265625
Answer to question c)
i) One's complement
Highest Value is 011
Lowest Value is 100
ii) Two's complement
Highest Value is 011
Lowest Value is 101
iii) Signed Magnitude
Highest Value is 011
Lowest Value is 111

6COMPUTER ORGANIZATION AND ARCHITECTURE
Question 2:
Answer to question a)
The expression for the logic diagram is: (a.b)’
Truth table of the expression is provided below:
A b a.b (a.b)’
0 0 0 1
0 1 0 1
1 0 0 1
1 1 1 0
The expression of the logic diagram
a’ + b’

7COMPUTER ORGANIZATION AND ARCHITECTURE
The truth table for the expression is provided below:
A b a’ b’ a’ + b’
0 0 1 1 1
0 1 1 0 1
1 0 0 1 1
1 1 0 0 0
Hence, LHS = RHS (Proved)
Answer to question b)
The expression derived from the given circuit
A’. B’ + A.B = X
The given circuit that can be minimized from given circuit is:

8COMPUTER ORGANIZATION AND ARCHITECTURE
Answer to question c)
X’ + Y’ + XYZ’
= X’ + Y’ + (X’ + Y’ + Z)’ [ by De-Morgan’s Law]
= (XY (X’ + Y’ + Z))’ [ by De-Morgan’s Law]
= (XX’Y + XYY’ + XYZ)
= (0 + 0 + XYZ)
= (XYZ)’
= X’ + Y’ + Z’ [ by De-Morgan’s Law]
Hence, X’ + Y’ + XYZ’ = X’ + Y’ + Z’ [PROVED]