# CS/ECE 252 Introduction to Computer Engineering

Spring 2013 Section 1 & 2
Instructors Mark D. Hill and Guri Sohi
TAs Preeti Agarwal, Mona Jalal, Rebecca Lam, Pradip Vallathol

URLs: http://www.cs.wisc.edu/~markhill/cs252/Spring2013/ and http://www.cs.wisc.edu/~sohi/cs252/Spring2013/

## Homework 2 [Due at lecture on Wed, Feb 6]

Primary contact for this homework: Pradip Vallathol [pradip16 at cs dot wisc dot edu]

You must do this homework in groups of two. Please write the full name and the student id of each member on every page and staple multiple pages together.

### Problem 1 (4 points)

1. Assume that 453 scientists have been invited to attend a conference. If every scientist is to be assigned a unique bit pattern, what is the minimum number of bits required to do this?
453 < 512 (29). So 9 bits.
2. How many more scientists can be invited to the conference, without requiring additional bits for each person's unique id?
512 - 453 = 59

### Problem 2 (8 points)

Using 7 bits to represent each number, write the representations of 25, -25 and 0 in signed magnitude, 1's complement and 2's complement notations.

 Number Sign-Magnitude 1's Complement 2's Complement 25 001 1001 001 1001 001 1001 -25 101 1001 110 0110 110 0111 0 000 0000OR100 0000 000 0000OR111 1111 000 0000

### Problem 3 (4 points)

The following binary numbers are 5-bit 2's complement binary numbers. Which of the following operations generate overflow? Justify your answers by translating the operands and results into decimal.

1. 00111 + 00110
01101 = 13
7 + 6 = 13
No overflow
2. 10111 - 11110
11001 = -7
(-9) - (-2) = -7
No overflow
3. 11000 - 00011
10101 = -11
(-8) - 3 = -11
No overflow
4. 10110 + 10011
01001 = 9
(-10) + (-13) = -23
Overflow

### Problem 4 (2 points)

Compute the followings:

1. (10100 OR 01101) AND (NOT(10101))
= 11101 AND 01010 = 01000
2. (10101 OR 00100) AND (NOT(00101) OR 01010)
= 10101 AND 11010 = 10000

### Problem 5 (4 points)

Convert the following decimal numbers into 6-bit 2's complement binary numbers. Explain any problem that you encounter for these conversions.

1. 23
01 0111
2. -33
-33 < -32 (largest negative integer that can be represented 2's complement form using 6 bits)
Therefore, cannot be represented.
3. 32
32 > 31 (largest positive integer that can be represented 2's complement form using 6 bits)
Therefore, cannot be represented.
4. -32
10 0000

### Problem 6 (2 points)

Perform the specified arithmentic operation for the following 2's complement binary numbers:

1. 11001 + 1011
11001 + 11011 = 10100
2. 00111 - 010
00111 - 00010 = 00111 + 11110 = 00101
Note:
1. All numbers beginning with 1 are negative.
2. Don't forget to sign-extend when required.

### Problem 7 (4 points)

Write the decimal equivalents for the following IEEE floating point numbers.

1. 1 01111110 01000000000000000000000
(-1)1 x 2126-127 x (1.01)2 = -0.625
2. 0 10000001 10100000000000000000000
(-1)0 x 2129-127 x (1.101)2 = 6.5

### Problem 8 (2 points)

Write the IEEE floating point representation of the decimal number 2.50.
(2.5)10 = (10.1)2 = (1.01)2 x 21 = (-1)0 x 2(128-127) x (1.01)2
0 10000000 01000000000000000000000

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