Deck 2: Data Representation in Computer Systems

Question 1

Perform the following base conversions:

Answer 1

(a) 20012 3 = ____ 10
To convert a number from base 3 to base 10, we need to multiply the individual number in the given number with ascending powers of positive integers of 3 from right to left. Let us see how to convert the given number,
20012= 2×3 4 + 0×3 3 + 0×3 2 + 1×3 1 + 2×3 0
=162 + 0 + 0 + 3 + 2
=167
Hence, the equivalent value of 20012 3 is 167 10.
(b) 4103 5 = ____ 10
To convert a number from base 5 to base 10, we need to multiply the individual number in the given number with ascending powers of positive integers of 5 from right to left. Let us see how to convert the given number,
4103= 4×5 3 + 1×5 2 + 0×5 1 + 3×5 0
=500 + 25 + 0 + 3
=528
Hence, the equivalent value of 4103 5 is 528 10.
(c) 3236 7 = ____ 10
To convert a number from base 7 to base 10, we need to multiply the individual number in the given number with ascending powers of positive integers of 7 from right to left. Let us see how to convert the given number,
3236= 3×7 3 + 2×7 2 + 3×7 1 + 6×7 0
=1029 + 98 + 21 + 6
=1154
Hence, the equivalent value of 3236 7 is 1154 10.
(d) 1378 9 = ____ 10
To convert a number from base 9 to base 10, we need to multiply the individual number in the given number with ascending powers of positive integers of 9 from right to left. Let us see how to convert the given number,
1378=1×9 3 + 3×9 2 + 7×9 1 + 8×9 0
=729 + 243 + 63 + 8
=1043
Hence, the equivalent value of 1378 9 is 1043 10.

Question 2

Convert the hexadecimal number to binary.

Answer 2

Convert Hexadecimal number into Binary
The table of hexadecimal to 4-bit binary digits: Take equivalent 4-bit binary value to AC12(Hexadecimal) from above table.
Then, Hence the binary value of given hexadecimal number as follows:

Question 3

What decimal value does the 8-bit binary number 00010001 have if:
a) it is interpreted as an unsigned number?
b) it is on a computer using signed-magnitude representation?
c) it is on a computer using one's complement representation?
d) it is on a computer using two's complement representation?
e) it is on a computer using excess-127 representation?

Answer 3

a.
The provided 8-bit unsigned binary number is 00010001. Here, the 0 th and 4 th position of the binary number is occupied by the value 1.
Therefore, the conversion of the unsigned binary number 00010001 into a decimal number can be made as follows: Hence, the equivalent decimal value for the provided unsigned number binary number 00010001 is .
b.
The following explains about the signed-magnitude representation of a binary number.
• If the most significant bit (MSB) of a binary number is considered as a sign bit, then the representation is said to be known as signed-magnitude representation. The sign bit will determine whether the binary number is a positive number or a negative number.
• If the MSB of the binary value is 1, then the binary value is said to be a negative number.
• If the MSB of the binary value is 0, then the binary value is said to be a positive number.
Here, the provided signed-magnitude representation of the binary value is 00010001. The MSB of the binary number is 0. Hence the binary number is said to be a positive number.
The 0 th and 4 th position of the binary number is occupied by the value 1. Hence, the equivalent decimal value for the provided signed-magnitude representation of the binary number 00010001 is .
c.
Converting all the binary values 1 into 0 and 0 into 1 in a binary number is said to be known as one's complement representation.
An 8-bit binary number 00010001 is provided such that it is the one's complement representation.
The conversion of the one's complement representation of a binary value 00010001 into a decimal value can be made as follows: The equivalent decimal value of the one's complement representation of a binary number 00010001 is .
d.
Converting all the binary values 1 into 0 and 0 into 1 in a binary number is said to be known as one's complement representation. Adding 1 to the resultant binary value of the one's complement representation is said to be the two's complement representation.
An 8-bit binary number 00010001 is provided such that it is used as the two's complement representation.
The conversion of the two's complement representation of the binary number 00010001 into a decimal value can be done as follows: The equivalent decimal value of the two's complement representation of the binary number 00010001 is .
e.
The decimal value 127 named as k is added with the provided binary value to produce a result. This is said to be the excess-127 representation.
Here, the binary number 00010001 is the excess-127 representation, and then the equivalent decimal number can be found as follows: Therefore, the equivalent decimal value for the excess-127 representation of the binary number 0001001 is .

Question 4

To add 2 two's complement numbers together, what must be true?

Question 5

Perform the following binary divisions, assuming unsigned integers:

Question 6

Given that the ASCII code for A is 1000001, what is the ASCII code for J ?
b) Given that the EBCDIC code for A is 1100 0001, what is the EBCDIC code for J ?

Question 7

Name two ways in which Reed-Solomon coding differs from Hamming coding.

Question 8

What is a radix?

Question 9

What is overflow, and how can it be detected? How does overflow in unsigned numbers differ from overflow in signed numbers?

Question 10

Why is there always some degree of error in floating-point arithmetic when performed by a binary digital computer?

Question 11

What is systematic error detection?

Question 12

Use the double-dabble method to convert directly to decimal. (Hint: You have to change the multiplier.)

Question 13

The ASCII code for the letter A is 1000001, and the ASCII code for the letter a is 1100001. Given that the ASCII code for the letter G is 1000111, without looking at Table 2.7, what is the ASCII code for the letter g ?
b) The EBCDIC code for the letter A is 1100 0001, and the EBCDIC code for the letter a is 1000 0001. Given that the EBCDIC code for the letter G is 1100 0111, without looking at Table 2.6, what is the EBCDIC code for the letter g ?
c) The ASCII code for the letter A is 1000001, and the ASCII code for the letter a is 1100001. Given that the ASCII code for the letter Q is 1010001, without looking at Table 2.7, what is the ASCII code for the letter q ?
d) The EBCDIC code for the letter J is 1101 0001, and the EBCDIC code for the letter j is 1001 0001. Given that the EBCDIC code for the letter Q is 1101 1000, without looking at Table 2.6, what is the EBCDIC code for the letter q ?
e) In general, if you were going to write a program to convert uppercase ASCII characters to lowercase, how would you do it? Looking at Table 2.6, could you use the same algorithm to convert uppercase EBCDIC letters to lowercase?
f) If you were tasked with interfacing an EBCDIC-based computer with an ASCII or Unicode computer, what would be the best way to convert the EBCDIC characters to ASCII characters?

Question 14

When would you choose a CRC code over a Hamming code? A Hamming code over a CRC?

Question 15

Perform the following base conversions:

Question 16

Convert the hexadecimal number to binary.

Question 17

What decimal value does the 8-bit binary number 10110100 have if:
a) it is interpreted as an unsigned number?
b) it is on a computer using signed-magnitude representation?
c) it is on a computer using one's complement representation?
d) it is on a computer using two's complement representation?
e) it is on a computer using excess-127 representation?

Question 18

What is the most common representation used in most computers to store signed integer values and why?

Question 19

Using signed-magnitude representation, complete the following operations:

Question 20

Assume a 24-bit word on a computer. In these 24 bits, we wish to represent the value 295.
? a) How would the computer represent the decimal value 295?
? b) If our computer uses 8-bit ASCII and even parity, how would the computer represent the string 295?
? c) If our computer uses packed BCD with zero padding, how would the computer represent the number 1295?

Question 21

Find the quotients and remainders for the following division problems modulo 2.

Question 22

How many of the "numbers to remember" (in all bases) from Table 2.1 can you remember?
Reference of Table 2.1:

Question 23

If a computer is capable only of manipulating and storing integers, what difficulties present themselves? How are these difficulties overcome?

Question 24

How many bits long is a double-precision number under the IEEE-754 floating-point standard?

Question 25

What is a Hamming code?

Question 26

Suppose a computer uses 4-bit one's complement representation. Ignoring overflows, what value will be stored in the variable j after the following pseudocode routine terminates?

Question 27

Decode the following ASCII message, assuming 7-bit ASCII characters and no parity:
1001010 1001111 1001000 1001110 0100000 1000100 1001111 1000101

Question 28

Find the quotients and remainders for the following division problems modulo 2.

Question 29

Convert the following decimal fractions to binary with a maximum of six places to the right of the binary point:
? a) 26.78125
? b) 194.03125
? c) 298.796875
? d) 16.1240234375

Question 30

Convert the hexadecimal number to binary.

Question 31

Given the following two binary numbers: 11111100 and 01110000.
a) Which of these two numbers is the larger unsigned binary number?
b) Which of these two is the larger when it is being interpreted on a computer using signed two's complement representation?
c) Which of these two is the smaller when it is being interpreted on a computer using signed-magnitude representation?

Question 32

You have stumbled on an unknown civilization while sailing around the world. The people, who call themselves Zebronians, do math using 40 separate characters (probably because there are 40 stripes on a zebra). They would very much like to use computers, but would need a computer to do Zebronian math, which would mean a computer that could represent all 40 characters. You are a computer designer and decide to help them. You decide the best thing is to use BCZ, Binary-Coded Zebronian (which is like BCD except it codes Zebronian, not Decimal). How many bits will you need to represent each character if you want to use the minimum number of bits?

Question 33

Perform the following binary multiplications using Booth's algorithm, assuming signed two's complement integers:

Question 34

Why would a system designer wish to make Unicode the default character set for their new system? What reason(s) could you give for not using Unicode as a default? (Hint: Think about language compatibility versus storage space.)

Question 35

Find the quotients and remainders for the following division problems modulo 2.

Question 36

What does overflow mean in the context of unsigned numbers?

Question 37

What are the goals of Booth's algorithm?

Question 38

What is EBCDIC, and how is it related to BCD?

Question 39

What is meant by Hamming distance, and why is it important? What is meant by minimum Hamming distance?

Question 40

Using arithmetic shifting, perform the following:
a) double the value
b) quadruple the value
c) divide the value in half

Question 41

Assume we wish to create a code using 3 information bits, 1 parity bit (appended to the end of the information), and odd parity. List all legal code words in this code. What is the Hamming distance of your code?

Question 42

Find the quotients and remainders for the following division problems modulo 2.

Question 43

Convert the following decimal fractions to binary with a maximum of six places to the right of the binary point:
a) 25.84375
b) 57.55
c) 80.90625
d) 84.874023

Question 44

Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement, two's complement, and excess-127 representations.
? a) 77
? b) -42
c) 119
d) -107

Question 45

Using a "word" of 3 bits, list all the possible signed binary numbers and their decimal equivalents that are representable in:
a) Signed magnitude
b) One's complement
c) Two's complement

Question 46

Add the following unsigned binary numbers as shown.

Question 47

If the floating-point number representation on a certain system has a sign bit, a 3-bit exponent, and a 4-bit significand:
a) What is the largest positive and the smallest positive number that can be stored on this system if the storage is normalized? (Assume that no bits are implied, there is no biasing, exponents use two's complement notation, and exponents of all zeros and all ones are allowed.)
b) What bias should be used in the exponent if we prefer all exponents to be nonnegative? Why would you choose this bias?

Question 48

Suppose we are given the following subset of code words, created for a 7-bit memory word with one parity bit: 11100110, 00001000, 10101011, and 11111110. Does this code use even or odd parity? Explain.

Question 49

Using the CRC polynomial 1011, compute the CRC code word for the information word, 1011001. Check the division performed at the receiver.

Question 50

Name the four ways in which signed integers can be represented in digital computers, and explain the differences.

Question 51

How does carry differ from overflow?

Question 52

What is ASCII, and how did it originate?

Question 53

How is the number of redundant bits necessary for code related to the number of data bits?

Question 54

Using the model in the previous question, including your chosen bias, add the following floating-point numbers and express your answer using the same notation as the addend and augend:
Calculate the relative error, if any, in your answer to the previous question.

Question 55

Are the error-correcting Hamming codes systematic? Explain.

Question 56

Using the CRC polynomial 1101, compute the CRC code word for the information word, 01001101. Check the division performed at the receiver.

Question 57

Perform the following base conversions using subtraction or division-remainder:
Perform the following base conversions using subtraction or division-remainder:

Question 58

Convert the following decimal fractions to binary with a maximum of six places to the right of the binary point:
a) 27.59375
b) 105.59375
c) 241.53125
d) 327.78125

Question 59

Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement, two's complement, and excess-127 representations:
a) 60
b) -60
c) 20
d) -20

Question 60

Using a "word" of 4 bits, list all the possible signed binary numbers and their decimal equivalents that are representable in:
a) Signed magnitude
b) One's complement
c) Two's complement

Question 61

Add the following unsigned binary numbers as shown.

Question 62

Assume we are using the simple model for floating-point representation as given in the text (the representation uses a 14-bit format, 5 bits for the exponent with a bias of 15, a normalized mantissa of 8 bits, and a single sign bit for the number):
a) Show how the computer would represent the numbers 100.0 and 0.25 using this floating-point format.
b) Show how the computer would add the two floating-point numbers in part a by changing one of the numbers so they are both expressed using the same power of 2.
c) Show how the computer would represent the sum in part b using the given floating- point representation. What decimal value for the sum is the computer actually storing? Explain.

Question 63

Compute the Hamming distance of the following code:
0011010010111100
0000011110001111
0010010110101101
0001011010011110

Question 64

Using the CRC polynomial 1101, compute the CRC code word for the information word, 1100011. Check the division performed at the receiver.

Question 65

The word bit is a contraction for what two words?

Question 66

Which one of the four representations for signed integers is used most often by digital computer systems?

Question 67

What is arithmetic shifting?

Question 68

Explain the difference between ASCII and Unicode.

Question 69

What is a burst error?

Question 70

What causes divide underflow, and what can be done about it?

Question 71

Compute the Hamming distance of the following code:
0000000101111111
0000001010111111
0000010011011111
0000100011101111
0001000011110111
0010000011111011
0100000011111101
1000000011111110

Question 72

Using the CRC polynomial 1101, compute the CRC code word for the information word, 01011101. Check the division performed at the receiver.

Question 73

Perform the following base conversions using subtraction or division-remainder:

Question 74

Convert the following binary fractions to decimal:
a) 10111.1101
b) 100011.10011
c) 1010011.10001
d) 11000010.111

Question 75

Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement, two's complement, and excess-127 representations:
a) 97
b) -97
c) 44
d) -44

Question 76

From the results of the previous two questions, generalize the range of values (in decimal) that can be represented in any given x number of bits using:
a) Signed magnitude
b) One's complement
c) Two's complement

Question 77

Subtract the following signed binary numbers as shown using two's complement arithmetic.

Question 78

Why do we usually store floating-point numbers in normalized form? What is the advantage of using a bias as opposed to adding a sign bit to the exponent?

Question 79

In defining the Hamming distance for a code, we choose to use the minimum (Hamming) distance between any two encodings. Explain why it would not be better to use the maximum or average distance.

Question 80

Pick an architecture (such as 80486, Pentium, Pentium IV, SPARC, Alpha, or MIPS). Do research to find out how your architecture approaches the concepts introduced in this chapter. For example, what representation does it use for negative values? What character codes does it support?