The Essentials of Computer Organization and Architecture
Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003
Chapter 2
Instructor's Manual
______________________________________________________________________________
Chapter Objectives Chapter 2, Data Representation, provides thorough coverage of the various means computers use to represent both numerical and character information. Addition, subtraction, multiplication, and division are covered once the reader has been exposed to number bases and the typical numeric representation techniques, including one’s complement, two’s complement, and BCD. In addition, EBCDIC, ASCII, and Unicode character representations are addressed. Fixed and floating point representation are also introduced. Codes for data recording and error detection and correction are covered briefly. This chapter should be covered after Chapter 1, but before Chapters 4 through 11. Lectures should focus on the following points: • • • •
• • • •
Number systems. Most students have been exposed to positional number systems and different bases. However, these concepts are crucial to understanding the remainder of Chapter 2, so they should be covered in detail. Decimal to binary conversions. Because the binary number system translates easily into electronic circuitry, it is important to become familiar with how computer represent values. Signed versus unsigned numbers. Representing unsigned numbers in binary form is much less complicated than dealing with signed numbers. Signed integer representation. There are basically three methods for representing signed numbers: signed magnitude, one's complement, and two's complement. Each of these methods should be covered, with the focus on signed magnitude and two's complement notations. Binary arithmetic. Although people do not often add binary values, performing binary addition and subtraction helps to reinforce the concepts necessary for understanding data representation. In particular, these operations illustrate the dangers of overflow conditions. Floating point representation. Computers must be able to represent floating point numbers, and there are numerous possible formats for doing so. Potential errors that may result from the limitations of the representation are also important to discuss. Character representation. ASCII, EBCDIC, Unicode and BCD are all important character codes. Lectures should emphasize the similarities and differences among these codes. Codes for data recording and transmission. When binary data is written to some sort of medium or transmitted over long distances, the binary one's and zero's can become Page 1 Last Updated: November 2003
•
blurred. Some sort of encoding is necessary to ensure that characters are properly encoded in these situations. Error detection and correction. Regardless of the coding method used, no communications channel or storage medium is error-free. Although simple parity bits can help to detect errors, more complicated codes, including cyclic redundancy checks and Hamming codes, and are often necessary for sophisticated error detection and correction.
Required Lecture Time Chapter 2 can typically be covered in 6 lecture hours, depending on how detailed one wishes to go into recording and transmission codes and error detection and correction. We suggest that the focus be on integer, floating-point, and character representation, with emphasis given to complement notation. If time permits, data recording and transmission codes and error detection and correction codes can be covered.
Lecture Tips Teachers should spend time explaining the ranges allowed by the different representation formats. For example, if we are using 4 bits to represent unsigned integers, we can represent values from 0 to 15. However, if we are using signed magnitude representation, the bits are interpreted differently, and the possible range is from -7 to +7. Make sure students understand why it is not possible to represent +9 or +10, as these will be seen as -1 and -2 respectively. Converting unsigned whole numbers tends to be relatively straight-forward, but signed numbers and fractions tend to be more difficult. In particular, complement systems confuse students. They often think that all numbers have to be negated to be represented. For example, if a student is asked how a computer, using two's complement representation, would represent -6 (assuming 4-bit values), they answer: 1010 (take 0110, toggle each bit and add 1). This is, of course, the correct answer. However, if that student is asked how the computer using two's complement representation would represent +6, they will often do exactly the same thing, giving the same answer. Teachers need to be sure that students realize representing positive numbers does not require any 'conversion'. It is only when a number is negative that two's complement, signed magnitude, and one's complement representations are necessary. Instructors should spend time on overflow and look at several examples so that students can fully appreciate the consequences. Students find it difficult to understand why one time a carry results in overflow, but yet another carry may not result in overflow. Floating point representation is fairly straight-forward, but students often have difficulty with the format and the bias.
Answers to Exercises 1. Perform the following base conversions using subtraction or division-remainder: a. 45810 = ________ 3 b. 67710 = ________ 5 Page 2 Last Updated: November 2003
c. 151810 = _______ 7 d. 440110 = _______ 9 Ans. a. 1212223
b. 102025
c. 42667
d. 60309
______________________________________________________________________________ 2. Perform the following base conversions using subtraction or division-remainder: a. b. c. d.
58810 = _________ 3 225410 = ________ 5 65210 = ________ 7 310410 = ________ 9
Ans. a. 2102103
b. 330045
c. 16217
d. 42289
______________________________________________________________________________ 3. 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 Ans. a. 11010.11001 b. 11000010.00001 c. 100101010.110011 d. 10000.000111
d. 16.1240234375
______________________________________________________________________________ 4. 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 Ans. a. 11001.11011 b. 111001.100011 c. 1010000.11101 d. 1010100.110111
d. 84.874023
______________________________________________________________________________ 5. Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement and two's complement: a. 77 b. -42 Ans. a. Signed magnitude: 01001101 One's complement: 01001101 Two's complement: 01001101
c. 119
d. –107
b. Signed magnitude: 10101010 One's complement: 11010101 Two's complement: 11010110 c. Signed magnitude: 01110111 One's complement: 01110111 Two's complement: 01110111 Page 3 Last Updated: November 2003
d. Signed magnitude: 11101011 One's complement: 10010100 Two's complement: 10010101
______________________________________________________________________________ 6. 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 b. One's complement Ans. a. 011 to 111, or +3 to -3 b. 011 to 100, or +3 to -3 c. 011 to 100, or +3 to -4
c. Two's complement
______________________________________________________________________________
7. Using a 'word' of 4 bits, list all of the possible signed binary numbers and their decimal equivalents that are representable in: a. Signed magnitude b. One's complement Ans. a. 0111 to 1111, or +7 to -7 b. 0111 to 1000, or +7 to -7 c. 0111 to 1000, or +7 to -8
c. Two's complement
______________________________________________________________________________ 8. 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 Ans. a. -(2x-1-1) to +(2x-1-1) b. -(2x-1-1) to +(2x-1-1) c. -(2x-1) to +(2x-1)
b. One's complement
c. Two's complement
______________________________________________________________________________ 9. Given a (very) tiny computer that has a word size of 6 bits, what are the smallest negative numbers and the largest positive numbers that this computer can represent in each of the following representations? a. One's complement b. Two's complement Ans. a. Largest Positive: 0111112 (31) Smallest Negative: 1000002 (-31) b. Largest Positive: 0111112 (31) Smallest Negative: 1000002 (-32)
______________________________________________________________________________ 10. 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 Page 4 Last Updated: November 2003
codes Zebronian, not Decimal). How many bits will you need to represent each character if you want to use the minimum number of bits? Ans. 40 characters need to be represented by binary coded Zebronian (BCZ), so you will need 6 bits. 5 bits would only give you 32 (25) unique characters. Note that 6 bits would allow you to represent 64 characters.
______________________________________________________________________________ 11. Perform the following binary multiplications: a.
1100 × 101
b. 10101 × 111
Ans. a. 111100
c.
11010 × 1100
b. 10010011
c. 100111000
______________________________________________________________________________ 12. Perform the following..