Pre-Class Learning Goals By the start of this class you should be able to CPSC 121: Models of Computation Convert unsigned integers from decimal to binary and back. Take two's complement of a binary integer. Unit 3 Convert signed integers from decimal to binary and back. Representing Numbers and Other Values Convert integers from binary to hexadecimal and back. in a Computer Add two binary integers. Based on slides by Patrice Belleville and Steve Wolfman Unit 3: Representing Values 2 Quiz 3 Feedback In-Class Learning Goals Overall: By the end of this unit, you should be able to: Issues: Critique the choice of a digital representation scheme, including describing its strengths, weaknesses, and flaws (such as imprecise representation or overflow), for a given type of data and purpose, such as fixed- width binary numbers using a two’s complement Clock Arithmetic and Can you be 1/3rd Scottish? scheme for signed integer arithmetic in computers We will get back to these questions soon. hexadecimal for human inspection of raw binary data. Unit 3: Representing Values Unit 3: Representing Values 3 4 1
Motivating Problem Where We Are in The Big Stories Theory: Hardware: Understand and avoid cases like those at: Photo by Philippe Semanaz How do we model How do we build devices to (CC by/sa) http://www.ima.umn.edu/~arnold/455.f96/disasters.html computational systems? compute? Death of 28 people caused by failure of an anti-missile system, caused in turn by the misuse of one Now: Now: representation for fractions. Showing that our logical Enabling our hardware to Explosion of a $7 billion space vehicle caused by failure models can connect work with numbers. smoothly to models of And once we have of the guidance system, caused in turn by misuse of a number systems. numbers, we can represent 16-bit signed binary value. pictures, words, sounds, and everything else! (Both representations are discussed in these slides.) Unit 3: Representing Values Unit 3: Representing Values 5 6 Outline Representing Numbers We can choose any arrangement of Ts and Fs to represent numbers... Unsigned and signed binary integers. If we use F -> 0 and T->1, the representation is a true Characters. base 2 numbers: Real numbers. Number V1 V2 V3 V4 Number b 3 b 2 b 1 b 0 0 F F F F 0 0 0 0 0 1 F F F T 1 0 0 0 1 2 F F T F 2 0 0 1 0 3 F F T T 3 0 0 1 1 4 F T F F 4 0 1 0 0 5 F T F T 5 0 1 0 1 6 F T T F 6 0 1 1 0 7 F T T T 7 0 1 1 1 8 T F F F 8 1 0 0 0 9 T F F T 9 1 0 0 1 Unit 3: Representing Values Unit 3: Representing Values 7 8 2
Representing Unsigned Integers Number Systems A number can be written using any base. The binary value What can you say about the following numbers? b b ... b b b 100001 2 n 1 n 2 2 1 0 represents the integer 33 10 n 1 n 2 2 1 1020 3 b 2 + b 2 + ... + b 2 + b 2 + b n 1 n 2 2 1 0 21 16 or written differently A. First number is different than the rest B. Second number is different than the rest i b 2 i C. Third number is different than the rest where each b i is either 0 or 1 D. All are the same number E. All are different numbers Unit 3: Representing Values Unit 3: Representing Values 9 10 Clock Arithmetic Clock Arithmetic Problem: It’s 05:00h. How 05:00 is five hours from midnight. many hours until midnight? 19:00 is five hours to midnight. 5 and 19 are “additive inverses” in Give an algorithm that clock arithmetic: 5 + 19 = 0. requires a 24-hour clock, a level, and no arithmetic. So. what is the additive inverse of 15? How did you find it? How else can you write 15? That’s even true for 12. Its additive inverse is itself! Unit 3: Representing Values Unit 3: Representing Values 11 12 3
Clock Arithmetic Open-Ended Quiz # Question Suppose it is 15:00 (3:00PM, that is). If we wanted negative What time was it 8* 21 hours ago? numbers on the clock, Don’t multiply 21 by 8! 0 we’d probably put them Must not use numbers larger than 24 in -3 “across the clock” from your calculation' the positives. A. 21:00 -6 B. 4:00 After all, if 3 + 21 is C. 8:00 already 0, why not put -3 -9 D. 15:00 where 21 usually goes? -12 E. None of these Unit 3: Representing Values Unit 3: Representing Values 13 14 Open-Ended Quiz # Question Unsigned Binary Clock Here’s a 3 -bit Suppose it is 15:00 (3:00PM, that is). What time will it be 13 * 23 hours unsigned binary from now? clock, numbered from 000 Don’t multiply 13 by 23! 0 (000) to 7 (111). 111 001 A. 13:00 110 010 B. 2:00 C. 22:00 101 011 D. 15:00 100 E. None of these Unit 3: Representing Values Unit 3: Representing Values 15 16 4
Two’s complement for Negative Integers Crossing the Clock To “cross the clock”, In a 3-bit signed numbers: go as many ticks left 2 3 is the same as 000 000 from the top as you the negation of a number x is the number 2 3 – x. (1000) previously went right -1 001 Why does this make sense? (111) from the top. For 3-bit integers, what is 111 + 001? -2 010 Here’s a clock labelled A. 110 (110) with 0 (000) to 3 (011) B. 111 -3 011 and -1 (111) to -4 C. 1000 (101) -4 (100). D. 000 (100) E. Error: we can not add these two values. Unit 3: Representing Values Unit 3: Representing Values 17 18 Two's Complement for n-Bit Integers Exercice Unfolding the Also note that clock into a line: What is 10110110 in decimal, assuming it's a signed first bit determines the sign 10…00 -2 n-1 8-bit binary integer? o 0-positive, 1- negative A. 182 -x has the same binary n B. -74 -2 2 x 11…01 representation as -1 C. 74 11… 11 And 0 00… 00 D. -182 1 1 00…01 n n x = x + 2 2 1 E. None of the above 00…10 2 Add 1 Flip bits from 0 to 1 and from 1 to 0 2 n-1 -1 01…11 Unit 3: Representing Values Unit 3: Representing Values 19 20 5
Converting Decimal to Binary Converting Binary to Decimal How do we convert a positive decimal integer n to Theorem: for n-bit signed integers, the binary value b b ... b b b binary? n 1 n 2 2 1 0 The last bit is 0 if n is even, and 1 if n is odd. represents the integer To find the remaining bits, we divide n by 2, ignore n 1 n 2 2 1 b 2 + b 2 + ... + b 2 + b 2 + b n 1 n 2 2 1 0 the remainder, and repeat. What do we do if n is negative? or written differently n 1 i b 2 + b 2 n 1 i Examples: Examples: Unit 3: Representing Values Unit 3: Representing Values 21 22 Summary Questions More Summary Questions With n bits, how many distinct values can we represent? How do we tell quickly if a unsigned binary integer is negative, positive, or zero? What are the smallest and largest n-bit unsigned binary How do we tell quickly if a signed binary integer is negative, integers? positive, or zero? How do we negate a signed binary integer? What are the smallest and largest n-bit signed binary integers? There is one signed n-bit binary integer that we should not try to negate. Which one? Why are there more negative n-bit signed integers than positive ones? What do we get if we try negating it? Unit 3: Representing Values Unit 3: Representing Values 23 24 6
Outline Representing Characters How do computers represent characters? Unsigned and signed binary integers. They use sequences of bits (like for everything else). Characters. Integers have a “natural” representation of this kind. There is no natural representation for characters. Real numbers. So people created arbitrary mappings: o EBCDIC: earliest, now used only for IBM mainframes. o ASCII: American Standard Code for Information Interchange • 7-bit per character, sufficient for upper/lowercase, digits, punctuation and a few special characters. o UNICODE: • 16+ bits, extended ASCII for languages other than English Unit 3: Representing Values Unit 3: Representing Values 25 26 Representing Characters Outline Unsigned and signed binary integers. What may the binary value 11111000 represent? Characters. A. -8 B. The character ø Real numbers. C. 248 D. More than one of the above E. None of the above. Unit 3: Representing Values Unit 3: Representing Values 27 28 7
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