Module 3: Representing Values in a Computer

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1 Module 3: Representing Values in a Computer

2 Due Dates Assignment #1 is due Thursday January 19th at 4pm Pre-class quiz #4 is due Thursday January 19th at 7pm. Assigned reading for the quiz: Epp, 4th edition: 2.3 Epp, 3rd edition: 1.3 Come to my office hours (ICCS X563)! Wednesday 4-5pm, Thursday 2-3pm, and Friday 12-1pm 2

3 Learning goals: pre-class By the start of this class you should be able to Convert unsigned integers from decimal to binary and back. Take two's complement of a binary integer. Convert signed integers from decimal to binary and back. Convert integers from binary to hexadecimal and back. Add two binary integers. 3

4 Learning goals: in-class By the end of this module, you should be able to: 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 scheme for signed integer arithmetic in computers hexadecimal for human inspection of raw binary data. 4

5 CPSC 121: the BIG questions: We will make progress on two of them: How does the computer (e.g. Dr. Racket) decide if the characters of your program represent a name, a number, or something else? How does it figure out if you have mismatched " " or ( )? How can we build a computer that is able to execute a user-defined program? 5

6 Module 3 outline Unsigned and signed binary integers. Characters. Real numbers. Hexadecimal. 8

7 Review question What does it mean for a binary integer to be unsigned or signed?

8 Recall the 7-segment display problem Mapping unsigned integers between decimal and binary Number X1 X2 X3 X

9 Unsigned binary à decimal The binary value Represents the decimal value Examples: = = = = = = 50 11

10 Decimal à unsigned binary Divide x by 2 and write down the remainder The remainder is 0 if x is even, and 1 if x is odd. Repeat this until the quotient is 0. Write down the remainders from right (first one) to left (last one).

11 Decimal à unsigned binary Example: Convert 50 to a 8-bit binary integer 50 / 2 = 25with remainder 0 25 / 2 = 12 with remainder 1 12 / 2 = 6 with remainder 0 6 / 2 = 3 with remainder 0 3 / 2 = 1 with remainder 1 1 / 2 = 0 with remainder 1 Answer is

12 To negate a (signed) binary integer The algorithm (also called taking two s complement of the binary number): Flip all bits: replace every 0 bit by a 1, and every 1 bit by a 0. Add 1 to the result. Why does it make sense to negate a binary integer by taking its two s complement? 14

13 Additive Inverse Two numbers x and y are additive inverses of each other if they sum to 0. Examples: 3 + (-3) = 0. (-7) + 7 = 0. 15

14 Clock arithmetic Pre-class quiz #3b: It is currently 18:00, that is 6pm. Without using numbers larger than 24 in your calculations, what time will it be 22 * 7 hours from now? (Don t multiply 22 by 7!) 16

15 Clock arithmetic 3:00 is 3 hour from midnight. 21:00 is 3 hours to midnight. 21 and 3 are additive inverses in clock arithmetic: = 0. Any two numbers across the clock are additive inverses of each other. 21 is equivalent to -3 in any calculation.

16 Clock arithmetic Pre-class quiz #3b: It is currently 18:00, that is 6pm. Without using numbers larger than 24 in your calculations, what time will it be 22 * 7 hours from now? (Don t multiply 22 by 7!) a. 0:00 (midnight) b. 4:00 (4am) c. 8:00 (8am) d. 14:00 (2pm) e. None of the above 18

17 Negative numbers in clock arithmetic Put negative numbers across the clock from the positive ones (where the additive inverses are) Since = 0, we could put -3 where 21 is

18 Binary clock arithmetic Suppose that we have 3 bits to work with. If we are only representing positive values or zero:

19 Binary clock arithmetic If we want to represent negative values... We can put them across the clock from the positive ones (where the additive inverses are)

20 Two s Complement Taking two s complement of B = b 1 b 2 b 3...b n : Flip the bits Add one b 1 b 2 b 3...b n x 1 x 2 x 3...x n B 22

21 Flip the bits Add one A Different View of Two s Complement Taking two s complement of B = b 1 b 2 b 3...b k : b 1 b 2 b 3...b k x 1 x 2 x 3...x k B b 1 b 2 b 3...b k B Equivalent to subtracting from with k 0s. 23

22 Two s Complement vs. Crossing the Clock Two s complement with k bits: b 1 b 2 b 3...b n B Equivalent to subtracting from with k 0s

23 Advantages of two s complement Which one(s) are the advantages of the two s complement representation scheme? a. There is a unique representation of zero. b. It is easy to tell a negative integer from a non-negative integer. c. Basic operations are easy. For example, subtracting a number is equivalent to adding the two s complement of the number. d. All of (a), (b), and (c). e. None of (a), (b), and (c).

24 Advantages of two s complement Why did we choose to let 100 represent -4 rather than 4?

25 What does two s complement mean? a. Taking two s complement of a binary integer means flip all of the bits and then add 1. b. Taking two s complement is a mathematical procedure to negate a binary integer. c. Two s complement is a representation scheme we use to map signed binary integers to decimal integers. d. 2 of (a), (b), and (c) are true. e. All of (a), (b), and (c) are true.

26 Negative binary à decimal Example: convert the 6-bit signed binary integer to a decimal number. 1. Convert the binary integer directly to decimal ( = 46) 2. Subtract 2^6 from it (2^6 because of 6-bit) (46 64 = -18). 3. Answer is

27 Negative binary à decimal The signed binary value represents the integer Example: convert the 6-bit signed binary integer to a decimal number = = -18. Answer is

28 Negative decimal à binary Example: convert -18 to a 6-bit signed binary number. 1. Add 2^6 to it (2^6 because of 6-bit) ( ^6 = = 46). 2. Convert the positive decimal integer directly to binary. 46 / 2 = 23 0, 23 / 2 = , 12 / 2 = , 6 / 2 = , 3 / 2 = , 1 / 2 = Answer is

29 Questions to ponder: With n bits, how many distinct values can we represent? What are the smallest and largest n-bit unsigned binary integers? What are the smallest and largest n-bit signed binary integers? 32

30 More questions to ponder: Why are there more negative n-bit signed integers than positive ones? How do we tell if an unsigned binary integer is: negative, positive, zero? How do we tell if a signed binary integer is: negative, positive, zero? How do we negate a signed binary integer? On what value does this negation not behave like negation on integers? How do we perform the subtraction x y? 33

31 Module 3 outline Unsigned and signed binary integers. Characters. Real numbers. Hexadecimal. 34

32 How do computers represent characters? It uses sequences of bits (like for everything else). Integers have a natural representation of this kind. There is no natural representation for characters. So people created arbitrary mappings: EBCDIC: earliest, now used only for IBM mainframes. ASCII: American Standard Code for Information Interchange7-bit per character, sufficient for upper/lowercase, digits, punctuation and a few special characters. UNICODE:16+ bits, extended ASCII for languages other than English 35

33 Representing characters What does the 8-bit binary value represent? a.-8 b.the character c.248 d.more than one of the above ø e.none of the above. Show Answer 36

34 Module 3 outline Unsigned and signed binary integers. Characters. Real numbers. Hexadecimal. 38

35 Can someone be 1/3rd Scottish? No, but what if I once wore a kilt? Not normally, since every person's genetic code is derived from two parents, branching out in halves going back. However, someone with a chromosome abnormality that gives them three chromosomes (trisomy), a person might be said to be one-third/two-thirds of a particular genetic marker. 39

36 Can someone be 1/3rd Scottish? I have come to the conclusion that no, you cannot be onethird Scottish. I will provide my reasoning with the following two examples. Say there are two progenitors, P and Q. P is Korean and Q is American. If P and Q have a child, PQ, he will be 1/2 Korean and 1/2 American. Now say that PQ grows up to the legal age (no pedophillia in here) and enters a romantic relationship with another progenitor, R, who is Scottish. PQ and R have a child, PQR, and he will be 1/2 Scottish, 1/4 Korean, and 1/4 American. The second example, we can say PQ conceives a child, PQUT, with someone named UT, who is 1/2 Ethiopian and 1/2 African. PQUT would be 1/4 of each ethnicity. We can see that you will only be (1/2)^n - where n is a nonnegative integer - of an ethnicity (n in this context means which generation it is). Notice that the denominator will always be a multiple of 2. Therefore, you can never be 1/3 of any ethnicity. 40

37 Can someone be 1/3rd Scottish? While debated, Scotland is traditionally said to be founded in 843AD, approximately 45 generations ago. Your mix of Scottish, will therefore be n/2 45 ; using 2 45 /3 (rounded to the nearest integer) as the numerator gives us /2 45 which give us approximately which is no more than 1/10 13 th away from 1/3. 41

38 Can someone be 1/3rd Scottish? If we assume that two Scots have a child, and that child has a child with a non-scot, and this continues in the right proportions, then eventually their Scottishness will approach 1/3: This is of course discounting the crazy citizenship laws we have these days, and the effect of wearing a kilt on one's heritage. 42

39 Can someone be 1/3rd Scottish? In a mathematical sense, you can create 1/3 using infinite sums of inverse powers of 2 1/2 isn't very close 1/4 isn't either 3/8 is getting there... 5/16 is yet closer, so is 11/32, 21/64, 43/128 etc 85/256 is , which is much closer, but which also implies eight generations of very careful romance amongst your elders. 5461/16384 is , which is still getting there, but this needs fourteen generations and a heck of a lot of Scots and non-scots. 43

40 Can someone be 1/3 rd Scottish? To answer this question, we need to make some assumptions. What does being Scottish mean? Nationality? Ethic identity? How does the Scottish-ness of a parent influence the Scottish-ness of a child? Our model of ancestry: Each parent gives you 50% of an ancestry.

41 Can someone be 1/3 rd Scottish? Suppose we start with people who are either 0% or 100% Scottish. After 1 generation, how Scottish can a child be? After 2 generations, how Scottish can a grandchild be? What about 3 generations? What about n generations? 45

42 Representing real numbers in binary Which of the following values have a finite binary representation? a.1/4 b.1/6 c.1/9 d.more than one of the above. e.none of the above. 46

43 Fractions in base 10 and base 2 5/8 = (in base 10) = (in base 2) Can be represented exactly in both. 1/3 = (in base 10) = (in base 2) Can be represented exactly in neither. Could you come up with a number that can be represented exactly in base 10 but not in base 2? Could you come up with a number that can be represented exactly in base 2 but not in base 10?

44 How does computer represent values of the form xxx.yyyy? Java uses scientific notation 1724 = x 10 3 But in binary, instead of decimal = x mantissa exponent Only the mantissa and exponent need to be stored. The mantissa has a fixed number of bits (24 for float, 53 for double). Scheme/Racket uses this for inexact numbers. 48

45 Floating point computations Computations involving floating point numbers are imprecise. The computer does not store 1/3, but a number that's very close to 1/3. The more computations we perform, the further away from the real value we are. 49

46 Floating point computations Predict the output of: (* #i ) a. 1 b. Not 1, but a value close to 1. c. Not 1, but a value far from 1. d. I don t know. 50

47 Floating point computations Consider the following: (define (addfractions x) (if (= x 1.0) 0 (+ 1 (addfractions (+ x #i0.1))))) What value will (addfractions 0) return? a. 10 b. 11 c. Less than 10 d. More than 11 e. No value will be printed 51

48 Module 3 outline Summary Unsigned and signed binary integers. Characters. Real numbers. Hexadecimal. 52

49 Module 3.4: Hexadecimal As you learned in CPSC 110, a program can be Interpreted: another program is reading your code and performing the operations indicated. Example: Scheme/Racket Compiled: the program is translated into machine language. Then the machine language version is executed directly by the computer. 53

50 Module 3.4: Hexadecimal What does a machine language instruction look like? It is a sequence of bits! Y86 example: adding two values. In human-readable form: addl %ebx, %ecx. In binary: %ebx Addition Arithmetic operation %ecx 54

51 Module 3.4: Hexadecimal Long sequences of bits are painful to read and write, and it's easy to make mistakes. Should we write this in decimal instead? Decimal version: Problem: We can not tell what operation this is. Solution: use hexadecimal 6031 Addition Arithmetic operation %ecx %ebx 55

52 Module 3.4: Hexadecimal Another example: Suppose we make the text in a web page use color What color is this? Red leaning towards purple. Written in hexadecimal: F00084 Red Green Blue 56

CPSC 121: Models of Computation. Module 3: Representing Values in a Computer

CPSC 121: Models of Computation in a Computer The 4th online quiz is due Thursday, January 22nd at 19:. Assigned reading for the quiz: Epp, 4th edition: 2.3 Epp, 3rd edition: 1.3 Rosen, 6th edition: 1.5