QUIZ: Generations of computer technology. Hardware:

Transcription

1 QUIZ: Generations of computer technology Hardware:

2 QUIZ: Generations of computer technology Software:

3 Chapter 2 Binary Values and Number Systems

4 Numbers Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative numbers A value less than 0, with a sign Examples: -24, -1, ,

5 Integers A natural number, a negative number, zero Examples: 249, 0, , - 32 Rational numbers An integer or the quotient of two integers Examples: -249, -1, 0, 3/7, -2/5 Real numbers In general cannot be represented as the quotient of any two integers. They have an infinite # of fractional digits. Example: Pi =

6 2.2 Positional notation How many ones (units) are there in 642? x x x 1 6 x x x is called the base This is called the polynomial expansion of the number We also write

7 QUIZ Write the polynomial expansion of (642 in base nine), and convert it to decimal. Use the previous expansion as example: x x x 1 6 x x x

8 QUIZ How many ones (units) are there in (642 in base nine)? 6 x x x 9 0 = =

9 Positional Notation The base of a number determines how many digits are used and the value of each digit s position. To be specific: In base R, there are R digits, from 0 to R-1 The positions have for values the powers of R, from right to left: R 0, R 1, R 2, 9 5

10 Positional Notation Formula: R is the base of the number d n * R n-1 + d n-1 * R n d 2 * R + d 1 n is the number of digits in the number d is the digit in the i th position in the number 10 7

11 Positional Notation reloaded The text shows the digits numbered like this: d n * R n-1 + d n-1 * R n d 2 * R + d 1 but, in CS, the digits are numbered from zero, to match the power of the base: d n-1 * R n-1 + d n-2 * R n d 1 * R 1 + d 0 * R

12 QUIZ What is 642 in base 13? 12 68

13 QUIZ What is 642 in base 13? + 6 x 13 2 = 6 x 169 = x 13 1 = 4 x 13 = x 13º = 2 x 1 = 2 = 1068 in base =

14 Nota bene! In a given base R, the digits range from 0 up to R 1 R itself cannot be a digit in base R Trick problem: Convert the number 473 from base 6 to base 10 14

15 Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 15 9

16 QUIZ: Converting Binary to Decimal What is the decimal equivalent of the binary number ? =???

17 What is the decimal equivalent of the binary number ? 1 x 2 6 = 1 x 64 = x 2 5 = 1 x 32 = x 2 4 = 0 x 16 = x 2 3 = 1 x 8 = x 2 2 = 1 x 4 = x 2 1 = 1 x 2 = x 2º = 0 x 1 = 0 = 110 in base 10 =

18 Base 8 = octal system What is the decimal equivalent of the octal number 642? =???

19 Converting Octal to Decimal What is the decimal equivalent of the octal number 642? 6 x 8 2 = 6 x 64 = x 8 1 = 4 x 8 = x 8º = 2 x 1 = 2 Add the above = 418 in base 10 =

20 Bases Higher than 10 How are digits in bases higher than 10 represented? Base 16 (hexadecimal, a.k.a. hex) has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F 20 10

21 Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number DEF? D x 16 2 = 13 x 256 = E x 16 1 = 14 x 16 = F x 16º = 15 x 1 = 15 = 3567 in base 10 21

22 QUIZ: 2AF 16 =??? 10 22

23 Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics? 23

24 QUIZ: Convert to decimal = C7 16 = 42 6 = 71 8 = 24

25 Today we ve covered pp of text (stopped before Arithmetic in Other Bases) Solve in notebook as individual work for next class: 1, 2, 3, 4, 5, 20, 21 25

26 QUIZ: Convert to decimal = AB7 16 = = = 26

27 The inverse problem: Converting Base 10 to Other Bases Algorithm for converting a number in base 10 to any other base R: While (the quotient is not zero) Divide the decimal number by R Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient Known as repeated division (by the base) 27 19

28 Converting Decimal to Binary Example: Convert to binary = 89 rem. 1 2 = 44 rem. 1 2 = 22 rem. 0 2 = 11 rem. 0 MSB LSB 2 = 5 rem. 1 2 = 2 rem. 1 2 = 1 rem = = 0 rem. 1 Notes: The first bit obtained is the rightmost (a.k.a. LSB) The algorithm stops when the quotient (not the remainder!) becomes zero 28 19

29 Repeated division QUIZ Convert to binary 42 2 = rem =

30 The repeated division algorithm can be used to convert from any base into any other base (but normally we use it only for 10 2) For next time: Read text example on p.43: Converting Decimal to Hex using repeated division 30

31 Addition in Binary Remember that there are only 2 digits in binary, 0 and is 0 with a carry Carry Values 31 14

32 Addition QUIZ Carry values go here Check in base ten! 32 14

33 Putting it all together! Computer hardware Python (or another high-level language) Convert decimal to binary Compute in binary Convert binary to decimal 33

34 Direct conversions between bases that are powers of 2 binary hexadecimal octal 34

35 Converting Binary to Octal Mark groups of three (from right) Convert each group is 253 in base

36 Converting Binary to Hexadecimal Mark groups of four (from right) Convert each group A B is AB in base

37 Extra-credit QUIZ: 37

38 Converting Octal to Hexadecimal End-of-chapter ex. 25: Explain how base 8 and base 16 are related A B 253 in base 8 = AB in base

39 End-of-chapter ex.37 Perform the following octal additions: a b

40 Read and take notes: Binary SUBTRACTION (with borrow bits) p.40 of text 40

41 Today we ve covered pp of text (stopped before Binary Values and Computers) Solve in notebook as individual work for next class: 6 through 11 41

42 Addition QUIZ Carry values go here Check in base ten! 42 14

43 End-of-chapter ex.37 Perform the following octal additions: c

44 Counting 44

45 Basic skill: counting (in any base!) 0, 1, 2, 3, 4, 5, 0, 1, 10, 11, 100, 101, 45

46 Hex 46

47 Conclusion: In order to represent any octal digit, we need at most bits In order to represent any hex digit, we need at most bits 47

48 Binary and Computers Word = group of bits that the computer processes at a time The number of bits in a word determines the word length of the computer. It is usually a multiple of 8. 1 Byte = 8 bits 8, 16, 32, 64-bit computers 128? 256? 48 23

49 6-bit computers: an Evolutionary dead-end? Motivated by the 6-bit codes for printable graphic patterns created by the U.S. Army and Navy 6, 18, 24, 36, 48-bit words Some history: 18-Bit Computers from DEC 36-bit Wikipedia Edged out of the market by the need for floatingpoint numbers IBM System/360 (1965) 8-bit microprocessors (1970s) Not in text 49

50 6-bit computers: an Evolutionary dead-end? Unisys is still successful with their 36- and 48-bit machines : Clearpath Dorado line of 36-bit CISC high-end servers Clearpath Libra line of 48-bit mainframes Not in text although they are being transitioned to Intel Xeon chips (64-bit): see article 50

51 Grace Murray Hopper Ph.D. in mathematics Wrote A-0, the world s first compiler, in 1952! Co-invented COBOL Rear-admiral of the US Navy Nanosecond wires 51

52 Grace also liked telling this story Harvard University Mark II Aiken Relay Calculator 52

53 Ethical Issues Tenth Strand What do the following acronyms stand for: ACM? IEEE? What is the tenth strand? Why the tenth? 53

54 Why the tenth? 54 A: In the 1989 ACM report ( Computing as a Discipline p.12), the following 9 areas (strands) of CS were defined: Algorithms and data structures Programming languages Computer Architecture Numerical and symbolic computations Operating systems (OS) Software engineering Databases Artificial intelligence (AI) Human-computer interaction

55 Ethical Issues Tenth Strand The latest official release of the IEEE/ACM CS Curriculum was in 2001: 3 levels of organization: areas knowledge units topics There are now 14 areas, and the tenth strand is listed in position 12 What does SP stand for? How many knowledge units does the SP area have? Name two of these units! 55

56 Chapter Review questions Describe positional notation (polynomial in the base) Convert numbers in other bases to base 10 Convert base-10 numbers to numbers in other bases Add and subtract in binary Convert between bases 2, 8, and 16 using groups of digits Count in binary Explain the importance to computing of bases that are powers of

57 Chapter Review questions IEEE/ACM CS Curriculum and the Tenth Strand

58 Homework for Ch.2 End-of-ch. 23, 24, 25, 28, 29, 30, 33, 38 Due next Wednesday, Sep.9 The latest homework assigned is always available on the course webpage 58

59 FYI: Subtraction in Binary Remember borrowing? Borrow values Check in base ten! 59 15

60 Subtraction QUIZ Borrow values go here Check in base ten! 60 15

61 Subtraction QUIZ Borrow values go here Check in base ten! 61 15

62 Another subtraction QUIZ Borrow values go here Check in base ten! 62 15