1. Chapter 2 - Data Types

2. BYU CS/ECEn 124Chapter 2 - Data Types2 Today… Unit 1 slides and homework in JFSB B115 Problems with CCE on Talmage Window’s machines… Drivers have to be installed with administrator privileges Ready shortly… Help Sessions begin Monday @4:00 pm Reading Assignments on-line under the Schedule Tab Concerns or problems??

3. BYU CS/ECEn 124Chapter 2 - Data Types3 Concepts to Learn… Binary Digital System Data Types Conversions Binary Arithmetic Overflow Logical Operations Floating Point Hexadecimal Numbers ASCII Characters

4. BYU CS/ECEn 124Chapter 2 - Data Types4 What are Decimal Numbers? “Decimal” means that we have ten digits to use in our representation the symbols 0 through 9 What is 3,546? 3 thousands + 5 hundreds + 4 tens + 6 ones. 3,546 10 = 3  10 3 + 5  10 2 + 4  10 1 + 6  10 0 How about negative numbers? Use two more symbols to distinguish positive and negative, namely, + and -. Digital Binary System

5. BYU CS/ECEn 124Chapter 2 - Data Types5 What are Binary Numbers? “Binary” means that we have two digits to use in our representation the symbols 0 and 1 What is 1011? 1 eights + 0 fours + 1 twos + 1 ones 1011 2 = 1  2 3 + 0  2 2 + 1  2 1 + 1  2 0 How about negative numbers? We don’t want to add additional symbols So… Digital Binary System

6. BYU CS/ECEn 124Chapter 2 - Data Types6 Binary Digital System Binary (base 2) because there are two states, 0 and 1. Digital because there are a finite number of symbols. Basic unit of information is the binary digit, or bit. Bit values are represented by various physical means. Voltages Residual magnetism Light Electromagnetic Radiation Polarization Values with more than two states require multiple bits. A collection of 2 bits has 4 possible states: 00, 01, 10, 11 A collection of 3 bits has 8 possible states: 000, 001, 010, 011, 100, 101, 110, 111 A collection of n bits has 2 n possible states. Digital Binary System

7. BYU CS/ECEn 124Chapter 2 - Data Types7 Electronic Representation of a Bit Relies only on approximate physical values. A logical ‘1’ is a relatively high voltage (2.4V - 5V). A logical ‘0’ is a relatively low voltage (0V - 1V). Analog processing relies on exact values which are affected by temperature, age, etc. Analog values are never quite the same. Each time you play a vinyl album, it will sound a bit different. CDs sound the same no matter how many times you play them. Digital Binary System

8. BYU CS/ECEn 124Chapter 2 - Data Types8 The Power of the Bit… Bits rely on approximate physical values that are not affected by age, temperature, etc. Music that never degrades. Pictures that never get dusty or scratched. By using groups of bits, we can achieve high precision. 8 bits => each bit pattern represents 1/256. 16 bits => each bit pattern represents 1/65,536 32 bits => each bit pattern represents 1/4,294,967,296 64 bits => each bit pattern represents 1/18,446,744,073,709,550,000 Disadvantage: bits only represent discrete values Digital = Discrete Digital Binary System

9. BYU CS/ECEn 124Chapter 2 - Data Types9 Binary Nomenclature Binary Digit: 0 or 1 Bit (short for binary digit): A single binary digit LSB (least significant bit): The rightmost bit MSB (most significant bit): The leftmost bit Data sizes 1 Nibble (or nybble) = 4 bits 1 Byte = 2 nibbles = 8 bits 1 Kilobyte (KB) = 1024 bytes 1 Megabyte (MB) = 1024 kilobytes = 1,048,576 bytes 1 Gigabyte (GB) = 1024 megabytes = 1,073,741,824 bytes Digital Binary System

10. BYU CS/ECEn 124Chapter 2 - Data Types10 What Kinds of Data? All kinds… Numbers – signed, unsigned, integers, floating point, complex, rational, irrational, … Text – characters, strings, … Images – pixels, colors, shapes, … Sound – pitch, amplitude, … Logical – true / false, open / closed, on / off, … Instructions – programs, … … Data type: representation and operations within the computer We’ll start with numbers… Data Types

11. BYU CS/ECEn 124Chapter 2 - Data Types11 Some Important Data Types Unsigned integers only non-negative numbers 0, 1, 2, 3, 4, … Signed integers negative, zero, positive numbers …, -3, -2, -1, 0, 1, 2, 3, … Floating point numbers numbers with decimal point PI = 3.14159 x 10 0 Characters 8-bit, unsigned integers ‘0’, ‘1’, ‘2’, …, ‘a’, ‘b’, ‘c’, …, ‘A’, ‘B’, ‘C’, …, ‘@’, ‘#’, Data Types

12. BYU CS/ECEn 124Chapter 2 - Data Types12 Unsigned Integers 329 10 2 10 1 10 0 1012 2121 2020 3x100 + 2x10 + 9x1 = 3291x4 + 0x2 + 1x1 = 5 most significant least significant What do these unsigned binary numbers represent? 0000 0110 1111 1010 0001 1000 0111 1100 1011 1001 Weighted positional notation “3” is worth 300, because of its position, while “9” is only worth 9 Data Types

13. BYU CS/ECEn 124Chapter 2 - Data Types13 Unsigned Integers (continued…) 2 2121 2020 0000 0011 0102 0113 1004 1015 1106 1117 Data Types

14. BYU CS/ECEn 124Chapter 2 - Data Types14 Unsigned Binary Arithmetic Base 2 addition – just like base 10! add from right to left, propagating carry 10010100101111 +1001 +1011+1 110111110110000 10111 +111 carry Subtraction, multiplication, division,… 01111 Data Types

15. BYU CS/ECEn 124Chapter 2 - Data Types15 Signed Integers With n bits, we have 2 n distinct values. assign about half to positive integers (1 through 2 n-1 ) and about half to negative (- 2 n-1 through -1) that leaves two values: one for 0, and one extra Positive integers just like unsigned – zero in most significant (MS) bit 00101 = 5 Negative integers sign-magnitude – set MS bit to show negative 10101 = -5 one’s complement – flip every bit to represent negative 11010 = -5 MS bit indicates sign: 0=positive, 1=negative Data Types

16. BYU CS/ECEn 124Chapter 2 - Data Types16 Sign-Magnitude Integers Representations 01111 binary => 15 decimal 11111=> -15 00000=> 0 10000=> -0 Problems Difficult addition/subtraction check signs convert to positive use adder or subtractor as required The left-bit encodes the sign: 0 = + 1 =  Data Types

17. BYU CS/ECEn 124Chapter 2 - Data Types17 1’s Complement Integers Representations 00110 binary => 6 decimal 11001=> -6 00000=> 0 11111=> -0 Problem Difficult addition/subtraction no need to check signs as before cumbersome logic circuits end-around-carry To negate a number, Invert it, bit-by-bit. The left-bit still encodes the sign: 0 = + 1 =  Data Types

18. BYU CS/ECEn 124Chapter 2 - Data Types18 2’s Complement Problems with sign-magnitude and 1’s complement two representations of zero (+0 and –0) arithmetic circuits are complex How to add two sign-magnitude numbers? e.g., try 2 + (-3) How to add to one’s complement numbers? e.g., try 4 + (-3) Two’s complement representation developed to make circuits easy for arithmetic. Data Types

19. BYU CS/ECEn 124Chapter 2 - Data Types19 2’s Complement (continued…) Simplifies logic circuit construction because addition and subtraction are always done using the same circuitry. there is no need to check signs and convert. operations are done same way as in decimal right to left with carries and borrows Bottom line: simpler hardware units! Data Types

20. BYU CS/ECEn 124Chapter 2 - Data Types20 2’s Complement (continued…) If number is positive or zero, normal binary representation If number is negative, start with positive number flip every bit (i.e., take the one’s complement) then add one 00101 (5) 01001 (9) 11010 (1’s comp)(1’s comp) +1+1 11011 (-5)(-9) 10110 10111 Data Types

21. BYU CS/ECEn 124Chapter 2 - Data Types21 2’s Complement (continued…) Positional number representation with a twist the most significant (left-most) digit has a negative weight n-bits represent numbers in the range  2 n  1 … 2 n  1  1 What are these? 0110 = 2 2 + 2 1 = 6 1110 = -2 3 + 2 2 + 2 1 = -2 0000 0110 1111 1010 0001 1000 0111 1100 1011 1001 Data Types

22. BYU CS/ECEn 124Chapter 2 - Data Types22 2’s Complement Shortcut To take the two’s complement of a number: copy bits from right to left until (and including) the first “1” flip remaining bits to the left 011010000 100101111 (1’s comp) +1100110000 (copy)(flip) Data Types

23. BYU CS/ECEn 124Chapter 2 - Data Types23 2’s Complement Negation To negate a number, invert all the bits and add 1 6 1010 7 1001 0 0000 -1 0001 4 1100 -8 1000 (??) Data Types

24. BYU CS/ECEn 124Chapter 2 - Data Types24 Quiz 00100110 (unsigned int) +10001101 (signed magnitude) +11111101 (1’s complement) +00001101 (2’s complement) +10111101 (2’s complement) (decimal)

25. BYU CS/ECEn 124Chapter 2 - Data Types25 Quiz 00100110(unsigned int) +10001101(signed magnitude) (unsigned int) +11111101(1’s complement) (signed int) +00001101(2’s complement) (2’s complement) +10111101(2’s complement) Decimal 00011001 00010111 00100100 -31 11100001 (2’s complement) 11100000 (1’s complement) 10011111 (signed magnitude) 38 +-13 25 + -2 23 + 13 36 +-67 -31

26. BYU CS/ECEn 124Chapter 2 - Data Types26 Continually divide the number by 2 and track the remainders. 1  2 5 + 0  2 4 + 1  2 3 + 0  2 2 + 1  2 1 + 1  2 0 32 + 0 + 8 + 0 + 2 + 1 = 43 432 2 2 2 2 2 Decimal to Binary Conversion For negative numbers, do above for positive number and negate result 5 R 0 0 2 R 1 1 21 R 1 1 10 R 1 1 1 R 0 0 0 R 1 1 Conversions

27. BYU CS/ECEn 124Chapter 2 - Data Types27 Decimal to Binary Conversion 0101 0110 01111011 00100011 11011101 01111101111 Conversions

28. BYU CS/ECEn 124Chapter 2 - Data Types28 Sign-Extension in 2’s Complement You can make a number wider by simply replicating its leftmost bit as desired. 0110 = 000000000000000110 = 1111 = 11111111111111111 = 1 = 6 6 What do these represent? Conversions

29. BYU CS/ECEn 124Chapter 2 - Data Types29 Word Sizes In the preceding slides, every bit pattern was a different length (15 was represented as 01111). Every real computer has a base word size our machine (MPS430) is 16-bits Memory fetches are word-by-word even if you only want 8 bits (a byte) Instructions are packed into words Numeric representations are word-sized 15 is represented as 0000000000001111 Conversions

30. BYU CS/ECEn 124Chapter 2 - Data Types30 Rules of Binary Addition 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0, with carry Two's complement addition follows the same rules as binary addition Two's complement subtraction is the binary addition of the minuend to the 2's complement of the subtrahend adding a negative number is the same as subtracting a positive one Binary Arithmetic 5 + (-3) = 2 0000 0101 = +5 + 1111 1101 = -3 --------- -- 0000 0010 = +2

31. BYU CS/ECEn 124Chapter 2 - Data Types31 Adding 2’s Complement Integers c 00110 +00101 01011 b1 00110 -00101 00001 Issues Overflow: the result cannot be represented by the number of bits available 0110 +0101 1011 Hmmm. 6 + 5  -5. Obviously something went wrong. This is a case of overflow. You can tell there is a problem - a positive plus a positive cannot give a negative. Binary Arithmetic

32. BYU CS/ECEn 124Chapter 2 - Data Types32 Overflow Revisited Overflow = the result doesn’t fit in the capacity of the representation ALU’s are designed to detect overflow It’s really quite simple if the carry in to the most significant position (MSB) is different from the carry out from the most significant position (MSB), then overflow occurred. Generally, overflows represented in CPU status bit Overflow

33. BYU CS/ECEn 124Chapter 2 - Data Types33 Logical Operations on Bits A B AND 0 0 0 0 1 0 1 0 0 1 1 1 A B OR 0 0 0 0 1 1 1 0 1 1 1 1 A NOT 0 1 1 0 a = 001100101 b = 110010100 a AND b = ? a OR b = ? NOT a = ? A XOR b = ? a AND b = 000000100 a = 001100101 b = 110010100 a OR b = 111110101 NOT a = 110011010 A B XOR 0 0 0 0 1 1 1 0 1 1 1 0 A XOR b = 111110001 Logical Operations

34. BYU CS/ECEn 124Chapter 2 - Data Types34 Examples of Logical Operations AND useful for clearing bits AND with zero = 0 AND with one = no change OR useful for setting bits OR with zero = no change OR with one = 1 NOT unary operation -- one argument flips every bit 11000101 AND 00001111 00000101 11000101 OR 00001111 11001111 NOT 11000101 00111010 Logical Operations

35. BYU CS/ECEn 124Chapter 2 - Data Types35 Floating Point Numbers Binary scientific notation 32-bit floating point Exponent is biased Implied leading 1 in mantissa sexponentmantissa 1823 Floating Point

36. BYU CS/ECEn 124Chapter 2 - Data Types36 Floating Point Numbers Why the leading implied 1? Always normalize after an operation shift mantissa until leading digit is a 1 can assume it is always there, so don’t store it Why the biased exponent? To avoid signed exponent representations sexponentmantissa 1823 Floating Point

37. BYU CS/ECEn 124Chapter 2 - Data Types37 Floating Point Numbers What does this represent? Positive number Exponent is 128 which means the real exponent is 1 Mantissa is to be interpreted as 1.1 This is 2 0 + 2 -1 = 1 + 1/2 = 1.5 The final number is 1.5 x 2 1 = 3 0 10000000 10000000000000000000000 Floating Point

38. BYU CS/ECEn 124Chapter 2 - Data Types38 Floating Point Numbers What does this represent? Negative number Exponent is 129 which means the real exponent is 2 Mantissa is to be interpreted as 1.10101 This is 2 0 + 2 -1 + 2 -3 + 2 -5 = 1 + 1/2 + 1/8 + 1/32 = 1.65625 The final number is -1.65625 x 2 2 = -6.625 1 10000001 10101000000000000000000 Floating Point

39. BYU CS/ECEn 124Chapter 2 - Data Types39 Hexadecimal Notation Binary is hard to read and write by hand Hexadecimal is a common alternative 16 digits are 0123456789ABCDEF 0100 0111 1000 1111 = 0x478F 1101 1110 1010 1101 = 0xDEAD 1011 1110 1110 1111 = 0xBEEF 1010 0101 1010 0101 = 0xA5A5 Binary Hex 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F 0x is a common prefix for writing numbers which means hexadecimal 1.Separate binary code into groups of 4 bits (starting from the right) 2.Translate each group into a single hex digit Hexadecimal

40. BYU CS/ECEn 124Chapter 2 - Data Types40 Binary to Hex Conversion Every four bits is a hex digit. start grouping from right-hand side 011101010001111010011010111 7D4F8A3 This is not a new machine representation, just a convenient way to write the number. Hexadecimal

41. BYU CS/ECEn 124Chapter 2 - Data Types41 Decimal to Hex Conversion Positive numbers Œ start with empty result  next digit to prepend is number modulo 16 Ž divide number by 16, throw away fractional part  if new number is non-zero, go back to  else you are done Negative numbers do above for positive version of number and negate result. Hexadecimal

42. BYU CS/ECEn 124Chapter 2 - Data Types42 Decimal to Hex Examples 12 decimal = 1100 = 0xc 21 decimal = 0001 0101 = 0x15 55 decimal = 0011 0111 = 0x37 256 decimal = 0001 0000 0000 = 0x100 47 decimal = 0010 1111 = 0x2f 3 decimal = 0011 = 0x3 127 decimal = 0111 1111 = 0x7f 1029 decimal = 0100 0000 0101 = 0x405 Hexadecimal

43. BYU CS/ECEn 124Chapter 2 - Data Types43 ASCII Codes How do you represent characters? ‘A’ ASCII is a set of standard 8-bit, unsigned integers (codes) ' ' = 32, '0' = 48, '1' = 49, 'A' = 65, 'B' = 66 Zero-extended to word size To convert an integer digit to ASCII character, add 48 (=‘0’) 1 + 48 = 49 => ‘1’ ASCII Characters

44. BYU CS/ECEn 124Chapter 2 - Data Types44 ASCII Characters NULDLESP0@P`p SOHDC1!1AQaq STXDC2“2BRbr ETXDC3#3CScs EOTDC4$4DTdt ENQNAK%5EUeu ACKSYN&6FVfv BELETB‘7GWgw BSCAN(8HXhx HTEM)9IYiy LFSUB*:JZjz VTESC+;K[k{ FFFS,<L\l| CRGS-=M]m} SORS.>N^n~ SIUS/?O_oDEL 0123456789abcdef0123456789abcdef 0 1 2 3 4 5 6 7 8-9 a-f More controls More symbols ASCII Characters

45. BYU CS/ECEn 124Chapter 2 - Data Types45 Properties of ASCII Code What is relationship between a decimal digit ('0', '1', …) and its ASCII code? What is the difference between an upper-case letter ('A', 'B', …) and its lower-case equivalent ('a', 'b', …)? Given two ASCII characters, how do we tell which comes first in alphabetical order? What is significant about the first 32 ASCII codes? Are 128 characters enough? (http://www.unicode.org/) ASCII Characters

46. BYU CS/ECEn 124Chapter 2 - Data Types46 Displaying Characters 48 Decimal 58 Decimal 116 Decimal 53 Decimal ASCII Characters

47. BYU CS/ECEn 124Chapter 2 - Data Types47 MSP430 Data Types Words and bytes are supported directly by the Instruction Set Architecture. add.b add.w 8-bit and 16-bit 2’s complement signed integers Other data types are supported by interpreting variable length values as logical, text, fixed-point, etc., in the software that we write. Data Types

48. BYU CS/ECEn 124Chapter 2 - Data Types48 Review: Representation Everything is stored in memory as one’s and zero’s integers, floating point numbers, characters program code Data Type = Representation + Operations You can’t tell what is what just by looking at the binary representation memory could have multiple meanings it is possible to execute your Word document Review

49. BYU CS/ECEn 124Chapter 2 - Data Types49 Review: Numbers… 7 6 5 4 3 2 1 0 -2 -3 -4 111 110 101 100 011 010 001 000 011 010 001 000, 100 101 110 111 011 010 001 000, 111 110 101 100 011 010 001 000 111 110 101 100 Un-signed Signed Magnitude 1’s Complement 2’s Complement Range:0 to 7-3 to 3 -4 to 3 Review

50. BYU CS/ECEn 124Chapter 2 - Data Types50

51. BYU CS/ECEn 124Chapter 2 - Data Types51 ASCII Characters