1. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20011 Representing Information in Computers:  numbers: counting numbers, integers: signed magnitude and 2’s complement (slides 2-29)  characters: ASCII, etc. (slides 30-33)  summary (slides 34) Data Representation

2. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20012  base: set of n atomic symbols + interpretation of each symbol as a “value” (quantity)  number: a string of symbols (digits) + interpretation rule  digits are numbered right to left, start at 0 e.g. 4-digit number: d 3 d 2 d 1 d 0  “value” of a digit depends on value of symbol and position of digit in string value of digit i = (value of symbol i )  n i  “value” of number is sum of value of digits Base-n Number Systems

3. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20013 (nothing new here!)  symbols = { 0, 1, 2, 3, , 9} – usual meaning  4-digit example: d 3 d 2 d 1 d 0 = 1436 = d 3  10 3 + d 2  10 2 + d 1  10 1 + d 0  10 0 = 1  10 3 + 4  10 2 + 3  10 1 + 6  10 0 = 1  1000 + 4  100 + 3  10 + 6  1 = 1000 + 400 + 30 + 6 = 1436 10 Base-10 Example

4. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20014  set of symbols = { 0, 1 } – usual meaning  4-digit number example: 1011 2 = 1  2 3 + 0  2 2 + 1  2 1 + 1  2 0 = 1  8 + 0  4 + 1  2 + 1  1 = 8 + 0 + 2 + 1 = 11 10 Base-2 (binary) Number System

5. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20015 Binary Decimal Binary Decimal 0000 0 1000 8 00011 1001 9 00102 101010 00113 101111 01004 110012 01015 110113 01106 111014 01117 111115 Binary – Decimal Conversion Table Memorize these!

6. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20016  What about the range of representation? n-bit values can represent (at most) 2 n different counting numbers (why?) if start representation at 0, then largest value represented is 2 n – 1 (why?) recall 4-bit example: ( 2 4 = 16 ) –range = 0.. 15 Range of Representation

7. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20017 Some Observations: binary representation of counting numbers often requires lots of bits –example: 601 10 = 1001011001 2 (10 bits!) converting binary  decimal is not trivial binary can be awkward and error prone for use by people! More about Binary

8. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20018  16 symbols = { 0.. 9, A, B, C, D, E, F} interpret: 0.. 9 – usual meaning A 10 B 11 C 12 D 13 E 14 F 15  Why use hexadecimal (hex) notation?  shorthand for binary notation  converting binary  hexadecimal is easy Hexadecimal (Base-16) Number System

9. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 20019 Converting Binary  Hex  4 binary digits make up the value of one hexadecimal digit: hex digits range 0.. 15 4-bit binary values range 0.. 15  2 Steps group bits in 4’s starting from least significant replace each group by corresponding hex digit Converting Hex  Binary  replace each hex digit by 4-digit binary rep Converting Binary  Hex

10. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200110 16-bits: 1000101001101110 2 (= 35,438 10 ) 1. group in 4’s  1000 1010 0110 1110 2. replace with hex  8 A 6 E 16 10-bits: 1001011001 2 (= 601 10 ) 1. group in 4’s  0010 0101 1001 2. replace with hex  2 5 9 16 Binary => Hex Examples

11. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200111 Binary Decimal Hex BinaryDecimal Hex 0000 0 0 1000 8 8 0001 1 1 1001 9 9 0010 2 2 1010 10 A 00113 3 1011 11 B 01004 4 1100 12 C 01015 5 1101 13 D 01106 6 1110 14 E 01117 7 1111 15 F Binary / Decimal / Hexadecimal Memorize these!

12. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200112  Encoding of More Types of Information  counting numbers, integers  what are they?  are these different from base-n systems?  Encoding of Counting Numbers (0, 1, 2, 3, …)  simple mapping: binary number system!  base-2 number system  Encoding of Integers  positive and negative values  must encode sign & magnitude  not so easy! Other Encodings

13. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200113  use most significant bit (leftmost) to encode sign  0 = positive, 1 = negative  use remaining n – 1 bits to encode magnitude using binary number system (as for counting numbers)  example: 8-bit signed-magnitude 10000001 2 magnitude = 1 sign bit = 1  negative number Thus this represents value – 1 Signed Magnitude Representation

14. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200114 Problems with Signed-Magnitude Encoding:  two representations of 0!? positive 0 (sign bit = 0) negative 0 (sign bit = 1) is zero positive or negative?  performing arithmetic operations with signed- magnitude values can be awkward  signed-magnitude not often used in practice Signed Magnitude (contd)

15. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200115  complement of one bit: b = 1 – b complement of 0 = 1; complement of 1 = 0  complement of an n-bit value: complement each bit  2’s complement of an n-bit value: complement each bit, and then add 1 ignore any carry out of most significant bit (why?) Binary Addition and Subtraction: 0 + 0 = 0; 1 + 0 = 1; 0 + 1 = 1; 1 + 1 = 0 (carry 1) 0 – 0 = 0; 1 – 0 = 1; 0 – 1 = 1 (borrow 1); 1 – 1 = 0 2’s Complement Encoding

16. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200116 1. original value = 100111010010 complement each bit: 011000101101 add 1 +1 2’s complement rep = 011000101110 2. original value = 0001 complement each bit: 1110 add 1 +1 2’s complement rep = 1111 2’s Complement Examples

17. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200117  Assume we have n bits available.  use half of binary values for negative values (i.e. 2 n–1 values), the rest for non-negative  encode positive values (and 0) the same way as counting numbers  range 0.. 2 n – 1 – 1  uses all values that start with 0  use 2’s complement as negation operation!  encoding of – “x” = 2’s complement of “x” Encoding Integers in 2’s Complement

18. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200118 1. 8-bit 2’s complement encoding of –1 +1 10 0000 0001 complement: 1111 1110 add 1 + 1 so – 1 10 =1111 1111  the largest binary representation is for the smallest negative number! Integer Encoding Examples

19. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200119 2. 8-bit 2’s complement encoding of –128 +128 10 1000 0000 complement: 0111 1111 add 1 + 1 –128 10 1000 0000  the smallest binary representation starting with 1 (indicating negative) is for the largest negative number that can be represented!  do we have two values with the same representation? –128 10 = +128 10 ? No, as +128 10 cannot be represented in 8-bit 2’s complement (see next page!) Integer Encoding Examples (contd)

20. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200120  range of numbers that can be represented is: – 2 n – 1, …, – 1, 0, …, 2 n – 1 – 1  e.g. for 8-bit 2’s complement encodings:  n = 8: 2 8 = 256, 2 7 = 128  range: – 2 n – 1, …, 2 n – 1 – 1 = – 128, …, 127  remember: all negative value encodings start with most significant bit = 1 We assume 8 bit representation in the following slides! 2’s Complement Range of Representation

21. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200121 Encoding of negative numbers seems strange!? Why do it like this?! We want  A. Negating a negative number = same positive number we started with  B. A single representation of zero  C. Math (addition, subtraction, etc.) to work without keeping track of representation being used (i.e. regular counting numbers vs. 2’s complement) More on 2’s Complement

22. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200122 A. Negating a negated number should give the original number:  with 2’s complement does – (– x) = x? Example 1: (8 bits) – 1 10 1111 1111 complement 0000 0000 add 1 + 1 +1 10 0000 0001 – ( – 1 ) = 1it works! 2’s Complement (contd)

23. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200123 A. (continued) Negating a negated number should give the original number: Special Case! Example 2: (8 bits) – 128 10 1000 0000 complement 0111 1111 add 1 + 1 – 128 10 1000 0000(we saw this already!) Should be +128 10 (which cannot be represented in 8 bits 2’s complement)  What happened? Overflow (more later!) 2’s Complement (contd)

24. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200124 B. How many representations are there for 0 in 2’s complement? 0 10 0000 0000 complement 1111 1111 add 1 + 1 (ignore carry out of “1”) 0 10 0000 0000 2’s Complement (contd)

25. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200125 C. (Example) Add: 18 10 + 75 10 (should be 93 10 ) Convert to Hex and Binary: 18 10 = 1*16 + 2 = 12h = 0001 0010 2 75 10 = 4*16 + 11 = 4Bh = 0100 1011 2 Add in Hex and Convert back to Decimal: 12h + 4Bh = 5Dh = 5*16 + 13 = 93 10 Add in Binary and Convert back to Decimal: 0001 0010 2 + 0100 1011 2 = 0101 1101 2 = 1*64 + 1*16 + 1*8 + 1*4 + 1*1 = 93 10 BinaryAddition worked! Binary Math Example

26. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200126 C. (continued) What if the binary numbers on the previous page were in 2’s complement representation? 0001 0010 2 0100 1011 2 0101 1101 2  No difference as all these numbers are positive (most significant bit is “0”) 2’s Complement Math (contd)

27. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200127 Add: 10 10 + 237 10 (should be 247 10 ) Convert to Hex and Binary: 10 10 = 0*16 + 10 = 0Ah = 0000 1010 2 237 10 = 14*16 + 13 = EDh = 1110 1101 2 Add in Hex and Convert back to Decimal: 0Ah + EDh = F7h = 15*16 + 7 = 247 10 Add in Binary and Convert back to Decimal: 0000 1010 2 + 1110 1101 2 = 1111 0111 2 = 1*128 + 1*64 + 1*32 + 1*16 + 1*4 + 1*2 + 1*1 = 247 10 As expected, BUT... More Binary Arithmetic

28. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200128 C. (continued) What if these binary numbers were in 2’s complement representation? 0000 1010 2 positive number so no change = 10 10 1110 1101 2 negative number so complement and add 1 = – (0001 0010 2 + 1) = – 0001 0011 2 = – (1*16 + 1*2 + 1*1) 10 = – 19 10 Thus we would want the answer to be: 10 10 + – 19 10 = – 9 10 converting – 9 10 to 2’s complement gives: 0000 1001 2 + 1 = 1111 0110 2 + 1 = 1111 0111 2 Which is exactly what we had on the previous page! More 2’s Complement Arithmetic

29. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200129 C. (continued) This “proves”:  addition works if the (Hex or) binary numbers represent counting numbers or 2’s complement number!  left as an exercise: subtraction works, too! Math works regardless of representation - this is why 2’s complement is used! Why 2’s Complement

30. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200130 Encoding of Characters: (slides 30-33)  want to represent “displayable” characters:  characters: { a, A, b, B,..., z, Z }  decimal (10) digits: {0, 1,..., 8, 9 }  punctuation: ! ,. – ? / : ;  math symbols: + * = ( – and / above)  brackets: ( ) [ ] { }  other symbols: @ # $ % ^ & \ | ~ “ ” etc.  about 90 characters need to be represented  various encoding schemes have been used  ASCII encoding – international standard  American Standard Code for Information Interchange Encoding of Characters

31. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200131 7-Bit ASCII Encoding 7 bits to encode each character (128 codes) often extended to 8-bit (byte) values by making most significant bit (msb) = 0 Note: all codes are given as Hex values 00 – 1Fnon-displayable control char’s  00NULL  07BELL  08backspace  09tab  others – often serve special purposes in communication applications Encoding of Characters  0Aline feed  0Cform feed  0Dcarriage return

32. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200132 7-Bit ASCII Encoding (continued)  20 = blank space “ ”; 21 = “!” ; 2E = “.”  30 – 39decimal digit characters  30 = “0”, …, 39 = “9”  41 – 5Aupper case letter characters  41 = “A”, …, 5A = “Z”  61 – 7Alower case letter characters  61 = “a”, …, 7A = “z” Encoding of Characters (contd)

33. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200133 ASCII Example:“94.201 is FUN!” hex encoding: 39 34 2E 32 30 31 20 69 73 20 46 55 4E 21 binary encoding: … 00110010 …. 01110011 … 00100001  hex is often used a shorthand for binary  ASCII encoding is summarized in ASCII Table (on web site) Other Character Encoding Schemes:  IBM standardized an 8-bit scheme (256 characters) as defacto standard (PCs)  Java: unicode – 16-bit scheme (65,536 characters) multi-lingual character sets Encoding of Characters (contd)

34. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200134  fixed-width binary values are used to represent information in computers  the computer works with the binary representations (NOT the information!!)  the same binary value can be used to represent different information: Programmer must be aware! 8-bit example: 1111 0000 2 represents  unsigned integer 240 10  2s complement signed integer –16 10  ASCII “  ”  and others... Data Representation Summary (Important Concepts)

35. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200135 Manipulating Information in Computers:  always operate on fixed-width binary representation  arithmetic: add, subtract (slides 36-40)  logic: and, or, xor (slides 41-42)  shift and rotate (slides 43-45) Data Manipulation (slides 35-45)

36. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200136  1-bit operations – previous slides  bitwise add/subtract values of width n to give result of width n  8-bit Unsigned Integer Examples: 117 10 = 0111 0101 2 133 10 = 1000 0101 2 + 99 10 = 0110 0011 2 – 51 10 = 0011 0011 2 216 10 = 1101 1000 2 82 10 = 0101 0010 2  do exactly the same thing for 2’s complement signed integers!  for 8-bit result: ignore any carry/borrow out/in @ msb (why?) Addition and Subtraction

37. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200137  8-bit Signed Integer (2’s Complement) Examples: –1 10 = 1111 1111 2 32 10 = 0010 0000 2 + 1 10 = 0000 0001 2 – 65 10 = 0100 0001 2 0 10 = 1 0000 0000 2 – 33 10 = 1 1101 1111 2 = 0000 0000 2 = 1101 1111 2  in addition, ignore carry at msb for 8-bit result  in subtraction, ignore borrow at msb for 8-bit result  computers often implement subtraction using “negate and add”, i.e. X – Y = X + (– Y) 32 10 = 0010 0000 2 + – 65 10 = 1011 1111 2 – 33 10 = 1101 1111 2 Addition and Subtraction

38. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200138  Problem: limited representation range due to fixed width  overflow occurs when the result of an operation is outside the range that can be represented  8-bit Unsigned Integer Example: 255 10 = 1111 1111 2 + 1 10 = 0000 0001 2 0 10 = 0000 0000 2 ?? 256 10 ??  carry @ msb is important in this case!  need 9 bits to represent 256 10  used 8-bits: Overflow Occurred! Overflow

39. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200139 But there is another interpretation, in 2’s complement: –1 10 = 1111 1111 2 + 1 10 = 0000 0001 2 0 10 = 1 0000 0000 2 = 0000 0000 2  The result is correct if the values are interpreted as signed integers! ( – 1 + 1 = 0 )  Note: Overflow depends on the interpretation of the values! Overflow (contd)

40. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200140 Another example: unsigned signed 0111 1111 2 127127 +0000 0001 2 + 1+ 1 1000 0000 2 128 – 128 OK Overflow  in this case there is overflow only in the signed representation Overflow (contd)

41. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200141 AND, OR, XOR (exclusive or): 1-bit truth tables: ab a AND b a OR b a XOR b 00000 01011 10011 11110 Logical Operations (slides 41-42)

42. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200142 8-bit examples: 1011 0110 1011 0110 AND1100 0011 OR1100 0011 1000 0010 1111 0111 1011 0110 XOR1100 0011 0111 0101 Logical Operations (contd)

43. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200143  values can be manipulated by shifting  left – towards msb  right – towards lsb (least significant bit)  shifting example:  value: 1001 1010  shifted left 4 bits:1010 0000  question: should injected values be 0 or 1?  shift left: always inject 0’s  shift right:  logical: inject 0’s  arithmetic:inject msb value Shifting and Rotating (Slides 43-45)

44. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200144  Shift left 2 bits value:1001 1010 shifted value:0110 1000  Logical shift right 2 bits value:1001 1010 shifted value:0010 0110  Arithmetic shift right 2 bits value:1001 1010 shifted value:1110 0110 More Shifting

45. 10-Sep-0194.201 - Fall 2001: copyright ©T. Pearce, D. Hutchinson, L. Marshall Sept. 200145  rotate: like shift, except bit that is shifted “out” of the value gets injected as the new bit  all bits in original value are saved  left – towards msb  right – towards lsb  rotating example: rotate right 3 bits value:1001 1110 rotated 1 st bit:0100 1111 rotated 2 nd bit:1010 0111 rotated 3 rd bit:1101 0011 Shifting and Rotating (contd)