Number Representation

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1 Number Representation

2 Number System :: The Basics We are accustomed to using the so-called decimal number system Ten digits ::,,,3,4,5,6,7,8,9 Every digit position has a weight which is a power of Base or radix is Example: 34 = x + 3 x + 4 x 5.67 = x + 5 x + x + 6 x x -

3 Binary Number System Two digits: and Every digit position has a weight which is a power of Base or radix is Example: = x + x + x. = x + x + x + x - + x - 3

4 Positional Number Systems (General) Decimal Numbers: Symbols {,,,3,4,5,6,7,8,9}, Base or Radix is 36.5 =

5 Positional Number Systems (General) Decimal Numbers: Symbols {,,,3,4,5,6,7,8,9}, Base or Radix is 36.5 = Binary Numbers: Symbols {,}, Base or Radix is. =

6 Positional Number Systems (General) Decimal Numbers: Symbols {,,,3,4,5,6,7,8,9}, Base or Radix is 36.5 = Binary Numbers: Symbols {,}, Base or Radix is. = Octal Numbers: 8 Symbols {,,,3,4,5,6,7}, Base or Radix is =

7 Positional Number Systems (General) Decimal Numbers: Symbols {,,,3,4,5,6,7,8,9}, Base or Radix is 36.5 = Binary Numbers: Symbols {,}, Base or Radix is. = Octal Numbers: 8 Symbols {,,,3,4,5,6,7}, Base or Radix is = Hexadecimal Numbers: 6 Symbols {,,,3,4,5,6,7,8,9,A,B,C,D,E,F}, Base is 6 6AF.3C =

8 Binary-to-Decimal Conversion Each digit position of a binary number has a weight Some power of A binary number: B = b n- b n-..b b. b - b -.. b -m Corresponding value in decimal: n- D = Σ i = -m b i i 8

9 Examples x 5 + x 4 + x 3 + x + x + x = 43 () = (43). x - + x - + x -3 + x -4 =.35 (.) = (.35). x + x + x + x - + x - = 5.75 (.) = (5.75) 9

10 Decimal to Binary: Integer Part Consider the integer and fractional parts separately. For the integer part: Repeatedly divide the given number by, and go on accumulating the remainders, until the number becomes zero. Arrange the remainders in reverse order. Base NumbRem (89) = ()

11 Decimal to Binary: Integer Part Consider the integer and fractional parts separately. For the integer part: Repeatedly divide the given number by, and go on accumulating the remainders, until the number becomes zero. Arrange the remainders in reverse order. Base NumbRem (89) = () (66) = ()

12 Decimal to Binary: Integer Part Consider the integer and fractional parts separately. For the integer part: Repeatedly divide the given number by, and go on accumulating the remainders, until the number becomes zero. Arrange the remainders in reverse order. Base NumbRem (89) = () (66) = () (39) = ()

13 Decimal to Binary: Fraction Part Repeatedly multiply the given fraction by. Accumulate the integer part ( or ). If the integer part is, chop it off. Arrange the integer parts in the order they are obtained. Example: x = x = x =.7.7 x = x =.88 : : (.634) = (. ) 3

14 Decimal to Binary: Fraction Part Repeatedly multiply the given fraction by. Accumulate the integer part ( or ). If the integer part is, chop it off. Arrange the integer parts in the order they are obtained. Example: x = x = x =.7.7 x = x =.88 : : Example: x =.5.5 x =.5.5 x =.5.5 x =. (.65) = (.) (.634) = (. ) 4

15 Decimal to Binary: Fraction Part Repeatedly multiply the given fraction by. Accumulate the integer part ( or ). If the integer part is, chop it off. Arrange the integer parts in the order they are obtained. Example: x = x = x =.7.7 x = x =.88 : : (.634) = (. ) Example: x =.5.5 x =.5.5 x =.5.5 x =. (.65) = (.) (37) = () (.65) = (.) (37.65) = (.) 5

16 Hexadecimal Number System A compact way of representing binary numbers 6 different symbols (radix = 6) 8 9 A 3 B 4 C 5 D 6 E 7 F 6

17 Binary-to-Hexadecimal Conversion For the integer part, Scan the binary number from right to left Translate each group of four bits into the corresponding hexadecimal digit Add leading zeros if necessary For the fractional part, Scan the binary number from left to right Translate each group of four bits into the corresponding hexadecimal digit Add trailing zeros if necessary 7

18 Example. ( ) = (B43) 6. ( ) = (A) 6 3. (. ) = (.84) 6 4. (. ) = (5.5E) 6 8

19 Hexadecimal-to-Binary Conversion Translate every hexadecimal digit into its 4-bit binary equivalent Examples: (3A5) 6 = ( ) (.3D) 6 = (. ) (.8) 6 = (. ) 9

20 Unsigned Binary Numbers An n-bit binary number B = b n- b n-. b b b n distinct combinations are possible, to n -. For example, for n = 3, there are 8 distinct combinations,,,,,,, Range of numbers that can be represented n=8 to 8 - (55) n=6 to 6 - (65535) n=3 to 3 - ( )

21 Signed Integer Representation Many of the numerical data items that are used in a program are signed (positive or negative) Question:: How to represent sign? Three possible approaches: Sign-magnitude representation One s complement representation Two s complement representation

22 Sign-magnitude Representation For an n-bit number representation The most significant bit (MSB) indicates sign positive negative The remaining n- bits represent magnitude b n- b n- b b Sign Magnitude

23 Contd. Range of numbers that can be represented: Maximum :: + ( n- ) Minimum :: ( n- ) A problem: Two different representations of zero

24 One s Complement Representation Basic idea: Positive numbers are represented exactly as in sign-magnitude form Negative numbers are represented in s complement form How to compute the s complement of a number? Complement every bit of the number ( and ) MSB will indicate the sign of the number positive negative 4

25 Example :: n= To find the representation of, say, -4, first note that +4 = -4 = s complement of = 5

26 Contd. Range of numbers that can be represented: Maximum :: + ( n- ) Minimum :: ( n- ) A problem: Two different representations of zero Advantage of s complement representation Subtraction can be done using addition Leads to substantial saving in circuitry 6

27 Two s Complement Representation Basic idea: Positive numbers are represented exactly as in sign-magnitude form Negative numbers are represented in s complement form How to compute the s complement of a number? Complement every bit of the number ( and ), and then add one to the resulting number MSB will indicate the sign of the number positive negative 7

28 Example : n= To find the representation of, say, -4, first note that +4 = -4 = s complement of = + = Rule : Value = msb* (n ) + [unsigned value of rest] Example: = + 6 = 6 = = 8

29 Contd. Range of numbers that can be represented: Maximum :: + ( n- ) Minimum :: n- Advantage: Unique representation of zero Subtraction can be done using addition Leads to substantial saving in circuitry Almost all computers today use the s complement representation for storing negative numbers 9

30 Contd. In C short int 6 bits + ( 5 -) to - 5 int or long int 3 bits + ( 3 -) to - 3 long long int 64 bits + ( 63 -) to

31 Adding Binary Numbers Basic Rules: += += += += (carry ) Example:

32 Subtraction Using Addition :: s Complement How to compute A B? Compute the s complement of B (say, B ). Compute R = A + B If the carry obtained after addition is Add the carry back to R (called end-around carry) That is, R = R + The result is a positive number Else The result is negative, and is in s complement form 3

33 Example :: 6 s complement of = 6 :: A - :: B R +4 Assume 4-bit representations Since there is a carry, it is added back to the result The result is positive End-around carry 33

34 Example :: 3 5 s complement of 5 = 3 :: -5 :: - A B R Assume 4-bit representations Since there is no carry, the result is negative is the s complement of, that is, it represents 34

35 Subtraction Using Addition :: s Complement How to compute A B? Compute the s complement of B (say, B ) Compute R = A + B If the carry obtained after addition is Ignore the carry The result is a positive number Else The result is negative, and is in s complement form 35

36 Example :: 6 s complement of = + = 6 :: - :: A B R Assume 4-bit representations Presence of carry indicates that the result is positive Ignore carry +4 No need to add the endaround carry like in s complement 36

37 Example :: 3 5 s complement of 5 = + = 3 :: -5 :: - A B R Assume 4-bit representations Since there is no carry, the result is negative is the s complement of, that is, it represents 37

38 s complement arithmetic: More Examples Example : 8- =? 8 is represented as is represented as s complement of is s complement of is Add 8 to s complement of (with a carry of which is ignored) is 7 38

39 Example : 7-9 =? 7 is represented as 9 is represented as s complement of 9 is s complement of 9 is Add 7 to s complement of (with a carry of which is ignored) is - 39

40 Overflow/Underflow: Adding two +ve (-ve) numbers should not produce a ve (+ve) number. If it does, overflow (underflow) occurs 4

41 Overflow/Underflow: Adding two +ve (-ve) numbers should not produce a ve (+ve) number. If it does, overflow (underflow) occurs Another equivalent condition : carry in and carry out from Most Significant Bit (MSB) differ. 4

42 Overflow/Underflow: Adding two +ve (-ve) numbers should not produce a ve (+ve) number. If it does, overflow (underflow) occurs Another equivalent condition : carry in and carry out from Most Significant Bit (MSB) differ. (64) ( 4) (68) carry (out)(in) 4

43 Overflow/Underflow: Adding two +ve (-ve) numbers should not produce a ve (+ve) number. If it does, overflow (underflow) occurs Another equivalent condition : carry in and carry out from Most Significant Bit (MSB) differ. (64) ( 4) (68) (64) (96) (-96) carry (out)(in) carry out in differ: overflow 43

44 Floating-point Numbers The representations discussed so far applies only to integers Cannot represent numbers with fractional parts We can assume a decimal point before a signed number In that case, pure fractions (without integer parts) can be represented We can also assume the decimal point somewhere in between This lacks flexibility Very large and very small numbers cannot be represented 44

45 Representation of Floating-Point Numbers A floating-point number F is represented by a doublet <M,E> : F = M x B E B exponent base (usually ) M mantissa E exponent M is usually represented in s complement form, with an implied binary point before it For example, In decimal,.35 x 6 In binary,. x 45

46 Example :: 3-bit representation M E 4 8 M represents a s complement fraction > M > - E represents the exponent (in s complement form) 7 > E > -8 Points to note: The number of significant digits depends on the number of bits in M 6 significant digits for 4-bit mantissa The range of the number depends on the number of bits in E 38 to -38 for 8-bit exponent. 46

47 A Warning The representation for floating-point numbers as shown is just for illustration The actual representation is a little more complex Example: IEEE 754 Floating Point format 47

48 IEEE 754 Floating-Point Format (Single Precision) S (3) E (Exponent) (3 3) M (Mantissa) ( ) S: Sign ( is +ve, is ve) E: Exponent (More bits gives a higher range) M: Mantissa (More bits means higher precision) [8 bytes are used for double precision] Value of a Normal Number: (-) S (. +.M) (E 7) 48

49 An example S (3) E (Exponent) (3 3) M (Mantissa) ( ) Value of a Normal Number: = (-) S (. +.M) (E 7) = (-) (. +.) ( ) =. = = 54. ( in decimal) 49

50 Representing.3 S (3) E (Exponent) (3 3) M (Mantissa) ( ).3 (decimal) =. =. =. = (-) S (. +.M) (E 7) 5 7 What are the largest and smallest numbers that can be represented in this scheme? 5

51 Representing S (3) E (Exponent) (3 3) M (Mantissa) ( ) Representing Inf ( ) Representing NaN (Not a Number) Non zero Non zero 5

52 Representation of Characters Many applications have to deal with non-numerical data. Characters and strings There must be a standard mechanism to represent alphanumeric and other characters in memory Three standards in use: Extended Binary Coded Decimal Interchange Code (EBCDIC) Used in older IBM machines American Standard Code for Information Interchange (ASCII) Most widely used today UNICODE Used to represent all international characters. Used by Java 5

53 ASCII Code Each individual character is numerically encoded into a unique 7-bit binary code A total of 7 or 8 different characters A character is normally encoded in a byte (8 bits), with the MSB not been used. The binary encoding of the characters follow a regular ordering Digits are ordered consecutively in their proper numerical sequence ( to 9) Letters (uppercase and lowercase) are arranged consecutively in their proper alphabetic order 53

54 Some Common ASCII Codes A :: 4 (H) 65 (D) B :: 4 (H) 66 (D).. Z :: 5A (H) 9 (D) :: 3 (H) 48 (D) :: 3 (H) 49 (D).. 9 :: 39 (H) 57 (D) a :: 6 (H) 97 (D) b :: 6 (H) 98 (D).. z :: 7A (H) (D) ( :: 8 (H) 4 (D) + :: B (H) 43 (D)? :: 3F (H) 63 (D) \n :: A (H) (D) \ :: (H) (D) 54

55 Character Strings Two ways of representing a sequence of characters in memory 5 H e l l o The first location contains the number of characters in the string, followed by the actual characters H e l l o The characters follow one another, and is terminated by a special delimiter 55

56 String Representation in C In C, the second approach is used The \ character is used as the string delimiter Example: Hello H e l l o \ A null string occupies one byte in memory. Only the \ character 56

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