How do computers represent data?

Transcription

1 Data Representation How do computers represent data? 1

2 Data and Computers Computers are multimedia devices,dealing with a vast array of information categories Computers store, present, and help us modify Numbers Text Audio Images and graphics Video 2

3 Analog and Digital Information Computers are finite! How do we represent an infinite it world? We represent enough of the world to satisfy our computational needs and our senses of sight and sound 3

4 Analog and Digital Information Information can be represented in one of two ways: analog or digital Analog data A continuous representation, analogous to the actual information it represents Digital data A discrete representation, breaking the information up into separate elements 4

5 Analog and Digital Information Thermometer is an analog device A mercury thermometer continually rises in direct proportion to the temperature 5

6 Analog and Digital Information Computers cannot work well with analog data, so we digitize the data Digitizeiti Breaking data into pieces and representing those pieces separately Why do we use binary to represent digitized data? 6

7 Electronic Signals Important facts about electronic signals An analog signal continually fluctuates in voltage up and down A digital signal has only a high or low state, corresponding to the two binary digits All electronic signals (both analog and digital) degrade as they move down a line The voltage of the signal fluctuates due to environmental effects 7

8 Electronic Signals (Cont d) Periodically, a digital signal is reclocked to regain its original shape An analog signal and a digital signal Degradation of analog and digital signals 8

9 Binary Numbers Each binary digit (called bit) is either 1 or 0 Bits have no inherent meaning, can represent Unsigned and signed integers Characters Floating-point numbers Most Significant Bit Least Significant Bit Images, sound, etc. Bit Numbering Least significant bit (LSB) is rightmost (bit 0) Most significant bit (MSB) is leftmost (bit 7 in an 8-bit number)

10 Binary Representations One bit can be either 0 or 1 One bit can represent two things (Why?) Two bits can represent four things (Why?) How many things can three bits represent? How many things can four bits represent? How many things can eight bits represent? 10

11 Binary Representations Counting with binary bits 11

12 Binary Representations How many things can bits represent? Why? What happens every time you increase the number of bits by one? 12

13 Binary Representations A bit (Binary digit) is the most basic unit of information in a computer. It is a state t of on or off in a digital it circuit. it Sometimes these states are high or low voltage instead of on or off.. A byte is a group of eight bits. A byte is the smallest possible addressable (can be found via its location) unit of computer storage. A word is a contiguous group of bytes. Words can be any number of bits (16, 32, 64 bits are common). A group of four bits is called a nibble (or nybble). Bytes, therefore, consist of two nibbles: a high-order nibble, and a low-order nibble.

14 Converting Binary to Decimal Each bit represents a power of 2 Every binary number is a sum of powers of 2 Decimal Value = (d n-1 2 n-1 ) (d ) + (d ) Binary ( ) 2 = = Some common powers of 2

15 Questions 1. Convert the following binary numbers to decimal

16 Questions 1. Convert the following binary numbers to decimal

17 Convert Unsigned Decimal to Binary Repeatedly divide the decimal integer by 2 Each remainder is a binary digit in the translated value least significant ifi bit 37 = (100101) 2 most significant bit stop when quotient is zero

18 Decimal -> Binary Questions 2. Convert the following decimal numbers to binary:

19 Decimal -> Binary Questions 2. Convert the following decimal numbers to binary:

20 Hexadecimal Integers 16 Hexadecimal Digits: 0 9, A F More convenient to use than binary numbers Binary, Decimal, and Hexadecimal Equivalents

21 Converting Binary to Hexadecimal Each hexadecimal digit corresponds to 4 binary bits Example: Convert the 32-bit binary number to hexadecimal Solution: E B 1 6 A

22 Converting Hexadecimal to Decimal Multiply each digit by its corresponding power of 16 Value = (d n-1 16 n-1 ) + (d n-2 16 n-2 ) (d 1 16) + d 0 Examples: (1234) 16 = ( ) + ( ) + (3 16) + 4 = Decimal Value 4660 (3BA4) 16 = ( ) + ( ) + (10 16) + 4 = Decimal Value 15268

23 Converting Decimal to Hexadecimal Repeatedly divide the decimal integer by 16 Each remainder is a hex digit in the translated value least significant digit most significant digit stop when quotient is zero Decimal 422 = 1A6 hexadecimal

24 Integer Storage Sizes Half Word Byte 8 16 Storage Sizes Word 32 Double Word 64 Storage Type Unsigned Range Powers of 2 Byte 0 to to (2 8 1) Half Word 0 to 65,535 0 to (2 16 1) Word 0 to 4,294,967,295 0 to (2 32 1) Double Word 0 to 18,446,744,073,709,551, t to (2 64 1) What is the largest 20-bit unsigned integer? Answer: = 1,048,575

25 0 + 0 = = = = 0 (carry 1) Binary Arithmetic Rules = 1 (carry 1) 25

26 Binary Addition Start with the least significant bit (rightmost bit) Add each pair of bits Include the carry in the addition, if present carry (54) (29) (83) bit position:

27 Hexadecimal Addition Start with the least significant hexadecimal digits Let Sum = summation of two hex digits If Sum is greater than or equal to 16 Sum = Sum 16 and Carry = 1 Example: carry: 1 1 C A A F C D A + B = = E 8 4 B Since Sum = = 5 Carry = B 5

28 Representing Negative Values Signed-magnitude number representation The sign represents the ordering, and the digits represent the magnitude of the number 28

29 Signed Magnitude Representation Unsigned: - and + are the same. In signed magnitude the left-most bit represents the sign of the integer. 0for positive numbers. 1for negative numbers. The remaining bits represent to magnitude of the numbers. 29

30 Example Suppose is a signed magnitude representation. The sign bit is 1, then the number represented is negative position Bit pattern contribution The magnitude is with a value = 29 Then the number represented by is 29.

31 Exercise has in signed magnitude notation. Find the signed magnitude of 37 10? 2. Using the signed magnitude notation find the 8-bit binary representation of the decimal value and Find the signed magnitude of 63 using 8-bit binary g g g y sequence?

32 Exercise has in signed magnitude notation. Find the signed magnitude of 37 10? Using the signed magnitude notation find the 8-bit binary representation of the decimal value and Find the signed magnitude of 63 using 8-bit binary 3. Find the signed magnitude of 63 using 8-bit binary sequence

33 Signed-Summary In signed magnitude notation, The most significant bit is used to represent the sign. 1 represents negative numbers 0 represents positive numbers. The unsigned value of the remaining bits represent The magnitude. Advantages: Represents positive and negative numbers Disadvantages: two representations of zero, Arithmetic operations are difficult. 33

34 Excess Notation In excess notation: The value represented is the unsigned value with a fixed value subtracted from it. For n-bit binary sequences the value subtracted fixed value is 2 (n-1). Most significant bit: 0for negative numbers 1 for positive numbers 34

35 Excess Notation with n bits represent 2 n-1 is the decimal value in unsigned notation. Decimal value In unsigned notation -2 n-1 = Decimal value In excess notation Therefore, in excess notation: will represent 0. 35

36 Example (1) - excess to decimal Find the decimal number represented by in excess notation. Unsigned value = = = Excess value: excess value = = =

37 Example (2) - decimal to excess Represent the decimal value 24 in 8-bit excess notation. We first add, 2 8-1, the fixed value = = 152 then, find the unsigned value of = (unsigned notation) = (excess notation) 37

38 example (3) Represent the decimal value -24 in 8-bit excess notation. We first add, 2 8-1, the fixed value = = 104 then, find the unsigned value of = (unsigned notation) = (excess notation) 38

39 Example (4) (10101) Unsigned = = The value represented in unsigned notation ti is 21 Sign Magnitude The sign bit is 1, so the sign is negative The magnitude is the unsigned value = 5 10 So the value represented in signed magnitude is Excess notation As an unsigned binary integer = subtracting = 2 4 = 16, we get = So the value represented in excess notation is

40 Advantages of Excess Notation It can represent positive and negative integers. There is only one representation for 0. It is easy to compare two numbers. When comparing the bits can be treated as unsigned integers. Excess notation is not normally used to represent integers. It is mainly used in floating point representation for representing fractions (later floating point rep.). 40

41 Excess notation - Summary In excess notation, the value represented is the unsigned value with a fixed value subtracted from it. ie i.e. fornbit n-bit binary sequences the value subtracted is 2 (n-1). Most significant bit: 0 for negative numbers. 1 positive numbers. Advantages: Only one representation of zero. Easy for comparison. 41

42 Representing Negative Values Two s Complement (Vertical line is easier to read) = =

43 Representing Negative Values Addition and subtraction are the same as in 10 s complement arithmetic Do you notice something interesting about the left-most bit? 43

44 Forming the Two's Complement starting value = +36 step1: reverse the bits (1's complement) step 2: add 1 to the value from step sum = 2's complement representation = -36 Sum of an integer and its 2's complement must be zero: = (8-bit sum) Ignore Carry Another way to obtain the 2's complement: Start at the least significant 1 Leave all the 0s to its right unchanged Complement all the bits to its left Binary Value = 's Complement = least significant 1

45 Advantages of Two s Complement Notation It is easy to add two numbers Subtraction can be easily performed. Multiplication is just a repeated addition. Division is just a repeated subtraction Two s complement is widely used in ALU

46 Highest bit indicates the sign 1 = negative Sign bit Sign Bit 0 = positive Negative Positive For Hexadecimal Numbers, check most significant digit If highest digit is > 7, then value is negative Examples: 8A and C5 are negative bytes B1C42A00 is a negative word (32-bit signed integer)

47 Binary Subtraction When subtracting A B, convert B to its 2's complement Add A to ( B) borrow: carry: (2's complement) (same result) Final carry is ignored, because Negative number is sign-extended with 1's You can imagine infinite 1's to the left of a negative number Adding the carry to the extended 1's produces extended zeros

48 Ranges of Signed Integers For n-bit signed integers: Range is -2 n 1 to (2 n 1 1) Positive range: 0 to 2 n 1 1 Negative range: -2 n 1 to -1 Storage Type Unsigned Range Powers of 2 Byte 128 to to (2 7 1) Half Word 32,768 to +32, to (2 15 1) Word 2,147,483,648 to +2,147,483, to (2 31 1) Double Word 9,223,372,036,854,775,808 to +9,223,372,036,854,775, to (2 63 1) Practice: What is the range of signed values that may be stored in 20 bits?

49 Exercise Determine the decimal value represented by in each of the following four systems. 1. Unsigned notation? 2. Signed magnitude notation? 3. Excess notation? 4. Tow s complements? 49

50 Number Overflow What happens if the computed value won't fit? Overflow If each value is stored using eight bits, adding 127 to 3 overflows Problems occur when mapping an infinite world onto a finite machine! 50

51 Carry and Overflow Carry is important when Adding or subtracting unsigned integers Indicates that the unsigned sum is out of range Either < 0 or >maximum unsigned n-bit value Overflow is important when Adding or subtracting signed integers Indicates that the signed sum is out of range Overflow occurs when Adding two positive numbers and the sum is negative Adding two negative numbers and the sum is positive Can happen because of the fixed number of sum bits

52 Carry and Overflow Examples We can have carry without overflow and vice-versa Four cases are possible (Examples are 8-bit numbers) (-8) Carry = 0 Overflow = 0 Carry = 1 Overflow = (-38) 157 (-99) (-113) Carry = 0 Overflow = 1 Carry = 1 Overflow = 1

53 Representing Real Numbers Real numbers A number with a whole part and a fractional part , , 357.0, and Positions to the right of the decimal point are the tenths position: 10-1, 10-2,

54 Representing Real Numbers Same rules apply in binary as in decimal Decimal point is actually the radix point Positions to the right of the radix point in binary are 2-1 (one half), 2-2 (one quarter), 2-3 (one eighth) 54

55 Examples: Representing fractional numbers Find the binary representations of: = = 0.11 Using only 8 binary digits find the binary representations of:

56 Representing Real Numbers A real value in base 10 can be defined by the following formula The representation is called floating point because the number of digits is fixed but the radix point floats 56

57 Floating-Point Representation Computer representation of a floating-point number consists of three fixed-size fields: The one-bit sign field is the sign of the stored value. The size of the exponent field, determines the range of values that can be represented. The size of the significand determines the precision of the representation.

58 Floating-Point Representation The IEEE-754 single precision floating point standard uses an 8-bit exponent and a 23-bit significand. The IEEE-754 double precision standard uses an 11-bit exponent and a 52-bit significand. For illustrative purposes, (and to keep life VERY simple) For illustrative purposes, (and to keep life VERY simple) we will use a 14-bit model with a 5-bit exponent and an 8- bit significand.

59 Floating-Point Representation The significand of a floating-point number is always preceded by an implied binary point. The significand contains a fractional binary value. The exponent is the power of 2 to which the significand is raised. Example: Express in the simplified 14-bit floating-point model. We know that is 2 5. So in (binary) scientific notation = 1.0 x 2 5 = 0.1 x 2 6. Using this information, we put 110 (= 6 10 ) in the exponent field and 1 in the significand as shown.

60 Floating-Point Representation BUT WAIT It s not that simple!! The illustrations shown at the right are all equivalent representations for 32 using our simplified model. Not only do these synonymous representations waste space, but they can also cause confusion. Another problem with our system is that we have made no allowances for negative exponents. We have no way to express 0.5 (=2-1 )! (There is no sign in the exponent field!) All of these problems can be fixed with no changes to our basic model. 60

61 Floating-Point Representation To resolve the issue of synonymous forms, we ll use a rule that t the first digit of the significand must be 1. This gives a unique pattern for each number. In the IEEE-754 standard, this 1 is implied meaning that a 1 is assumed after the binary point. By using an implied 1, we increase the precision of the representation by a power of two. (Why?) In our simple instructional model, we will not use the implied bits mechanism the 1 will BE there. To provide for negative exponents, we ll use a biased exponent We have a 5-bit exponent. We ll use 16 for our bias. This is called excess-16 representation. In our model, exponent values less than 16 are negative, meaning that the number we re representing is negative.

62 IEEE Example 1 Convert 6.75 to 32 bit IEEE format. 1.The Mantissa. The Integer first. 6 / 2 = 3 r 0 3 / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next..75 * 2 = * 2 = 1.0 = = put the two parts together Now normalise * 2 2

63 IEEE Biased 127 Exponent To generate a biased 127 exponent Take the value of the signed exponent and add 127. Example then = and my value for the exponent would be 143 = So it is simply now an unsigned value... 63

64 Binary Possible Representations of an Exponent Sign Magnitude 2's Complement Biased 127 Exponent {reserved} {reserved} 64

65 Floating-Point Representation From Decimal to Floating Point. Example: Express in the revised 14-bit floating-point model. We know that 32 = 1.0 x 2 5 = 0.1 x 2 6. To use our excess 16 biased exponent, we add 16 to 6,,giving g22 10 (= ) ). Example: Express in the revised 14-bit floating-point model. We know that is 2-4. So in (binary) scientific notation = 1.0 x 2-4 = 0.1 x 2-3. To use our excess 16 biased exponent, we add 16 to -3, giving (= ).

66 2.5 Floating-Point Representation From Decimal to Floating Point. Example: Express in the revised 14-bit floating-point model. We find = Normalizing, we have: = x 2 5. To use our excess 16 biased exponent, we add 16 to 5, giving (= ). We also need a 1 in the sign bit.

67 Floating-Point Representation From Floating Point to Decimal Exponents will be stored using excess-16 Sign bit = 0 (positive) Exponent = 5 ( = 5) Significand = We shift the decimal point 5 positions giving us = +17 Sign bit = 0 (positive) Exponent = -2 ( = -2) Significand = We shift the decimal point 2 positions to the left, giving us = Sign bit = 1 (negative) Exponent = 3 ( = 3) Significand = We shift the decimal point 3 positions to the right, giving =

68 Representation in IEEE 754 single sign bit: precision 0 for positive and, 1for negative numbers 8biased exponent by bit normalised mantissa Sign Exponent Mantissa 68

69 Example (2) which number does the following IEEE single precision notation represent? The sign bit is 1, hence it is a negative number. The exponent is = It is biased by 127, hence the real exponent is = 1. The mantissa: It is normalised, hence the true mantissa is 1.01 = Finally, the number represented is: x 2 1 =

70 Floating-Point Representation Floating-point addition and subtraction are done the same as using pencil and paper. The first thing that we do is express both operands in the same exponential power, then add the numbers, keeping the exponent in the sum. If the exponent needs adjustment, t fix it at the end. Example: Find the sum of and using the 14-bit floating-point model. We find = x 2 4. And = x 2 1 = x 2 4. Thus, our sum is x 2 4.

71 Floating-Point Representation When discussing i floating-point i t numbers, it is important t to understand the terms range, precision, and accuracy. The range of a numeric integer format is the difference between the largest and smallest values that it can express. Accuracy refers to how closely a numeric representation approximates a true value. The precision of a number indicates how much information we have about a value

72 Range of numbers Normalized (positive range; negative e is symmetric) smallest (1+0) =2-126 largest ( ) ( ) Positive underflow Positive overflow 2014/8/25 PITT CS

73 Floating-Point Representation No matter how many bits we use in a floating-point representation, our model must be finite. The real number system is infinite, so F.P. can give nothing more than an approximation of a real value. At some point, every model breaks down, introducing errors into our calculations. By using a greater number of bits in our model, we can reduce these errors, but we can never totally eliminate them. Our job becomes one of reducing error, or at least being aware of the possible magnitude of error in our calculations. Errors can compound through repetitive arithmetic operations. For example, our 14-bit model cannot exactly represent the decimal value In binary, it is 9 bits wide: = %

74 Floating-Point Representation Floating-point overflow and underflow can cause programs to crash. Overflow occurs when there is no room to store the high-order bits resulting from a calculation. Underflow occurs when a value is too small to store, possibly resulting in division by zero. The hardware detects the error, causes an interrupt that is caught by the OS which terminates the process. It s better for a program to crash than to have it produce incorrect, but plausible, results.

75 IEEE 754 Encoding Single Precision Double Precision Represented Object Exponent Fraction Exponent Fraction non-zero 0 non-zero 1~254 anything 1~2046 anything +/- denormalized number +/- floating-point numbers /- infinity it 255 non-zero 2047 non-zero NaN (Not a Number) 75 75

76 Representing Text What must be provided d to represent text? t? There are a finite number of characters to represent, so list them all and assign each a binary string. Character set A list of characters and the codes used to represent each one Computer manufacturers agreed to standardize 76

77 The ASCII Character Set ASCII stands for American Standard Code for Information Interchange ASCII originally used seven bits to represent each character, allowing for 128 unique characters Later extended ASCII evolved so that all eight bits were used How many characters could be represented? 77

78 78 ASCII Character Set Mapping

79 The ASCII Character Set The first 32 characters in the ASCII character chart do not have a simple character representation to print to the screen What do you think they are used for? 79

80 Character strings How to represent character strings? A collection of adjacent words (bit-string units) can store a sequence of letters 'H' 'e' ' 'l' 'l' o' ' ' 'W' 'o'' 'r'' 'l' 'd' '\0' Notation: o enclose strings in double quotes "Hello world" Representation convention: null character defines end of string Null is sometimes written as '\0' Its binary representation is the number 0 80

81 exercise Use the ASCII table to write the ASCII code for the following: CIS110 6=2*3 81

82 The Unicode Character Set Extended ASCII is not enough for international use One Unicode mapping uses 16 bits per character How many characters can this mapping represent? Unicode is a superset of ASCII The first 256 characters correspond exactly 82 to the extended ASCII character set

83 The Unicode Character Set Figure 3.6 A few characters in the Unicode character set 83

84 the end