Presentation on theme: "Data Representation How integers are represented in computers?"— Presentation transcript:
1Data Representation How integers are represented in computers? How fractional numbers are represented in computers?Calculate the decimal value represented by a binary sequence.How to determine the binary sequence that represent a decimal number?Explain how characters are represented in computers?Explain how colours, images and sound are represented in computers?
2Binary numbersBinary number is simply a number comprised of only 0's and 1's.Computers use binary numbers because it's easy for them to communicate using electrical current is off, 1 is on.You can express any base 10 (our number system -- where each digit is between 0 and 9) with binary numbers.The need to translate decimal number to binary and binary to decimal.There are many ways in representing decimal number in binary numbers. (later)
3How does the binary system work? It is just like any other systemIn decimal system the base is 10We have 10 possible values 0-9In binary system the base is 2We have only two possible values 0 or 1.The same as in any base, 0 digit has non contribution, where 1’s has contribution which depends at there position.
5Examples: decimal -- binary Find the binary representation ofFind the decimal value represented by the following binary representations:
6Examples: Representing fractional numbers Find the binary representations of:0.5100.7510Using only 8 binary digits find the binary representations of:0.2100.810
7Representation of Numbers Representing whole numbersUnsigned notationSigned magnitude notionExcess notationTwo’s complement notation.Representing fractionsFixed point representationFloating point representationNormalisationIEEE Standard.
8Integer Representations All data in computers are represented in terms of 1s and 0s.Any integer can be represented as a sequence of 1s and 0s.Several ways of doing this.
9Unsigned Representation Represents positive integers.Unsigned representation of 157:Addition is simple:=position7654321Bit patterncontribution2724232220
10Advantages and disadvantages of unsigned notation One representation of zeroSimple additionDisadvantagesNegative numbers can not be represented.The need of different notation to represent negative numbers.
11Representation of negative numbers Is a representation of negative numbers possible?Unfortunately you can not just stick a negative sign in from of a binary number. (it does not work like that)There are three methods used to represent negative numbers.Signed magnitude notationExcess notation notationTwo’s complement notation
12Signed Magnitude Representation Unsigned: - and + are the same.In signed magnitude the left most bit represents the sign of the integer.0 for positive numbers.1 for negative numbers.The remaining bits represent to magnitude of the numbers.
13Example - Suppose 10011101 is a signed magnitude representation. The sign bit is 1, then the number represented is negativeThe magnitude is with a value = 29Then the number represented by is –29.position7654321Bit patterncontribution-24232220
14Exercise 13710 has in signed magnitude notation. Find the signed magnitude of –3710 ?Using the signed magnitude notation find the 8-bit binary representation of the decimal value 2410 andFind the signed magnitude of –63 using 8-bit binary sequence?
15Disadvantage of Signed Magnitude Addition and subtractions are difficult.Signs and magnitude, both have to carry out the required operation.They are two representations of 0==To test if a number is 0 or not, the CPU will need to see whether it is or0 is always performed in programs.Therefore, having two representations of 0 is inconvenient.
16Signed-Summary In signed magnitude notation, Advantages: the most significant bit is used to represent the sign.1 represents negative numbers0 represents positive numbers.The unsigned value of the remaining bits represent The magnitude.Advantages:Represents positive and negative numbersDisadvantages:two representations of zero,difficult operation.
17Excess 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:0 for negative numbers1 for positive numbers
18Excess Notation with n bits 1000…0 represent 2n-1 is the decimal value in unsigned notation.Therefore, in excess notation:1000…0 will represent 0 .Decimal valueIn unsignednotationDecimal valueIn excessnotation- 2n-1 =
19Example (1) - excess to decimal Find the decimal number represented by in excess notation.Unsigned value= = = 15310Excess value:excess value = 153 – 27 = 152 – 128 = 25.
20Example (2) - decimal to excess Represent the decimal value 24 in 8-bit excess notation.We first add, 28-1, the fixed value= = 152then, find the unsigned value of 15215210 = (unsigned notation).= (excess notation)
21example (3) Represent the decimal value -24 in 8-bit excess notation. We first add, 28-1, the fixed value= = 104then, find the unsigned value of 10410410 = (unsigned notation).-2410 = (excess notation)
22Example (4) -- 10101 Unsigned Sign Magnitude Excess notation = = 2110The value represented in unsigned notation is 21Sign MagnitudeThe sign bit is 1, so the sign is negativeThe magnitude is the unsigned value = 510So the value represented in signed magnitude is -510Excess notationAs an unsigned binary integer = 2110subtracting = 16, we get = 510.So the value represented in excess notation is 510.
23Advantages 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.).
24Exercise 2 Find 10011001 is an 8-bit binary sequence. Find the decimal value it represents if it was in unsigned and signed magnitude.Suppose this representation is excess notation, find the decimal value it represents?Using 8-bit binary sequence notation, find the unsigned, signed magnitude and excess notation of the decimal value ?
25Excess notation - Summary In excess notation, the value represented is the unsigned value with a fixed value subtracted from it.i.e. for 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.
26Two’s Complement Notation The most used representation for integers.All positive numbers begin with 0.All negative numbers begin with 1.One representation of zeroi.e. 0 is represented as 0000 using 4-bit binary sequence.
27Two’s Complement Notation with 4-bits Binary pattern Value in 2’s coml.not.
28Properties of Two’s Complement Notation Positive numbers begin with 0Negative numbers begin with 1Only one representation of 0, i.e. 0000Relationship between +n and –n.
29Advantages 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 subtractionTwo’s complement is widely used in ALU
30Evaluating numbers in two’s complement notation Sign bit = 0, the number is positive. The value is determined in the usual way.Sign bit = 1, the number is negative. three methods can be used:Method 1decimal value of (n-1) bits, then subtract 2n-1Method 2- 2n-1 is the contribution of the sign bit.Method 3Binary rep. of the corresponding positive number.Let V be its decimal value.- V is the required value.
31Example- 10101 in Two’s Complement The most significant bit is 1, hence it is a negative number.Method 10101 = (+5 – 25-1 = 5 – 24 = 5-16 = -11)Method 2Method 3Corresponding + number is = = 11the result is then –11.
32Two’s complement-summary In two’s complement the most significant for an n-bit number has a contribution of –2(n-1).One representation of zeroAll arithmetic operations can be performed by using addition and inversion.The most significant bit: 0 for positive and 1 for negative.Three methods can the decimal value of a negative number:Method 1decimal value of (n-1) bits, then subtract 2n-1Method 2- 2n-1 is the contribution of the sign bit.Method 3Binary rep. of the corresponding positive number.Let V be its decimal value.- V is the required value.
33ExerciseDetermine the decimal value represented by in each of the following four systems.Unsigned notation?Signed magnitude notation?Excess notation?Tow’s complements?
34Fraction Representation To represent fraction we need other representations:Fixed point representationFloating point representation.
35Fixed-Point Representation old positionNew positionBit patternContribution =19.625Radix-point
36Limitation of Fixed-Point Representation To represent large number of very small numbers we need a very long sequences of bits.This is because we have to give bits to both the integer part and the fraction part.
37Floating Point Representation In decimal notation we can get around thisproblem using scientific notation or floatingpoint notation.NumberScientific notationFloating-point notation1,245,000,000,000
38Floating point format M B E Sign mantissa or base exponent significandSignExponentMantissa
39Floating Point Representation format SignThe exponent is biased by a fixed value b, called the bias.The mantissa should be normalised, e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence.ExponentMantissa
40IEEE 745 StandardThe most important floating point representation is defined in IEEE Standard 754.It was developed to facilitate the portability from one processor to another.The IEEE Standard defines both a 32-bit and 64-bit format.
41Representation in IEEE 754 single precision sign bit:0 for positive and,1 for negative numbers8 biased exponent by 12723 bit normalised mantissaSignExponentMantissa
42Biased ExponentThe exponent bit needs to represent both negative and positive numbers.Excess notation is used to represent negative and positive exponent.The exponent is represented with an excess of a fixed value bThe value b is called to bias. And we say that the exponent is biased by the value bIn IEEE single-precision format the the bias is We say the exponent is biased by 127
43Mantissa in IEEE single precision format The mantissa in this case has 23 bits.It should be normalised.For example: if the real mantissa is:then the normalised mantissa should beBecause of this we do not need to store the 1. And as a result of this can use 23 to represent 24 bits. Which will achieve a great precision. This is what happens in IEEE 754 single precision.
44Mantissa in IEEE single precision format The mantissa in this case has 23 bits.It should be normalised.For example: if the real mantissa is:then the normalised mantissa should beBecause of this we do not need to store the 1. And as a result of this can use 23 to represent 24 bits. Which will achieve a great precision. This is what happens in IEEE 754 single precision.
45Example (1) 5.7510 in IEEE single precision 5.75 is a positive number then the sign bit is 0.5.75 = * 20= * 21= * 22The real mantissa is , then the normalised mantissa isThe real exponent is 2 and the bias is 127, then the exponent is=12910 =The representation of 5.75 in IEE sing-precision is:
46Example (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 = 12810It is biased by 127, hence the real exponent is128 –127 = 1.The mantissa:It is normalised, hence the true mantissa is1.01 =Finally, the number represented is: x 21 = -2.501
47Representation in IEEE 754 double precision format It uses 64 bits1 bit sign11 bit biased exponent52 bit mantissaSignExponentMantissa
48IEEE 754 double precision Biased = 1023 11-bit exponent with an excess of 1023.For example:If the exponent is -1we then add 1023 to it = 1022We then find the binary representation of 1022Which isThe exponent field will now holdThis means that we just represent -1 with an excess of 1023.
49Floating Point Representation format (summary) the sign bit represents the sign0 for positive numbers1 for negative numbersThe exponent is biased by a fixed value b, called the bias.The mantissa should be normalised, e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence.SignExponentMantissa
50Character representation- ASCII ASCII (American Standard Code for Information Interchange)It is the scheme used to represent characters.Each character is represented using 7-bit binary code.If 8-bits are used, the first bit is always set to 0See (table 5.1 p56, study guide) for character representation in ASCII.
51ASCII – example Symbol decimal Binary 7 55 00110111 8 56 00111000 :;<=>?@ABC
52Character strings How to represent character strings? A collection of adjacent “words” (bit-string units) can store a sequence of lettersNotation: enclose strings in double quotes"Hello world"Representation convention: null character defines end of stringNull is sometimes written as '\0'Its binary representation is the number 0'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0'
53Layered View of Representation Text stringInformationDataSequence ofcharactersInformationDataCharacterBit string
54Working With A Layered View of Representation Represent “SI” at the two layers shown on the previous slide.Representation schemes:Top layer - Character string to character sequence: Write each letter separately, enclosed in quotes. End string with ‘\0’.Bottom layer - Character to bit-string: Represent a character using the binary equivalent according to the ASCII table provided.NEED TO PROVIDE PRINTED ASCII TABLES
55SolutionSI‘S’ ‘I’ ‘\0’The colors are intended to help you read it; computers don’t care that all the bits run together.
56exerciseUse the ASCII table to write the ASCII code for the following:CIS1106=2*3
57Unicode - representation ASCII code can represent only 128 = 27 characters.It only represents the English Alphabet plus some control characters.Unicode is designed to represent the worldwide interchange.It uses 16 bits and can represents 32,768 characters.For compatibility, the first 128 Unicode are the same as the one of the ASCII.
58Colour representation Colours can represented using a sequence of bits.256 colours – how many bits?Hint for calculatingTo figure out how many bits are needed to represent a range of values, figure out the smallest power of 2 that is equal to or bigger than the size of the range.That is, find x for 2 x => 25624-bit colour – how many possible colors can be represented?Hints16 million possible colours (why 16 millions?)
5924-bits -- the True colour 24-bit color is often referred to as the true colour.Any real-life shade, detected by the naked eye, will be among the 16 million possible colours.
61Image representationAn image can be divided into many tiny squares, called pixels.Each pixel has a particular colour.The quality of the picture depends on two factors:the density of pixels.The length of the word representing colours.The resolution of an image is the density of pixels.The higher the resolution the more information information the image contains.
62Bitmap ImagesEach individual pixel (pi(x)cture element) in a graphic stored as a binary numberPixel: A small area with associated coordinate locationExample: each point below is represented by a 4-bit code corresponding to 1 of 16 shades of gray
63Representing Sound Graphically X axis: timeY axis: pressureA: amplitude (volume): wavelength (inverse of frequency = 1/)
64Sampling Sampling is a method used to digitise sound waves. A sample is the measurement of the amplitude at a point in time.The quality of the sound depends on:The sampling rate, the faster the betterThe size of the word used to represent a sample.
65Digitizing Sound Capture amplitude at these points Lose all variation between data pointsZoomed Low Frequency Signal
66Summary Integer representation Fraction representation Unsigned,Signed,Excess notation, andTwo’s complement.Fraction representationFloating point (IEEE 754 format )Single and double precisionCharacter representationColour representationSound representation
67ExerciseRepresent +0.8 in the following floating-point representation:1-bit sign4-bit exponent6-bit normalised mantissa (significand).Convert the value represented back to decimal.Calculate the relative error of the representation.