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Integer and Fractional Representations in Computers

This chapter explains how integers and fractional numbers are represented in computers. It covers signed, unsigned, excess, and two's complement notations, as well as floating-point notation and representation of characters, colors, images, sound, and movies.

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Integer and Fractional Representations in Computers

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  1. Chapter 5 Data representation

  2. Aim • Explain how integers are represented in computers using: • Unsigned, signed magnitude, excess, and two’s complement notations • Explain how fractional numbers are represented in computers • Floating point notation (IEEE 754 single format) • Calculate the decimal value represented by a binary sequence in: • Unsigned, signed notation, excess, two’s complement, and the IEEE 754 notations. • Explain how characters are represented in computers • E.g. using ASCII and Unicode • Explain how colours, images, sound and movies are represented

  3. Integer Representations • Unsigned notation • Signed magnitude notion • Excess notation • Two’s complement notation.

  4. Unsigned Representation • Represents positive integers. • Unsigned representation of 157: • Addition is simple: 1 0 0 1 + 0 1 0 1 = 1 1 1 0.

  5. Advantages and disadvantages of unsigned notation • Advantages: • One representation of zero • Simple addition • Disadvantages • Negative numbers can not be represented. • The need of different notation to represent negative numbers.

  6. Signed Magnitude Representation • 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.

  7. Example • Suppose 10011101 is a signed magnitude representation. • The sign bit is 1, then the number represented is negative • The magnitude is 0011101 with a value 24+23+22+20= 29 • Then the number represented by 10011101 is –29.

  8. Exercise 1 • 3710has 0010 0101 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 and -2410. • Find the signed magnitude of –63 using 8-bit binary sequence?

  9. 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, • difficult operation.

  10. 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: • 0 for negative numbers • 1 for positive numbers

  11. Excess Notation with n bits • 1000…0 represent 2n-1is the decimal value in unsigned notation. • Therefore, in excess notation: • 1000…0 will represent 0 . Decimal value In unsigned notation Decimal value In excess notation - 2n-1 =

  12. Example (1) - excess to decimal • Find the decimal number represented by 10011001 in excess notation. • Unsigned value • 100110002 = 27 +24 + 23 +20 = 128 + 16 +8 +1 = 15310 • Excess value: • excess value = 153 – 27 = 152 – 128 = 25.

  13. Example (2) - decimal to excess • Represent the decimal value 24 in 8-bit excess notation. • We first add, 28-1, the fixed value • 24 + 28-1 = 24 + 128= 152 • then, find the unsigned value of 152 • 15210 = 10011000 (unsigned notation). • 2410 = 10011000 (excess notation)

  14. example (3) • Represent the decimal value -24 in 8-bit excess notation. • We first add, 28-1, the fixed value • -24 + 28-1 = -24 + 128= 104 • then, find the unsigned value of 104 • 10410 = 01101000 (unsigned notation). • -2410 = 01101000 (excess notation)

  15. Example (4) -- 10101 • Unsigned • 101012 = 16+4+1 = 2110 • The value represented in unsigned notation is 21 • Sign Magnitude • The sign bit is 1, so the sign is negative • The magnitude is the unsigned value 01012 = 510 • So the value represented in signed magnitude is -510 • Excess notation • As an unsigned binary integer 101012 = 2110 • subtracting 25-1 = 16, we get 21-16 = 510. • So the value represented in excess notation is 510.

  16. Excess 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.

  17. Two’s Complement Notation • The most used representation for integers. • All positive numbers begin with 0. • All negative numbers begin with 1. • One representation of zero • i.e. 0 is represented as 0000 using 4-bit binary sequence.

  18. Properties of Two’s Complement Notation • Positive numbers begin with 0 • Negative numbers begin with 1 • Only one representation of 0, i.e. 0000 • Relationship between +n and –n. • 0 1 0 0 +4 0 0 0 1 0 0 1 0 +18 • 1 1 0 0 -4 1 1 1 0 1 1 1 0 -18

  19. Two’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 zero • All 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:

  20. Exercise - 10001011 • Determine the decimal value represented by 10001011 in each of the following four systems. • Unsigned notation? • Signed magnitude notation? • Excess notation? • Tow’s complements?

  21. Fraction Representation • Floating point representation.

  22. Floating Point Representation format • The 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. Sign Exponent Mantissa

  23. Representation in IEEE 754 single precision • sign bit: • 0 for positive and, • 1 for negative numbers • 8 biased exponent by 127 • 23 bit normalised mantissa Sign Exponent Mantissa

  24. Example (1)5.7510 in IEEE single precision • 5.75 is a positive number then the sign bit is 0. 5.75 = 101.11 * 20 = 10.111 * 21 = 1.0111 * 22 • The real mantissa is 1.0111, then the normalised mantissa is 0111 0000 0000 0000 0000 000 • The real exponent is 2 and the bias is 127, then the exponent is 2 + 127=12910 = 1000 00012. • The representation of 5.75 in IEE sing-precision is: 0 1000 0001 0111 0000 0000 0000 0000 000

  25. 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 1000 0000 = 12810 • It is biased by 127, hence the real exponent is 128 –127 = 1. • The mantissa: 0100 0000 0000 0000 0000 000. • It is normalised, hence the true mantissa is 1.01 = 1.2510 • Finally, the number represented is: -1.25 x 21 = -2.50 1 1000 0000 0100 0000 0000 0000 0000 000

  26. Single Precision Format • The exponent is formatted using excess-127 notation, with an implied base of 2 • Example: • Exponent: 10000111 • Representation: 135 – 127 = 8 • The stored values 0 and 255 of the exponent are used to indicate special values, the exponential range is restricted to 2-126 to 2127 • The number 0.0 is defined by a mantissa of 0 together with the special exponential value 0 • The standard allows also values +/-∞ (represented as mantissa +/-0 and exponent 255 • Allows various other special conditions

  27. In comparison The smallest and largest possible 32-bit integers in two’s complement are only -232 and 231- 1 2020/1/1 27 PITT CS 1621

  28. Range of numbers Normalized (positive range; negative is symmetric) 00000000100000000000000000000000 01111111011111111111111111111111 smallest +2-126×(1+0) = 2-126 largest +2127×(2-2-23) 2127(2-2-23) 2-126 0 Positive overflow Positive underflow 2020/1/1 28 PITT CS 1621

  29. Representation in IEEE 754 double precision format • It uses 64 bits • 1 bit sign • 11 bit exponent biased by 1023. • 52 bit mantissa Sign Exponent Mantissa

  30. Character 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 0 • See (table 5.1 p56, study guide) for character representation in ASCII.

  31. ASCII – example Symbol decimal Binary 7 55 00110111 8 56 00111000 9 57 00111001 : 58 00111010 ; 59 00111011 < 60 00111100 = 61 00111101 > 62 00111110 ? 63 00111111 @ 64 01000000 A 65 01000001 B 66 01000010 C 67 01000011

  32. Character strings • How to represent character strings? • A collection of adjacent “words” (bit-string units) can store a sequence of letters • Notation: 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 'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0'

  33. Information Information Information Information Data Data Data Data Layered View of Representation Textstring Sequence of characters Character Bit string

  34. Unicode - 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.

  35. Colour representation • Colours can represented using a sequence of bits. • 256 colours – how many bits? • Hint for calculating • To 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 xfor 2 x => 256 • 24-bit colour – how many possible colors can be represented? • Hints • 16 million possible colours (why 16 millions?)

  36. 24-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.

  37. 4=22 choices 00 (off, off)=white 01 (off, on)=light grey 10 (on, off)=dark grey 11 (on, on)=black = (white) 0 0 = (light grey) 0 1 = (dark grey) 1 0 = (black) 1 1 Example: 2-bit per pixel

  38. Image representation • An 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.

  39. Representing Sound Graphically • X axis: time • Y axis: pressure • A: amplitude (volume) • : wavelength (inverse of frequency = 1/)

  40. Sampling • 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 better • The size of the word used to represent a sample.

  41. Digitizing Sound Capture amplitude at these points Lose all variation between data points Zoomed Low Frequency Signal

  42. Summary • Integer representation • Unsigned, • Signed, • Excess notation, and • Two’s complement. • Fraction representation • Floating point (IEEE 754 format ) • Single and double precision • Character representation • Colour representation • Sound representation

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