Lecture2 1'2 2'5

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Lecture-2 (1.2 2.5)

• CSCI130-03A
• Instructor Dr. Imad Rahal

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Outline

• Components of a computer
• Computer Memory
• Binary to decimal conversion
• Decimal to binary conversion
• Binary to Hexadecimal conversion
• Hexadecimal to binary conversion
• 2s Complement to decimal conversion
• Decimal to 2s complement conversion
• Scientific to decimal conversion
• Decimal to scientific conversion
• Floating point to decimal conversion
• Decimal to floating point conversion
• Text data

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Overview of Computer Design

• Computers composed of 4 main components
• CPU (brain)
• Main Memory (STM)
• I/O devices (Mouth Ears)
• Auxiliary/secondary storage (LTM)
• Draw picture
• theoretically not needed to have a functional
  computer

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Overview of Computer Design

• Data (math equation) is input through some input
  device
• Keyboard, mouse, etc
• Stored in main memory
• Think of it as our short term memory where we
  keep all things we are thinking of right now
• Stores data before and after CPU processes it
• Processed by CPU and result is stored again in
  main memory
• CPU only processes data in main memory
• Outputted to the user through some output device
• Screen, printer, etc
• Main memory is emptied when we turn off computer,
  if we want to save data for some period of time
• Transferred to some auxiliary storage
• Hard disk, CD, DVD, floppy disk
• Analogous to our long term memory

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Overview of Computer Design

• Memory is divided into locations/registers/words
• Each holds a fixed amount of data
• Numbers, characters
• Each has an address used to access its contents
• P.O. boxes number is address and the box is the
  register
• Draw memory example
• Words are made of a fixed number of memory units
  called bits (Binary digIT) each having one of two
  states
• 1 means electric current is strong
• 0 no/weak electric current
• Digital or analogue?
• A word is usually 8 bits (1 byte)
• Data and addresses are represented in binary
• Memory sizes are measures in bytes in groups of
  1024 210(KB) or 1,048,576220 (MB) or 230 (GB)

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Overview of Computer Design

• The number system that we use daily is called the
  decimal system
• In computers, we have only two possibilities (0
  or 1) ? we can use the binary system
• 104103102101100 becomes 2423222120
  ...168421
• 37 3x101 7x100
• 100101 1x25 0x24 0x23 1x22 0x21 1x20

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Binary to Decimal and Vice Versa

• Binary ? decimal
• 1100 10012 ?
• 1001 10012 ?
• 0010 10012 ?
• Decimal ? Binary
• 1410 ?
• 910 ?
• 129 10 ?

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Overview of Computer Design

• Binary is not very convenient for humans to use
• 1001 0000 1001 0010
• Instead we use the hexadecimal system (base 16)
• 163162161160 4096256161
• In base 10, we have 10 digits (0-9)
• In base 2, we have 2 digits (0-1)
• What about base 16 (hexadecimal)?
• 0, 1, 2, 3.. 9, A, B, C, D, E, and F
• A1F A162 1161 F160 101616
  116 15 259110

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Binary to Hexa and Vice Versa

• From binary ? hexadecimal group 4 bits at a time
  as one hexa digit
• 10011111100100102 1001 1111 1001 0010 9F9216
• 1100 10012 ?
• 1001 10012 ?
• 0010 10012 ?
• From hexa ? binary
• 9A9216 1001 1010 1001 0010 2
• A416 ?
• 916 ?
• BC16 ?

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Signed Integers

• Easy to store positive numbers
• Negative numbers?
• Signed-magnitude representation
• Use the last (leftmost) bit as the sign bit
• 1 for negative and 0 for positive
• 1000 0011 ?
• 0100 0011 ?
• 1000 0000 ?
• 0000 0000 ?

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Signed Integers

• Problems
• Two values for zero!
• Problems for comparisons
• Addition wont work well
• 1000 0011 (-3)
• 0100 0011 (67)
• -----------------1100 0110 (-70)
• Obviously, we need something better
• ? 2s complement representation

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2s Complement Representation

• Used by almost all computers today
• All places hold the value they would in binary
  except for the leftmost place which is negative
• 8 bit integer -128 64 32 16 8 4 2 1
• Range -128,127
• If last bit
• is 0, then positive ? looks just as before
• is 1, then negative ?add values for first 7
  digits and then subtract 128
• 1000 1101 148-128 -115

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2s Complement Representation

• Converting from decimal to 2s complement
• For positive numbers find the binary
  representation
• For negative numbers
• Find the binary representation for its positive
  equivalent
• Flip all bits
• Add a 1
• 43
• 43 0010 1011
• -43
• 43 0010 1011 ? 1101 0100 ? 1101 0101
• 1101 0101 -128641641 -43
• We can use the usual laws of binary addition

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2s Complement Representation

• 125 - 19 106
• 0111 1101 1110 1101 ------------- 0110 1010
  106 !
• What happened to the one that we carried at the
  end?
• Got lost but still we got the right answer!
• We carried into and carried out of the leftmost
  column

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2s Complement Representation

• 12565 190
• 0111 11010100 0001 -------------1111 1110 -
  2 !!!
• We only carried into ? overflow
• Number too big to fit in 8 bits since
  range-128,127
• 12565 190 gt 127

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2s Complement Representation

• -125-65 -190
• 1000 00101011 1111 -------------0100 0011 67
  !!!
• We only carried out of ? underflow
• -190 lt -127
• Solution
• use larger registers (more than 7 bits)
• Very large positive and very small negative we
  might still have a problem ? combine two
  registers (double precision)

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Conversion Practice

• Binary to decimal
• 1100 00102 0001 11002
• Decimal to binary
• 24510,10210 310
• Hexa to binary
• BA1016, CA0D16 12FE16
• Binary to hexa
• 1100 00102 0001 11002
• Decimal to 2s complement
• 102, -102, 3, -3. 245 -245
• 2s complement to decimal
• 1100 0010 0001 1100

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Real Number Representation

• We deal with a lot of real numbers in our lives
  (class average) and while using computers
• Fractions or numbers with decimal parts
• 3/100.3 or 82.34
• 82.34 810 21 31/10 41/100
• To represent this number in binary, we use a
  radix point instead of a decimal point
• 1101.11 8 4 1 ½ ¼ 13.75
• But how can we represent the radix point on the
  computer? 0? 1?

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Real Number Representation

• We resort to the floating-point representation
• We represent the number without the radix
  point!!!
• Based on a popular scientific notation that has
  three parts
• Sign
• Mantissa (one digit before the decimal point, and
  two or more after), and
• Exponent (written with an E following by a sign
  and an integer which represents what power to
  raise 10 to)
• 6.124E5
• 6.124 105 612,400
• Sign? Mantissa? Exponent?
• Scientific to decimal
• -9.14E-3 -9.14 10-3 -9.14/1000 - 0.00914

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Real Number Representation

• Decimal to scientific
• Divide (or multiply) by 10 until you have only
  one digit (usually non-zero) before the decimal
  point
• Add (or subtract) one to the exponent every time
  you divide (or multiply)
• 123.8 ?
• 1.238E2
• 0.2348 ?
• 2.348E-1

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Real Number Representation

• Floating-point numbers is very similar
• But use only 0s and 1s
• For an eight bit number
• 0 000 0000
• Sign, exponent, mantissa (not standard lab 2)
• Sign 0 for positive and 1 for negative
• The exponent has three digits 0,7, but can be
  positive or negative
• shift by 4 ? -4,3
• 000?-4, 001?-3, 010?-2, , 111?3
• i.e. subtract 4 from the real value
• Use base two now (i.e. 2 raised to the power of
  the exponent value)

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Real Number Representation

• The mantissa has one (non-zero) place before the
  decimal point and a number of places after it
• in binary our digits are 0 and 1 and so we always
  have 1.xxxx
• No need to represent the one (add one to the
  result)
• 11100010 ? 1 110 0010
• Negative
• 1106, -4 ? 2, exponent22
• Mantissa 0010 01/2 01/4 11/8 01/16
  1/8
• without the initial 1 which weve omitted ? 11/8
• Or 1 decimal equivalent/16 1 2/16
• - (22)(11/8) -4 1/2

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Real Number Representation

• Floating point to decimal conversion
• 1 break bit pattern into sign (1 bit), exponent
  (3 bits), mantissa (4 bits)
• 2 sign is if 1 and otherwise
• 3 exponent decimal equivalent -4
• 4 mantissa 1 decimal equivalent/16
• 5 number sign (mantissa2exponent)

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Real Number Representation

• What about from decimal to floating point?
• Format the number into the form 1.xxxx2exponent
• Multiply (or divide) by 2 until we have a 1
  before the decimal point
• Subtract (or add) 1 from (to) the exponent for
  every such multiplication (or division)
• Add 4 to result (in floating-to-decimal
  conversion we subtracted 4)
• Convert exponent to binary
• Subtract 1 from the resulting mantissa
• Multiply mantissa by 16
• Round to the nearest integer
• Convert mantissa to binary

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Real Number Representation

• -8.48
• Sign is negative ? 1
• 8.48 ? 4.24 ? 2.12? 1.06 ? 3 divisions
• exponent34 7 or 1112
• mantissa mantissa -1 ? .06
• multiply mantissa by 16 (write it in terms of
  16ths) ? 0.96 1 0r 0001
• 1 111 0001

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Real Number Representation

• Decimal to floating-point conversion
• Sign bit 1 if negative and 0 otherwise
• Exponent 0, mantissa absolute(number)
• While mantissa lt 1 do
• mantissa mantissa 2
• exponent exponent -1
• While mantissa gt 2 do
• mantissa mantissa / 2
• exponent exponent 1
• Exponent exponent 4
• Change to binary
• Mantissa (mantissa -1)16
• Round off to nearest integer and change to binary
• Assemble number sign exponent mantissa

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Real Number Representation

• 0.319
• Positive ? sign bit 02
• mantissa 0.319 2 0.638 ? exponent -1
• mantissa 0.638 2 1.276 ? exponent -2
• exponent -2 4 2 0102
• mantissa (mantissa -1)16 4.416 4 01002
• 0 010 0100
• - 0.319
• Negative ? sign bit 12
• exponent 0102
• mantissa 01002
• 1 010 0100
• 0
• Sign bit 02
• mantissa? Not going to change to 1.xxxx no matter
  what!

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Real Number Representation

• No representation for 0
• 0 000 0000 1.0 2-4 (we assumed there is a 1
  before the mantissa)
• Assume this to be zero!
• Another issue, is the method exact?
• rounding? truncation (close numbers give same
  floating point values)
• Mantissa 4.416 or 4.0123 or 4.400 are all 4
  01002
• Problem is also due to using only 8 bits (4-bit
  mantissa)
• Floating-point numbers require 32 or even 64 bits
  usually
• But still we will have to round off and truncate
• That happens regularly with us even when not
  using computers
• PI 22/7
• 2/3 or 1/3
• We approximate a lot but we should know it !
• Higher precision applications use much larger
  size registers

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Practice

• Scientific to decimal
• -1.23E-10 9.01E3
• Decimal to scientific
• 123.23 0.001911
• Floating point to decimal
• 1101 1010, 0001 1110 1000 1111
• Decimal to floating point
• 3.23 0.2911

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Non-numeric Data Representation

• Text data
• The most common type of data people use on
  computers
• How can we transform it to binary --- not
  intuitive as numbers!
• Words can be divided into characters
• Each character can then be encoded by some binary
  code
• Every language has its set of letters ? we will
  limit ourselves to the Latin alphabet
• Numbers were (relatively) easy to map to binary
• Decimal to binary change
• What about letters and other symbols?

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Non-numeric Data Representation

• Many transformations exist and all are arbitrary
• Two popular
• EBCDIC (Extended Binary Coded Decimal Interchange
  Code) by IBM
• ASCII (American Standard Code for Information
  Interchange) by American National Standards
  Institute (ANSI)
• Most widely used (table in book page 22)
• Every letter/symbol is represented by 7 bits
• How many letters/symbols do we have in total?
• A-Z (26) , a-z (26) , 0-9 (10), symbols (33),
  control characters (33)
• If using 1 byte/character ? we have one extra bit
• Extended ASCII-8 (more mathematical and graphical
  symbols or phonetic marks) --- 256

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Non-numeric Data Representation

• Given an integer 00110100
• 2s complement ? 52
• Floating point ? 0.625
• ASCII ? character 4 (check ASCII table in book)
• There must be some code blocks preceding such
  values informing the computer of the type
• Sometimes called meta-data
• What would you use if (2s, floating pt. or
  ASCII)
• Do calculations?
• Phone numbers?