Computer Science 1

1
Computer Science 1

• Erick Dube
• 327 H1 Block
• Westville
• Tel [phone]
• E-mail dubee2_at_ukzn.ac.za

2
Course Administration

• This Discrete Maths part of the module basically
  comprises of three components
• Number representation,
• Circuits and computer components.
• Simple Architecture Machine

3
Examination

• The module will be examined at the end of the
  semester by
• One 3-hour written exam that includes both
  Discrete Maths and Special Topics/Intro to
  Computer Sci.
• Discrete Maths makes up 50 of the entire
  assessment.
• The exam counts 50 of the final mark, with the
  class mark counting 50.
• In order to pass Computer Science 1 you need 50
  for your final mark,
• With a sub-minimum of 40 in the exam paper and
  the class mark

4
Discrete mathematics

• Number Representations
• Logic
• Circuits and Computers

5
Number Representations

• Number systems
• Positional Systems
• Other Bases
• The decimal number system that we use in everyday
  calculations is a positional system. In such a
  system the position of the digit within the
  number determines the weight attached to that
  digit
• When we write the number 1234 we really mean
• 1234 1 1000 2 100 3 10 4 1
• so that the right-most digit has weight 100 and
  the next digit has weight 101, the next 102, and
  so on. Writing it out this way is called expanded
  notation.

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• This system is called the decimal system because
  the weights
• are powers of 10.
• We say that the decimal system has base 10.
• The decimal system is not the only way of
  representing numbers.
• In fact we can use any positive integer greater
  than 1 as the base of a number system.
• In base 5 the number 2314 means
• 23145 253 3 52 1 51 4 50
• Note that in Base 5 we use the digits 1,2,3,4

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• In general, to construct a positional system for
• representing numbers we must select
• An integer b gt 1 which is called the base of
  the system.
• A set of b symbols to represent the digits
• 0, 1, 2, 3, , b-1
• Then a number such as
• N an bn an-1 bn-1 . a1 b a0
• where 0 ? ai lt b, is written as the sequence of
  digits
• N anan-1an-2 ..a2 a1 a0
• Here an is the most significant digit (provided
  its nonzero) and a0 is the least significant
  digit.

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• Determine the power of each digit for four-digit
  numbers in base 8.
• 83 82 81 80
• 512 64 8 1
• Determine the power of each digit for five digit
  numbers in base 7.
• 74 73 72 71 70
• 2401 343 79 7 1

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• To distinguish the numbers written in different
  bases, it is usual to subscript the base
  immediately after the number. Thus 1234 in base 7
  is written as 12347. If no subscript is given
  then the number is assumed to be in decimal.

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Common Bases

• The four most important bases are
• Decimal
• base 10
• Binary
• base 2. This uses only the digits 0 and 1 which
  are called bits.
• Hex (short for hexadecimal)
• base 16.
• This uses the digits 0, 1, 2, 3, 4,5, 6, 7, 8, 9,
  A, B,C,D,E, F. That is,A10, B11, C12, D13,
  E14 and F15.
• Octal
• base 8

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Common Bases
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Conversion between Bases

• First Method of conversion
• By weights
• Convert the number 137548 to base 10
• 83 82 81 80
• 4096 512 64 8 1
• 1 3 7 5 4
• 4096 1536 448 40 4
• Total 612410

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• By multiplication
• Convert the number 137548 to base 10
• 188
• (83)888
• (887)8760
• (7605)86120
• 61204612410

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• The method to convert from base 10 to another
  base ,
• Find the value corresponding to the weight of
  each digit such that the total will add up to the
  base 10 number being converted.
• The value for each digit must be the largest
  value that will not exceed the number being
  converted.

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• Convert 612410 to base 5.
• The weights of each digit in base 5 are as
  follows
• 15625 3125 625 125 25 5 1
• 15625 digit is too large, 3125 fits into 6124
  only once with a remainder of 2999 thus the
  first digit is a 1.
• Next digit , 625 goes into 2999 four times with a
  remainder of 499, 125 into 499 three times with
  remainder of 124, 25 into 124 four times, and so
  on. we get a final result of 1434445.

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• Second method of conversion
• From Base 10 to another Base
• Let B be the base to which we want to convert a
  number into. The number to convert , will be
  divided successively by the base, B, and each
  remainder is considered. We will do this until
  there is nothing left to divide. Each successive
  remainder represents the value of a digit in the
  new base , B, reading the new value from right to
  left

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612410 to Base 5
1434445
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Conversion between related Bases

• Let B and R be two bases such that R Bt The
  conversion from base B into base R consists of
  breaking the number in base B into groups of t,
  starting from the least significant digit , and
  converting each group independently .
• Eg 1624, converting from base 2 to base 16
  consists of breaking the binary number into
  groups of Four.

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• Let us convert 10010110111100102 to hexadecimal
• 1624
• grouping the binary number by fours, we have
• 1001 0110 1111 0010
• or
• 9 6 F 216
• The conversion in the reverse direction works
  identically

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Arithmetic in different Bases

• Decimal addition

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Arithmetic in different Bases

• Octal addition

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Arithmetic in different Bases

• Decimal Multiplication

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Arithmetic in different Bases

• Octal Multiplication

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Arithmetic in different Bases

• Binary Multiplication
• 1101101
• X 100110
• 1101101
• 1101101
• 110110100000
• 1000000101110

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Fractions
26
Fractions

• If we move the number point one place to the
  right in a number, the value of the number will
  be multiplied by the base.
• For example
• 129010 is 10 times as large as 129.010
• The number point is known by the name of its
  base, for example binary point or hexadecimal
  point.
• If we move the number point to the left one place
  , the value is divided by the base.

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Fractions
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Fractional Conversion Methods

• The way to convert a fractional number from some
  base B to base 10 is to determine the appropriate
  weights for each digit, multiply each digit by
  its weight, and add the values.
• Convert 0.1224 to base 10

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• the value is then
• 0.250.1250.015625 0.39062510
• To convert fractional numbers from 10 to another
  base, multiply the fraction by the base
  repeatedly, and record , then drop, the value
  that move to left of the radix point.
• We repeat this until the desired number of digits
  of accuracy is attained or until the value being
  multiplied is zero.

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• Convert 0.82812510 to base 2. Multiplying by 2,
  we get
• 0 .828125
• 2
• 1 .656250
• 2
• 1 .312500
• 2
• 0 .625000
• 2
• 1 .250000
• 2
• 0 .500000
• 2
• 1 .000000 The result is 0.1101012