Computer Math

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Computer Math

• CPS120 Data Representation

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Representing Data

• The computer knows the type of data stored in a
  particular location from the context in which the
  data are being used
• i.e. individual bytes, a word, a longword, etc
• 01100011 01100101 01000100 01000000
• Bytes 99(10, 101 (10, 68 (10, 64(10
• Two byte words 24,445 (10 and 17,472 (10
• Longword 1,667,580,992 (10

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Numbers
Natural Numbers Zero and any number obtained by
repeatedly adding one to it. Examples 100, 0,
45645, 32
Negative Numbers A value less than 0, with a
sign Examples -24, -1, -45645, -32
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Numbers (Contd)
Integers A natural number, a negative number,
zero Examples 249, 0, - 45645, - 32
Rational Numbers An integer or the quotient of
two integers Examples -249, -1, 0, ¼ , - ½
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Natural Numbers
How many ones are there in 642?
600 40 2 ? Or is it 384 32 2 ? --
Octal Or maybe 1536 64 2 ? -- Hexadecimal
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Natural Numbers
642 is 600 40 2 in BASE 10 The base of a
number determines the number of digits and the
value of digit positions
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Positional Notation
Continuing with our example 642 in base 10
positional notation is 6 x 10² 6 x
100 600 4 x 10¹ 4 x 10
40 2 x 10º 2 x 1 2
642 in base 10
The power indicates the position of the number
This number is in base 10
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Positional Notation
R is the base of the number
As a formula
 dn Rn-1 dn-1 Rn-2 ... d2 R d1
n is the number of digits in the number
d is the digit in the ith position in the
number
642 is  63 102  42 10  21
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Positional Notation
What if 642 has the base of 13? 642
in base 13 is equivalent to 1068 in base 10
6 x 13² 6 x 169 1014 4 x 13¹
4 x 13 52 2 x 13º 2 x 1
2 1068 in base 10
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Representing Real Numbers

• Real numbers have a whole part and a fractional
  part. For example 104.32, 0.999999, 357.0, and
  3.14159 the digits represent values according to
  their position, and those position values are
  relative to the base.
• The positions to the right of the decimal point
  are the tenths position (10-1 or one tenth), the
  hundredths position (10-2 or one hundredth), etc.

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Representing Real Numbers (Contd)

• In binary, the same rules apply but the base
  value is 2. Since we are not working in base 10,
  the decimal point is referred to as a radix
  point.
• The positions to the right of the radix point in
  binary are the halves position (2-1 or one half),
  the quarters position (2-2 or one quarter), etc.

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Representing Real Numbers (Contd)

• 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.

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Representing Real Numbers (Contd)

• Likewise, a binary floating point value is
  defined by the following formula
• sign mantissa 2exp

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Representing Real Numbers (Contd)

• Scientific notation is a term with which you may
  already be familiar, so we mention it here.
  Scientific notation is a form of floating-point
  representation in which the decimal point is kept
  to the right of the leftmost digit.
• For example, 12001.32708 would be written as
  1.200132708E4 in scientific notation.

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Representing Text

• To represent a text document in digital form, we
  simply need to be able to represent every
  possible character that may appear.
• There are finite number of characters to
  represent. So the general approach for
  representing characters is to list them all and
  assign each a binary string.
• A character set is simply a list of characters
  and the codes used to represent each one. By
  agreeing to use a particular character set,
  computer manufacturers have made the processing
  of text data easier.

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Alphanumeric Codes

• American Standard Code for Information
  Interchange (ASCII)
• 7-bit code
• Since the unit of storage is a bit, all ASCII
  codes are represented by 8 bits, with a zero in
  the most significant digit
• H e l l o W o r l d
• 48 65 6C 6C 6F 20 57 6F 72 6C 64
• Extended Binary Coded Decimal Interchange Code
  (EBCDIC)

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The ASCII Character Set

• ASCII stands for American Standard Code for
  Information Interchange. The ASCII character set
  originally used seven bits to represent each
  character, allowing for 128 unique characters.
• Later ASCII evolved so that all eight bits were
  used which allows for 256 characters.

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The ASCII Character Set (Contd)
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The ASCII Character Set (Contd)

• Note that the first 32 characters in the ASCII
  character chart do not have a simple character
  representation that you could print to the
  screen.

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The Unicode Character Set

• The extended version of the ASCII character set
  is not enough for international use.
• The Unicode character set uses 16 bits per
  character. Therefore, the Unicode character set
  can represent 216, or over 65 thousand,
  characters.
• Unicode was designed to be a superset of ASCII.
  That is, the first 256 characters in the Unicode
  character set correspond exactly to the extended
  ASCII character set.

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The Unicode Character Set (Contd)
A few characters in the Unicode character set