NumberING Systems: bInary, decImal, and hexadecImal

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DFSC 1317 Introduction to Digital Forensics and
Information Assurance

• 02
• NumberING Systems bInary, decImal, and
  hexadecImal

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Basic operation performed by a computer

• Arithmetic Operations Addition, subtraction,
  multiplication and division
• Logical operations the sign or the comparative
  magnitude of two numbers
• Data transfer Moving data from one location to
  another in the memory.
• Input-output operations Controlling the
  reading/writing of information into or out of the
  computer

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On Digital Computers

• Digital computers store numbers in an entity (or
  unit) called a word which consists of a string of
  binary digits, or bits. Various number systems
  are used to represent mathematical numbers. Some
  commonly used number systems are hexadecimal
  (base 16), decimal (base 10), octal (base 8), and
  binary (base 2). For example, in a decimal
  system the number 8,410 is represented in powers
  of ten as
• 8?103 4?102 1?101 0?100 8000 400 10
  0 8,410

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On Digital Computers (Contd)

• A method known as the doubling procedure is as
  follows. Given a decimal number N, it can be
  decomposed as
• N 2Q1 R1 (Q1 N/2 - remainder)
• Q1 2Q2 R2 (Q2 Q1/2 - remainder)
• . .
• . .
• . .
• Qk 0 Rk1 etc.
• The corresponding binary number is obtained by
  writing the remainders Rk1, Rk, ... , R1 in the
  reverse order as
• B Rk1RkRk-1 ... R1

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Example

• Convert the decimal number N 8,410, to a binary
  number.
• Solution
• Perform sequential division by 2 as follows
• 8,410 (2?4,205) 0 65 (2?32) 1
• 4,205 (2?2,102) 1 32 (2?16) 0
• 2,102 (2?1,051) 0 16 (2?8) 0
• 1,051 (2?525) 1 8 (2?4) 0
• 525 (2?262) 1 4 (2?2) 0
• 262 (2?131) 0 2 (2?1) 0
• 131 (2?65) 1 1 (2?0) 1
• The binary equivalent of 8,410 is then given by
  collecting the remainder digits from the last to
  the first
• 10000011011010 1?1213 0?212 0?211 0?210
  0?29 0?28 1?27 1?26 0?25 1?24 1?23
  0?22 1?21 0?20

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Representations of Numbers

• Numbers are usually represented using the
  normal form notation. That is,
• x m.10E for 10-1 lt ?m? lt 1
• where for x ? 0, m is called the mantissa and E
  is the exponent. By convention, the number zero
  has the normal notation, 0.100.

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Significant Digits

• If a number is written in standard decimal,
  floating-point form, or in normal form such that
• x 0.d1 d2 d3 ... dk?10n
• with d1 ? 0 and dk ? 0, we say that this number
  has k significant digits (or significant figures)
  which indicates those digits that can be used
  with a confidence relative to the true value of
  the number.

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Significant Digits (Contd)

• Note that the zeros which are used only to shift
  the decimal point are not counted as significant
  figures. The leading zeros may or may not be
  significant. For example,
• x 0.0002815 has 4 significant figures!
• x 1,200 may have 4 significant figures!
• Some examples are
• 46.45072800 0.46450728?102 (with 8
  significant digits)
• -335.12 -0.33512?103 (with 5
  significant digits)
• 0.00517 0.517?10-3 (with 3
  significant digits)
• 0.74 0.74?100 (with 2
  significant digits)

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Computer Representation of Numbers

• The decimal equivalent of the binary number
  represented in Figure 1 is given by
• -(0?26 0?25 0?24 1?23 0?22 1?21
  1?20)
• -(0 0 0 8 0 2 1)
• -11

• Figure 1 Binary representation of an integer
  using an
• 8 bit word (or Byte)

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Example

• Determine the largest integer that can be
  represented by an 8 bit machine.
• Solution
• Imax (1?26 1?25 1?24 1?23
  1?22 1?21 1?20)
• ( 64 32 16 8 4
  2 1 )
• (127)
• (27 - 1)
• In general
• Imax 2(n -1) - 1 Imin -2(n -1) -
  1
• For a binary computer utilizing 32 bit words,
• Imax 2,147,483,647

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Floating-Point Representation

• A floating-point number is written as
• x (sign)m.b(sign)E
• where m is the mantissa, b is the base (b 2
  for a binary system), and E is the exponent.

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Example

• Determine the smallest, positive, nonzero,
  floating point number that can be represented by
  an eight bit machine using binary system with one
  bit spared for the sign of the mantissa, one bit
  for the sign of the exponent, and two bits for
  the digits of the exponent

Solution m (0?23 0?22 0?21 1?20) m
( 0 0 0 1 ) 1 E
-(1?21) (1?20) -(21) - 3 Number 1?2-3
(which is equal to 0.1250 in decimal system)
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Decimal Numeral System

• Base 10 positional notation
• 1010 1103 0102 1101 0100
• 11000 0100 110 01
• 1010
• Number set (0, 1, 2, , 8, 9)
• Operations addition, multiplication, etc.
• Numbers of same value are longer than Hex but
  shorter than Bin
• Not so easy to convert into Bin or Hex

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Binary Numeral System

• Base 2 positional notation
• 1010 (B) 123 022 121 020
• 18 04 12 01
• 10 (in decimal)
• Number set (0 1)
• Operations addition, multiplication, etc.
• Addition 1001 1100 10101
• Good for computer systems logical gates with
  only two different values or states Can be
  easily converted into Hex (4 Bin bits ?1 Hex bit)

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Hexadecimal Numeral System

• Base 16 positional notation
• 1010 (H) 1163 0162 1161 0160
• 1 4096 0 256 1 16 0 1
• 4112 (in decimal)
• Number set (0 9, A, B, C, D, E, F)
• Operations addition, multiplication, etc.
• Numbers of same value are shorter (vs. binary and
  decimal)
• Can be easily converted into Binary (1 Hex bit ?
  4 Bin bits)

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Binary ?? Hexadecimal??Decimal
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File Extension ? Description ? Hex Signature BMP ? Windows Bitmap Image ? 42 4D xx xx GIF ? Graphics Interchange Format File?47 49 46 38 39 61 xx xx GIF ? GIF87a ? 47 49 46 38 37 61 xx xx GIF ? GIF87b ? Trailer 00 3B xx xx ICO ? Windows Icon File ? 00 00 01 00 xx xx JPEG, JPG ? Compressed graphics file ? FF D8 FF E0 xx xx 4A 46 49 46 00 xx xx FF D8 FF E1 xx xx 45 78 69 66 00 xx xx PNG ? Portable Network Graphics File ? 89 50 4E 47 0D xx xx TIFF, TIF ? Tagged Image File Format ? 49 49 2A 00 xx xx
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Hash Table