Number Systems

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Number Systems
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Number Systems

• Prehistory
• Unary, or marks /
• /////// 7
• /////// ////// /////////////
• Grouping lead to Roman Numerals
• VII V VVII XII
• Better, Arabic Numerals
• 7 5 12 1 x 10 2

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Arabic Numerals

• 345 is really
• 3 x 100 4 x 10 5 x 1
• 3 x 102 4 x 101 5 x 100
• 3 is the most significant symbol (carries the
  most weight)
• 5 is the least significant symbol (carries the
  least weight)
• Digits (or symbols) allowed 0-9
• Base (or radix) 10

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• Base 10 is a special case of positional number
  system
• First used over 4000 years ago in
  Mesopotamia(Iraq)
• Base 60 (Sexagesimal)
• Digits 0..59 (written differently)
• 5,4560 5 x 60 45 x 1 34510
• Positional number systems a great advance in
  mathematics
• Why?

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• Try multiplication in (non-positional) Roman
  numerals(!)
• XXXIII (33 in decimal)
• XII (12 in decimal)
• ---------
• XXXIII
• XXXIII
• CCCXXX
• -----------
• CCCXXXXXXXXXIIIIII
• CCCLXXXXVI
• CCCXCVI 396
• The Mesopotamians wouldnt have had this problem.

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• There are many ways to represent a number
• Representation does not affect computation result
• LIX XXXIII LXXXXII (Roman)
• 59 33 92 (Decimal)
• Representation affects difficulty of computing
  results
• Computers need a representation that works with
  fastelectronic circuits

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Binary positional numbers work great with
2-state devices

• Digits (symbols) allowed 0, 1
• Binary Digits, or bits
• Base (radix) 2
• 10012 is really
• 1 x 23 0 x 22 0 X 21 1 X 20
• 910
• 110002 is really
• ?
• ?
• Computers usually multiply Arabic numerals by
  convertingto binary, multiplying and converting
  back (much as us with Roman numerals)

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Octal number system

• Digits (symbols) 0 7
• Base (radix) 8
• 3458 is really
• 3 x 82 4 x 81 5 x 80
• 192 32 5
• 22910
• 10018 is really
• ?
• ?
• ?
• In C, octal numbers are represented with a
  leading 0 (0345 or 01001).

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Hexadecimal number system

• Digits (symbols) allowed 0 9, a f
• Base (radix) 16

10

• A316 is really
• A x 161 3 x 160
• 160 3
• 16310
• 3E816 is really
• 3 x 162 E x 161 8 x 160
• 3 x 256 14 x 16 8 x 1
• 768 224 8
• 100010
• 10C16 is really
• ?
• ?
• ?
• ?
• In C, hex numbers are represented with a leading
  0x(0xa3 or 0x10c).

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• For any positional number system
• Base (radix) b
• Digits (symbols) 0..b 1
• Sn-1Sn-2.S2S1S0
• Use sum to transform any base to decimal

n-1
Value S(Sibi)
i0
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Decimal ? Binary

• Divide decimal value by 2 until the value is 0
  (see book)
• Know your powers of two and subtract
• 256 128 64 32 16 8 4 2 1
• Example 42
• What is the biggest power of two that fits?
• What is the remainder?
• What fits?
• What is the remainder?
• What fits?
• What is the binary representation?

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Binary ? Octal

• Group into 3s starting at least significant
  symbol
• Add leading 0s if needed (why not trailing?)
• Write 1 octal digit for each group
• Example
• 100 010 111 (binary)
• 4 2 7 (octal)
• 10 101 110 (binary)
• 2 5 6 (octal)

• Octal ? Binary
• Write down the 3-bit binary code for each octal
  digit

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Binary ? Hex

• Group into 4s starting al least significant
  symbol
• Adding leading 0s if needed
• Write 1 hex digit for each group
• Example
• 1001 1110 0111 0000
• 9 e 7 0
• 0001 1111 1010 0011
• 1 f a 3

• Hex ? Binary
• Write down the 4 bit binary code for each hex
  digit
• Example
• 3 9 c 8
• 0011 1001 1100 1000

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• Hex ? Octal
• Do it in 2 steps, hex ? binary ? octal
• Decimal ? Hex
• Do it in 2 steps, decimal ?binary?hex
• Why use hex and octal?

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Negative Integers

• Most humans precede number with - (e.g., -2000)
• Accountants, however, use parentheses (2000)
• Sign-magnitude
• Example -1000 in hex?
• 100010 3 x 162 e x 161 8 x 160
• -3E816

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• Mesopotamians used positional fractions
• Sqrt(2)
• 1.24,51,1060 1 x 600 24 x 60-1 51 x 60-2
• 10 x 60-3
• 1.414222
• Most accurate approximations until the
  Renaissance
• What is 3E.8F16?
• How about 10.1012?

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f f . . . f f f .
f f f n-1 n-2 2 1
0 -1 -2 -3
Binary point
2-1 .5 2-2 .25 2-3 .125 2-4 .0625
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Converting decimal to binary fractions

• Consider left and right of the decimal point
  separately.
• The stuff to the left can be converted to binary
  as before.
• Use the following algorithm to convert the
  fraction

• Different bases have different repeating
  fractions.
• 0.810 0.1100110011002 0.11002
• Numbers can repeat in one base and not in another.

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• What is 2.2 in
• Binary
• Hex