Chapter 1.5 The Binary System

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Chapter 1.5 The Binary System

• CSCI 3

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Basic Concepts Behind the Binary System

• In the decimal system, things are organized into
  columns
• H T O
• 1 9 3
• such that "H" is the hundreds column, "T" is the
  tens column, and "O" is the ones column. So the
  number "193" is 1-hundreds plus 9-tens plus
  3-ones
• ones column meant 100, the tens column meant
  101, the hundreds column 102 and so on
• the number 193 is really (1102)(9101)(310
  0).

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The base ten system

• A. Base ten system (375)
• 3 7 5 Representation
• Hundred Ten One
• (102) (101) (100)
• Positions quantity
• Figure 1.15, Page 41

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The base binary system

• A. Base two system (1011)
• 1 0 1 1 Representation
• Eight Four Two One
• (23) (22) (21) (20)
• Positions quantity
• Figure 1.15, Page 41

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Basic Concepts Behind the Binary System

• The binary system works under the exact same
  principles as the decimal system, only it
  operates in base 2 rather than base 10.
• In other words, instead of columns being
  102101100 they are 222120

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Basic Concepts Behind the Binary System

• Instead of using the digits 0-9, we only use 0
  and 1.
• Examples What would the binary number 1011 be in
  decimal notation?
• It would be 230211
• Which will be 82111

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Basic Concepts Behind the Binary System

• Try converting these numbers from binary to
  decimal
• 10
• 111
• 10101
• 11110
• Remember
• 24 23 22 21 20

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Basic Concepts Behind the Binary System

• Try converting these numbers from binary to
  decimal
• 10(121) (020) 20 2
• 111 (122) (121) (120) 4217
• 10101 (124) (023) (122) (021)
  (120)16040121
• 11110 (124) (123) (122) (121)
  [phone]

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Basic Concepts Behind the Binary System

• Binary Addition
• Consider the addition of decimal numbers
• 23
• 48
• ___
• We begin by adding 3811. Since 11 is greater
  than 10, a one is put into the 10's column
  (carried), and a 1 is recorded in the one's
  column of the sum. Next, add (24) 1 (the one
  is from the carry)7, which is put in the 10's
  column of the sum. Thus, the answer is 71.

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Basic Concepts Behind the Binary System

• Binary addition works on the same principle, but
  the numerals are different. Begin with one-bit
  binary addition
• 0 0 1
• 0 1 0
• 0 1 1

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Basic Concepts Behind the Binary System

• 11 carries us into the next column. In decimal
  form, 112. In binary, any digit higher than 1
  puts us a column to the left (as would 10 in
  decimal notation). The decimal number "2" is
  written in binary notation as "10"
  (121)(020). Record the 0 in the ones column,
  and carry the 1 to the twos column to get an
  answer of "10." In our vertical notation,
• 1
• 1
• 10

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Basic Concepts Behind the Binary System

• Example
• 1010
• 1111
• Step one Column 20 011.Record the 1.
  Temporary Result 1 Carry 0
• Step two Column 21 1110. Record the 0,
  carry the 1.Temporary Result 01 Carry 1
• Step three Column 22 101 Add 1 from carry
  1110. Record the 0, carry the 1.Temporary
  Result 001 Carry 1
• Step four Column 23 1110. Add 1 from carry
  10111.Record the 11. Final result 11001

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Basic Concepts Behind the Binary System

• Always remember
• 000
• 101
• 1110

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Basic Concepts Behind the Binary System

• Try a few examples of binary addition
• 111110
• 101111
• 111111

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Basic Concepts Behind the Binary System

• Answer
• 1111101101
• 1011111100
• 1111111110

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Basic Concepts Behind the Binary System

• How to find the binary representation of a
  positive integer?

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Basic Concepts Behind the Binary System
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Basic Concepts Behind the Binary System

• Example
• 52 in decimal ? 110100 in binary
• What about 10 (1010)? Or other numbers?

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Basic Concepts Behind the Binary System

• To extend binary notation to accommodate
  fractional number values, we use a radix point.
• 1011.0111
• radix point

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Basic Concepts Behind the Binary System
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Homework 4