Introduction to Computer Science

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Introduction to Computer Science
Dr. Nagy Ramadan
E-mail Nagyrdo_at_yahoo.com
Lecture - 4
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Outline

• Part 1 - Binary Addition
• Part 2 - Binary Subtraction
• Part 3 - Subtraction Using the Complement Method
• Part 4 - Decimal Subtraction using 9s and 10s
  Complement
• Part 5 - Binary Subtraction using 1s and 2s
  Complement

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Binary Addition

• Rules for binary addition are
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0 with 1 to carry for the next
  column
• 1 1 1 1 with 1 to carry for the
  next column

Ex 1 Find the sum of the binary numbers 1101
110 and verify the result using
decimal numbers Solution
(10011)2 124 023 022 121 120
(19)10
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• Ex2 Perform the following binary addition
  operation then verify the result using decimal
  numbers 110101.101 10110.111
• Solution

(1001100.1)2 126 123 122 12-1
64 8 4 0.5
(76.5)10
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Binary Subtraction

• Rules for binary subtraction are
• 0 0 0
• 1 0 1
• 1 1 0
• 0 1 1 , with 1 borrowed from the next column

Ex 1 Use the direct binary subtraction to get
the result of 1100101 100111
Verify the result in decimal system. Solution
(10011)2 124 121 120 (19)10
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Subtraction Using the Complement Method

• The complement method allows performing
  binary subtraction in the form of binary addition
  which is much easier. This greatly simplifies the
  design of the electronic circuits of the digital
  computers.

• Examples
• Decimal Subtraction using 9s and 10s Complement
  
• Binary Subtraction using 1s and 2s Complement

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Decimal Subtraction using 9s and 10s Complement
Ex 1 Decimal subtraction using 9s complement
Solution
The number 6832 is the 9s complement of 3167
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Ex 2 Decimal subtraction using 10s complement
Solution
The 10s complement the 9s complement 1
The number 6833 is the 9s complement 1 of the
number 3167. Therefore, it is called the 10s
complement.
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Binary Subtraction using 1s and 2s Complement

• The 1s complement of a binary number is simply
  obtained by replacing every 1 by 0 , and every 0
  by 1.
• The 2s complement of a binary number can be
  obtained in two ways
• By adding 1 to the 1s complement.
• Start the binary number from right. Leave the
  binary digits unchanged until the first 1 appear,
  then replace every 1 by 0 , and every 0 by 1.

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Ex 1 Obtain the twos complement of the binary
number 1011010.110
First solution
Second solution
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Ex 2 Calculate the following binary
Subtraction 11101.101 1011.11 ,
then verify the result in decimal System.
Solution
(10001.111)2 16 1 0.5 0.125 (17. 875)10
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Important Note

• When using the complement methods in subtraction
  and having no additional 1 in the extreme left
  cell, then , this means a negative result.
• In this case, the solution is the negative of
  1s complement of the result (if using 1s
  complement initially), or the negative of 2s
  complement of the result (if using 2s complement
  initially).

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Ex 2 Calculate the following binary
Subtraction 1101.101 11011.11 ,
then verify the result in decimal System.
Solution
Therefore, the result - (the 1s complement of
10001.110) - 01110.001
Or - ( the 2s complement of 10001.111
) - 01110.001
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Questions