IST 220

Week 5: DATA REPRESENTATION

 

NUMBER CONVERSIONS

In chapter 4, you will learn how to convert between the decimal, binary, octal and hexadecimal numbering systems. The decimal numbering system is the numbering system we use in our daily lives. The binary numbering system is the method used to represent data in the computer system. The binary numbering system is made up of a serial of 1’s and 0’s. 

Since it is difficult to read a series of binary numbers to view the contents of a memory location or a register, two shorthand versions of the binary numbering systems were developed to make viewing binary numbers easier. The hexadecimal and octal numbering systems were designed for that purpose.

First you will look at the characteristics of all numbering systems and then you will learn how to convert from one numbering system to another. 

Topics:

List the characteristics of numbering systems

List the characteristics of each numbering system:

 
Decimal Numbering System - Base 10

Binary Numbering System - Base 2

Hexadecimal Numbering System - Base 16

Octal Numbering System - Base 8
 

 Convert each number type to the other three numbering systems:  
Decimal to Binary

Binary to Decimal

Binary to Octal

Octal to Binary

Binary to Hexadecimal

Hexadecimal to Binary 
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Characteristics of Numbering Systems

    1. The smallest symbol is 0.
    2. The largest symbol is the base minus 1.
    3. The total number of symbols equals the base.
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CHARACTERISTICS OF THE DECIMAL NUMBERING SYSTEM – BASE 10 

    1. Smallest symbol – 0
    2. Largest symbol – 9
    3. Total number of symbols – 10
List of symbols - (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
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CHARACTERISTICS OF THE BINARY NUMBERING SYSTEM – BASE 2

    1. Smallest symbol – 0
    2. Largest symbol – 1
    3. Total number of symbols – 2
List of symbols - (0, 1)
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CHARACTERISTICS OF THE HEXADECIMAL NUMBERING SYSTEM – BASE 16

    1. Smallest symbol – 0
    2. Largest symbol – 15 (F) Because 15 is actually 2 digits, 1 and 5, the 15 is changed to the letter F.
    3. Total number of symbols – 15
List of symbols – (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)

A = 10

B = 11

C = 12

D = 13

E = 14

F = 15
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CHARACTERISTICS OF THE OCTAL NUMBERING SYSTEM – BASE 8 

    1. Smallest symbol – 0
    2. Largest symbol – 7
    3. Total number of symbols – 8
List of symbols (0, 1, 2, 3, 4, 5, 6, 7)

Converting from the Decimal to the Binary Numbering System:

You will convert numbers to 8 bit binary numbers. The process to convert larger numbers is exactly the same.
      You need to first write out the positional values of the binary numbering system. But before you do, let’s first take a look at the positional values in the decimal numbering system. 

      We know the number 89210 (The 10 subscript indicates the number is a decimal number) has the value eight hundred ninety two because of the position of each digit in the number. The rightmost position is the 1’s position, the next position is the 10’s position, and the leftmost position is the 100’s position. The 1’s position is actually 100 – anything to the 0 power is always 1. The 10’s position is actually 101 - which is 10. The 100’s position is actually 102. If there was a number before the 8, it would be in the 1000’s position or 103. 

      The same principles are true for the other numbering systems. If you look at a binary number 1012 (The 2 subscript indicates the number is a binary number). The rightmost position is the 1’s position, the next position is the 2’s position and the leftmost position is the 4’s position. The 1’s position is actually 20 – anything to the 0 power is always 1. The 2’s position is actually 21 - which is 2. The 4’s position is actually 22 . If there was a number before the leftmost 1, it would be in the 8’s position or 23. 

      Let’s start converting: 
      Write out the positional values of the binary numbering system for 8 positions.

      27 26 25 24 23 22 21 20 which is actually 

      128 64 32 16 8 4 2 1 (you should be able to see a pattern – start with 1 and double the numbers for 8 positions.)
       

    1. Find the largest number that will divide into the decimal number from the 

      2. numbers listed (128, 64, 32, 16, 8, 4, 2, 1)
       
    2. Place a 1 in that position.
  1. Subtract the number from the decimal number.
  2. Continue with steps 2 – 4 until you have a remainder of 0.
  3. Place a 0 in all empty locations.
  4. The result will give you the binary equivalent of the decimal number.

Let’s look at an example:

    Convert 5910 to binary:
     
  • Write out the positional values for the binary numbering system.
       
      128
      64
      32 
      16
      8
      4 
       2
      1
  • The largest number that will divide into the decimal number 59 is 32.
  • Place a 1 in that position.
       
         
      1
      		          
      128
      64
      32 
      16
      8
      4 
       2
      1
    • Subtract 32 from 59. (59 – 32 = 27)
    • The largest number that will divide into the decimal number 27 is 16.
    • Place a 1 in that position.
       
         
      1
      1
      		        
      128
      64
      32 
      16
      8
      4 
       2
      1
    • Subtract 16 from 27. (27 – 16 = 11)
    • The largest number that will divide into the decimal number 11 is 8.
    • Place a 1 in that position.
       
         
      1
      1
      1
      		      
      128
      64
      32 
      16
      8
      4 
       2
      1
    • Subtract 8 from 11. (11 – 8 = 3)
    • The largest number that will divide into the decimal number 3 is 2.
    • Place a 1 in that position.
       
         
      1
      1
      1
      		  
      1
       
      128
      64
      32 
      16
      8
      4 
       2
      1
    • Subtract 2 from 3. (3 - 2 = 1)
    • The largest number that will divide into the decimal number 1 is 1.
    • Place a 1 in that position.
       
         
      1
       1
      1 
       
      1 
      1 
      128
      64
      32 
      16
      8
      4 
       2
      1
    • Subtract 1 from 1.
    • You now have a result of 0. Place a 0 in the remaining positions.
       
       0
      0 
      1
       1
      1 
       0
      1 
      1 
      128
      64
      32 
      16
      8
      4 
       2
      1

       
    • The binary equivalent of 5910 is 001110112 
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Converting from the Binary to the Decimal Numbering System:

To convert to a decimal number: 
    1. Place the positional values (128, 64, 32, 16, 8, 4, 2, 1) under the binary number. 
    2. In every location where there is a 1 bit, add the numbers together.
    3. The result will be the decimal equivalent.
Let’s look at an example:

Convert this number 101010112 to a decimal number.

  1. Place the positional values under the binary number.
       
      1
      0 
      1
       0
      1 
       0
      1 
      1 
      128
      64
      32 
      16
      8
      4 
       2
      1
  1. In every location where there is a 1 bit, add the numbers together.
    (128 + 32 + 8 + 2 + 1) = 17110
  1. Decimal equivalent of 101010112 is 17110.
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Converting from the Binary to the Octal Numbering System:

The Octal Numbering System breaks a binary number into groups of three and represents each group of three binary numbers with one octal number. The reason the binary number is broken up into groups of three is because the largest octal number that can be represented is 7 (the base 8 minus 1). It only takes 3 binary digits to represent the largest octal number 7. 
1 1 1 = (4 + 2 + 1) = 7

4 2 1 

To convert to an Octal number:
    1. Break the binary number into groups of three binary digits, starting with the rightmost bit.
    2. Convert each group of three binary digits into the octal equivalent separately.
Let’s look at an example:
      010101002
  1. Break the number into groups of three binary digits, starting with the rightmost bit.
      2.  
      01010100 = Group1 = 100

      Group 2 = 010

      Group 3 = 01 (you can add an extra 0 to the leftmost side, it will not change the value of the number.)

  1. Convert each group of three digits separately. Place the binary positional values below each bit. Every place there is a 1 bit, add the numbers together.
    Group 1 = 1 0 0 = 4

    4 2 1

    Group 2 = 0 1 0 = 2

    4 2 1

    Group 3 = 0 1 = 1

    4 2 1 
     
  1. The octal equivalent of 010101002 is 4218 
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Converting from the Octal to the Binary Numbering System:

To convert from a binary number to an octal number, you need to list the three digit binary equivalent for each octal number. 
    1. List the binary number base for 3 places.


      2. 22 21 20 or 4 2 1
       
       

    2. You want to add the numbers 4 or 2 or 1 to add up to the octal number.


      3. For example: 6 is 4 + 2. You need to place a 1 in the 4 and 2 positions and a 0 in the 1 position.
       

      1 1 0

      4 2 1

      For example: 5 is 4 + 1. You need to place a 1 in the 4 and 1 positions and a 0 in the 2 position.
       
      1 0 1

      4 2 1
       

    3. Convert each octal number into a group of three binary digits.
    For example:

    4528
     
    4 = 1 0 0 5 = 1 0 1 2 = 0 1 0

    4 2 1 4 2 1 4 2 1
     

  1. The binary equivalent of 4528 is 1001010102
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Converting from the Binary to the Hexadecimal Numbering System:

The Hexadecimal Numbering System breaks a binary number into groups of four and represents each group of four binary numbers with one hexadecimal number. The reason the binary number is broken up into groups of four is because the largest hexadecimal number that can be represented is 15 (the base 16 minus 1). It only takes 4 binary digits to represent the largest hexadecimal number 15. 
1 1 1 1 = (8 + 4 + 2 + 1) = 15

8 4 2 1 

To convert to a Hexadecimal number: 
  1. Break the binary number into groups of four binary digits, starting with the rightmost bit.
  2. Convert each group of four binary digits into the hexadecimal equivalent separately.
Let’s look at an example:
010101002 
  1. Break the number into groups of four binary digits, starting with the rightmost bit.


    2. 01010100 = Group1 = 0100

    Group 2 = 0101

    Convert each group of three digits separately. Place the binary positional values below each bit. Every place there is a 1 bit, add the numbers together.

    Group 1 = 0 1 0 0 = 4

    8 4 2 1

    Group 2 = 0 1 0 1 = ( 4 + 1 ) = 5

    8 4 2 1
     
     

  2. The hexadecimal equivalent of 010101002 is 4516
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Converting from the Hexadecimal to the Binary Numbering System:

To convert from a hexadecimal number to a binary number, you need to list the four digit binary equivalent for each hexadecimal number.
      List the binary number base for 4 places.

      23 22 21 20 or 8 4 2 1
       
       

    1. You want to add the numbers 8 or 4 or 2 or 1 to add up to the hexadecimal number.


      2. For example: A is 8 + 2. You need to place a 1 in the 8 and 2 positions and a 0 in the 1 and 4 positions.

      1 0 1 0

      8 4 2 1

      For example: 9 is 8 + 1. You need to place a 1 in the 8 and 1 positions and a 0 in the 2 and 4 positions.

      1 0 0 1

      8 4 2 1
       
       

    2. Convert each hexadecimal number into a group of four binary digits.
For example:   B316
    B = 1 0 1 1   3 = 0 0 1 1

    8 4 2 1 8 4 2 1 
     
     

  1. The binary equivalent of B316 is 101100112
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