Review: why computer must use binary numbers

The reason why a computer must use binary numbers to store all its information:

  • An (electronical) computer stores all its information in the (computer) memory

  • Computer memory consists of (electronical) switches (that can be in the on (= 1) or off (= 0)

  • Each cell of the computer can store a binary number as ON/OFF states:

        

    Therefore:

      • All information inside a computer must be stored as binary numbers

The Number Systems

  • Number (numeral) system: (From: Wikipedia: click here )

      • A number (numeral) system = a writing system for expressing numerical values

  • We will first study:

      1. How humans use the decimal number system to represent unsigned integer (whole number) values

      2. How computers use the binary number system to represent unsigned integer (whole number) values

  • Later, we will study how to represent other things:

      • Signed integer values, floating point numbers, letters, etc

The univeral way to represent numerical values zero, one, ..., nine

A universal way to represent the values zero, one, ..., nine is to use tally marks:

Note:

  • Numbers like 0, 1, 2, ..., 9 are not universal, but cultural ("Western Civilization)

  • Example: (ancient) Romans use different symbols:   I, II, III, IV, V, VI, ...

 

For simplicity, I will use number of dots to represent a numerical value - like this: 

 ( ) = zero   (•) = one   (••) = two  (•••) = three  (••••) = four    ...
 and so on...

The decimal number system and decimal numbers

  • The decimal (deca = 10) number system uses 10 symbols (= digits) to represent 10 different values.

  • The values that we need to represent in the decimal (= 10) number system are: 

        ( )   (•)   (••)   (•••)   (••••)   (•••••)   
  (••••••)   (•••••••)   (••••••••)   (•••••••••)

  • Western cultures use the following symbols to represent the 10 values: 

        ( )   (•)   (••)   (•••)   (••••)   (•••••)
   0     1      2      3       4         5        
   (••••••)   (•••••••)   (••••••••)   (•••••••••)   
      6           7           8             9

Positional number systems: the value represented by a given digit symbol

Positional number system:

  • The value of a digit (= symbol) depends on its position within the (decimal) number

The decimal number system:

  • The value of a digit is multiplied by 10 for each position further to the left

    Example: 

         2  represents the value (••)
   3  represents the value (•••)    

  23  represents the value

       (••  ••  ••  ••  ••  ••  ••  ••  ••  ••)   // = 10 times (••)   
       (•••)

   for a total of twenty three • !!!

The binary number system

  • The binary (bin = 2) number system is based on the number 2

  • The binary number system uses 2 digits to represent 2 different values: 

       digit  0  represents:  ( )     i.e.: the value zero 
 digit  1  represents:  (•)     i.e.: the value one

  • A binary number is made up of a number of binary digits

    Examples of binary numbers: 

          0
    1
   01  (same as 1 because leading 0's can be omitted) 
   10
  111

The binary number system

The value that is represented by a binary number:

  • The value of a digit in a binary number also depends on its position in the (binary) number:         

      • The value of a binary digit is multiplied by 2 for each position further to the left          

    Example: binary numbers 

         1  represents the quantity: (•)  (= ONE)     

  11  represents the quantity: (• • •)  (= THREE)

            (• •)           // = 2 times (•)
	     •

 111  represents the quantity: (• • • • • • •)  (= SEVEN)     

            (• •) (• •)      // = 22 times (•)
            (• •)            // = 2 times (•)
	     •

The value represented by a binary number - general expression

  • Formula to compute the decimal value that is represented of a binary number: 

       Given a binary number: 

      dndn-1...d1d0   

 where di is a binery digit (0 or 1)

 The decimal value represented by the binary number is:  

      dn×2n + dn-1×2n-1 ... + d1×21 + d0×20

    Example: 

        Binary number     Value represented in decimal   
 ---------------   --------------------------------
     101            1×22 + 0×21 + 1×20 = 4 + 1 = 5  
     110            1×22 + 1×21 + 0×20 = 4 + 2 = 6

Sample of some numerical values and their representation in decimal and in binary 

           Representation in the       Representation in the
  Value    Decimal number system       Binary number system
 -----------------------------------------------------------------
  Zero                0                     0
  One                 1                     1
  Two                 2                    10
  Three		      3			   11
  Four		      4			  100
  Five		      5			  101
  Six	              6			  110
  Seven		      7			  111
  Eight		      8			 1000
  Nine		      9			 1001
  Ten		     10		         1010
  Eleven	     11			 1011
  Twelve	     12			 1100
  Thirteen	     13			 1101
  Fourteen	     14			 1110
  Fifteen	     15			 1111

Question:   do you see an ambiguity problem ??? (Does 10 represent the value ten or two ?)

Notation used in CS255 to make a number unambiguous
 

Notation used in CS255 to make a number unambiguous:

  • We write 10(10) for decimal (base 10) number 10

    I.e.:   10(10) represents the value ten

  • We write 10(2) for binary (base 2) number 10

    I.e.:   10(2) represents the value two

Comment:

  • When I state that some number is a binary number or a decimal number, there would not be any ambiguity !!

You should now understand this sentence....

How do you read this sentence:

 

You should now understand this sentence....

How do you read this sentence:

There are 10(2) (= 2) types of people in the world: those who understand binary and those who don't