Binary

Binary

1. Why binary?

   • needed to represent data and program instructions based of the way in which computers work
   • data system
     • processors
     • transitsors
     • separate states: digital
       • on or off
         • 1 and 0
     • continuous range: analogue

2. Representing information

   • only two digits (1 and 0)
     • can be combined into groups
       • 3 bits = 8 combos
       • 4 bits = 16 combos
       • 2 bits = 4 combos
   • needs 52 digits (upper and lower case of the alphabet) + punctuation symbols

3. Number systems

   • Denary system
     • based on 0-9
     • works on powers of 10
   • Binary system
     • Converting binary to denary
     • Converting denary to binary
     • based on 1s and 0s
     • works in powers of 2
     • bite and bytes

4. Binary arithmetic

   • Denary addition
     • 'carry over' is used if result is greater than 9
   • Binary addition
     • 'carry over' is used if result is greater than 1
   • can be manipulated the same way as denary

5. Signed and unsigned numbers

   • Sign and magnitude
     • MSB (left-most bit)
       • used to sign integers
     • 1 = negative
       • 10011100 = -28
     • 0 = positive
       • 00011100 = +28
   • negative numbers
   • 'signed' def
     • when an integer is positive or negative
   • denary
     • '+' and '-'

6. Adding signed integers

7. Two's compliment

   • Finding the two's compliment of a binary number
     • 1. flip all the bits
     • 2. add 1 to the flipped result
   • Converting two's compliment numbers back into denary
     • Method 1 - reverse calculation
       • 1. use the result from when finding 2's compliment
       • 2. flip all values
       • 3. add 1 to the result
     • Method 2 - place values
       • convert the binary number directly into denary by using the place values
       • if there's a 1 in the 128 place, this will equal to -128
     • Adding -28 to +28
       • !overflow!
       • result = 0

8. Binary shifts

   • Logical
     • same way as in denary
     • shifts either to left or right side (depending on the question)
   • Arithmetic
     • Left
       • identical to a left logical shift
       • MSB not included
       • multiplying
     • Right
       • dividing
       • add the MSB at the left
     • used with signed numbers expressed in 2's complement format
   • multiplication and division by powers of 10

9. Hexadecimal numbers

   • 16-number system
     • 0-15
     • 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

10. Converting binary to hexadecimal

    • 1.8 bits
    • 2. split into 2 4 bits
    • 3. convert each nibble into its denary number
    • 4. add the denarys together
    • 5. convert the two added denary's to hexadecimal and combine

11. Converting hexadecimal to binary

    • reverse of
    • 1. convert each digit to denary
    • 2. convert the denary numbers to 4-bit binary numbers

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