Addition & Subtraction

Chapter 2: Arithmetic for Computers

Addition:

Binary Addition

·         The four basic rules for binary addition are:

1.      0 + 0 = 0           Sum of 0 with carry of 0

2.      0 + 1 = 1           Sum of 1 with carry of 0

3.      1 + 0 = 1           Sum of 1 with carry of 0

4.      1 + 1 = 10         Sum of 0 with carry of 1

Example:10+10=100 

  carry 1      1     0

+            1     0

        1    0     0

• ·         10 + 10 = 100, because  in the basic rule 0+0=0 and 1+1=10. Since 1 is largest digit in the binary system , any sum is greater that  1 requires a digit to be carried over. 
  
  
  
  
  Addition Of Hexadecimal Operations
  ·        The hexadecimal system(base 16) with its 16 digits (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}
  ·        If any additions of hexadecimal operations have digits more that F like G,H,I,J,K, the addition does not have answer because the digits more that F is undefined.
  ·        Example:  123EFG + 456ABC this type of questions is undefined questions .
  ·         Example of additions of hexadecimal operations:
  1.      66₁₆+37₁₆
  2.      CD₁₆+AB₁₆
  
  
   Solution:
  
  
  1.                          6           6₁₆
                +          3           7 ₁₆  
  
  
                                9           13₁₆ 
  
    9           13_{16   =}    9D₁₆
  
  
  
  
  2.                ^{1      1}C            D₁₆              (C=12₁₀ , D=13₁₀)
              +            A            B₁₆                 (A=10₁₀, B=11₁₀)
                                  1         7            8₁₆
                        D + B= 13₁₀ + 11₁₀ = 24₁₀
                                                           = 24₁₀ - 16₁₀ = 8₁₆ with a 1 carry
                       C+A+ carry from D+B that is 1₁₀ = 12₁₀ + 10₁₀ + 1₁₀ = 23₁₀
                                                            23₁₀ - 16₁₀ = 7₁₆ with a 1 carry
  
  
  Subtraction:
  Binary Subtraction
  ·        The four basis rules for binary subtraction are:
  1.     0 - 0 = 0
  2.     1 - 1 = 0
  3.     1 -0  = 1
  4.     10 - 1 = 1            0-1 with a borrow of 1   
  ·         A borrow is required in binary subtraction only when we need to subtract a 1 from a 0 .
  
  
  Example: a.)11-10
                   b.)100-011
                  Solution:  a.)             
     1          1
                            -      1         0
                                                       0          1
  
  
   
  
  
  b.)                      1 ⁰      0 ^{2 1    2}0
                      -     0         1          1
                             0      0          1
  
  
  • ·         Left column: When a 1 is borrowed, a 0 is left , so 0 – 0 = 0.
  • ·         Middle column: Borrow 1 from  left  column , making 2 which mean 10 in     binary,when 2 borrowed it become 1,then 1 – 1 =0 .
  • ·         Right column: Borrow 1 from middle column , making 10 then 10-1=1.
   
  
  
  
  
   Subtraction Of Hexadecimal Operations
  ·        The hexadecimal system(base 16) with its 16 digits (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}
  
  
  ·        If any subtraction of hexadecimal operations have digits more that F like G,H,I,J,K, the addition does not have answer because the digits more that F is undefined.
  ·        Example:  123EFG  -  456ABC this type of questions is undefined questions .
  ·        Example of subtraction of hexadecimal operations:
                  1. 75₁₆ - 21₁₆
                     2. 55₁₆ - 3B₁₆
   
• Solutions:
  1.                     7          5₁₆
              -           2          1₁₆
                           5         4₁₆
  
  
  2.                     5  ^{4    16+}5₁₆
              -           3          B₁₆
                          1          A₁₆       
  
  
  ·         Left  column   : When 5 is borrowed, a 4 is left, so 4 – 1 = 1 .
  ·         Right column :  Borrow 1 from left column , making 5₁₆+16₁₆=21₁₆  
                              ,so 21₁₆ – B₁₆ = A₁₆  
  
  
  
  
  
  
  CHOO YIE YUNG(B031210026)