Subtraction

• Perform the following in base 10: 4291-2892

• 
    4 2 7 2
  - 2 8 7 3    
    -------
  
  2 < 4  so we need to borrow 1 from the 7 to the left of the 2.
  We add the 10 we borrowd from the 7 to the 2 to get 12 
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
  
  We can subtract 3 from 12 to get 9
  
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
          9
  
  again 6 < 7 so we need to borrow 1 from the 2 to the left of the 6 
  to get 1 and 16
  
     1 16
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
          9
  
  16 - 7 = 9
  
  
     1 16
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
        9 9
  
  Again we need to borrow 1 from the 4.
  And 11-8 = 3
  
   3 11 
     1 16
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
      3 9 9
  
  Finally we can subtract 2 from 3 to get 1
  
   3 11 
     1 16
        6 12
    4 2 7 1
  - 2 8 7 3    
    -------
    1 3 9 9
       

• Is there any difference in binary?
  • 0-0 = 0
  • 1-0 = 1
  • 1-1 = 0
  • 0-1 = 1 borrow a 1
• Perform the following $1001010110_2 - 100011101_2$

  
    1 0 0 1 0 1 0 1 1 02 
  -   1 0 0 0 1 1 1 0 12
   --------------------
                        2
  
  0-1 = 0, we need to borrow a 1.
  It becomes 10-1 = 1
  
                    0 10
    1 0 0 1 0 1 0 1 1 02 
  -   1 0 0 0 1 1 1 0 12
   --------------------
                      1 2
  
  The new problem is 0-0 = 0
  
                    0 10
    1 0 0 1 0 1 0 1 1 02 
  -   1 0 0 0 1 1 1 0 12
   --------------------
                    0 1 2
  
  The next one is fine, 1-1 = 0
  
                    0 10
    1 0 0 1 0 1 0 1 1 02 
  -   1 0 0 0 1 1 1 0 12
   --------------------
                  0 0 1 2
  
  For the next one, we need to borrow a 10 again
  and the problem becomes 10-1 = 1
  
              0 10  0 10
    1 0 0 1 0 1 0 1 1 02 
  -   1 0 0 0 1 1 1 0 12
   --------------------
                1 0 0 1 2
  
  Now we need to borrow again, but we need to borrow from two places away
  We borrow a 10 but borrow a 10 from that
            1  10
          0 10 0 10 0 10
    1 0 0 1 0  1 0 1 1 02 
  -   1 0 0 0  1 1 1 0 12
   --------------------
               1 1 0 0 1 2
  
  1-0 = 1
  
            1  10
          0 10 0 10 0 10
    1 0 0 1 0  1 0 1 1 02 
  -   1 0 0 0  1 1 1 0 12
   --------------------
            1  1 1 0 0 1 2
  
  twice 0-0 = 0
            1  10
          0 10 0 10 0 10
    1 0 0 1 0  1 0 1 1 02 
  -   1 0 0 0  1 1 1 0 12
   --------------------
        0 0 1  1 1 0 0 1 2
  
  Finally 10-1 = 1 (with a borrow)
  
            1  10
     0 10   0 10 0 10 0 10
    1 0 0 1 0  1 0 1 1 02 
  -   1 0 0 0  1 1 1 0 12
   --------------------
      1 0 0 1  1 1 0 0 1 2
       

• How would you do 127-983?
  • I probably would
    • note that the 983 is larger so the result will be negative
    • Subtract 983-127 = 864
    • Since the negative number was larger, the answer is -864