Addition in other bases

Objectives

1. Be able to describe the addition algorithm
2. Be able to add in binary

Notes

• If you had to add two longer numbers by hand how would you do this?

  •    3251 First add 1+6
      +4976 From basic number facts you know this is 7
      -----
    
        1
       3251    Now add the 5 and the 7,
      +4976    again from basic number facts you know this is 12
      -----      But now we have another operation, we need to "carry" the 1
         27
    
       11
       3251    Now add  1, 2 and 9,
      +4976    Number fact 1+2 =3 , 3+9 = 12 
      -----     
        227
    
       11
       3251    Now add  1, 3 and 4,
      +4976    Number fact 1+3 =4 , 4+4 = 8 
      -----     
       8227
           

• What would we have done if the last sum produced a carry?
• This entire system works because we know the addition table for decimal.

  +	0	1	2	3	4	5	6	7	8	9
  0	0	1	2	3	4	5	6	7	8	9
  1	1	2	3	4	5	6	7	8	9	10
  2	2	3	4	5	6	7	8	9	10	11
  3	3	4	5	6	7	8	9	10	11	12
  4	4	5	6	7	8	9	10	11	12	13
  5	5	6	7	8	9	10	11	12	13	14
  6	6	7	8	9	10	11	12	13	14	15
  7	7	8	9	10	11	12	13	14	15	16
  8	8	9	10	11	12	13	14	15	16	17
  9	9	10	11	12	13	14	15	16	17	18

• To do the same in binary, we need to memorize the binary addition table.

  +	0	1
  0	0	1
  1	0	10

• Add 1011111₂ + 1101010₂

  
    10111112   1 + 0 = 1
  + 11010102
    -------
  
    10111112   1 + 1 = 10
  + 11010102
    -------
          1
  
        1
    10111112   1 + 1 = 10
  + 11010102
    -------
         01
  
       11
    10111112   1 + 1 + 1 = 11
  + 11010102
    -------
        001
  
      111
    10111112   1 + 1 = 10
  + 11010102
    -------
       1001
  
     1111
    10111112   1 + 1 = 10
  + 11010102
    -------
      01001
  
    11111
    10111112   1 + 1 + 1  = 11
  + 11010102
    -------
     001001
  
    11111
    10111112   1 + 1 + 1  = 11
  + 11010102
    -------
   110010012 

• You should be able to add in binary.
• We could do the same in octal and hex, what would we need?
• Let's think about addition in a fixed number of bits.
  • What could go wrong?
  • We could have overflow.
  • How do we detect this? (A carry out from the last bit)
• You should be able to add numbers in binary.
• The addition algorithm

  ADDITION
     Input A, B two strings of digit in the same base.
    
     Output c = a+b in the same base.
  
     Assume A[0] is the least significant digit of A, B[0] of B.
     Assume that Sum(digit, digit, carry) will compute the sum in the given base
  
  
   pad either a or b on the left with 0 so that they are the same size.
   carry ← 0
   For i ← 0 to A.length()-1
      (C[i],carry) ← Sum(A[i],B[i], carry)
   if carry > 0
      C[A.length()] = carry