Addition

• To do addition an any base, we need two thing
  • The addition table.
  • The addition algorithm.
• Addition Table
  • You may have had to memorize this table in elementary school

    +	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

  • The red ones represent a carry out to the next digit.
    • 4 + 9 = 10 + 3
  • The addition algorithm is

    • Align the two numbers at the decimal point.
      Pad to the left and right if required.
      Start at the LSD of each number.
      Start with a carry in of 0
      While there are digits left
         Add the two digits and the carry in.
         Write down the "ones" result
         Make the carry out the carry in of the next digit.
         Move to the left
      If there is a non-zero carry out, write it down

  • Let's Try

    0    +
• Addition in Binary
  • The algorithm stays the same.
  • The Addition table is MUCH easier

    +	0	1
    0	0	1
    1	1	1 0

  • Everything else stays the same.
• Counting in binary
  • The first two are easy, 0, 1
  • What comes after 1? 1+1 = 10
  • What comes after 10? 10+1 = 11.
  • Next 11 + 1 = 100 (use the adder above, or do it by hand)
  • Then 101, 110, 111
  • Add 1 to 111 and you get 1000
  • And the process repeats.
• Other bases
  • The rule remains the same
  • The table changes.
  • Try

     25427 +16657 
                               
    5+2 = 7, but we don't have seven in base 7 so it is 10
    4+6+1 = 7+4 = 147
    6+5+1 = 7+5 = 157
    2+1+1 = 4
    
     111
     25427 
    +16657 
     4540
            

  • Note the biggest carry will be a 1
    • In base 10: 9+9+1 = 19
    • In base 2: 1+1+1 = 11
    • In base n:
      • n-1+n-1+1 = 2n-1
      • 2n-1 = n + n-1
      • which is 1(n-1)
      • Which matches the pattern above
• Count to 20 in base 7
• Count to 20 in base 16