Math Forum - Ask Dr. Math:

I am trying to figure out why the method below, called the "method of
complements" by my university professor, will give me the correct
answer all of the time. It is used in place of regrouping in
subtraction problems. 

     623     999
    -465    -465
    -----   -----
     158     534
            +623
            -----
            1157

            /157     (Cross off the 1 in the thousands column and add

              +1      this 1 to the ones column. 
            -----
             158

Nines must always be used, and the order of operations depicted above
must be followed. If these conditions are met, the problem will always
work no matter how many numbers are in the problem. This will also
work with standard subtraction when regrouping is not necessary. I
would like to know why this method works. What mathematics are lurking
behind the scenes? I wish to explain this to a third grade child I am
student teaching.