File:Adding integers by decrementing and incrementing.ogv

English: As an alternative to memorizing a table of sums of 110 pairs of numbers, the method of incrementing and decrementing can be used to calculate the sum of two numbers between 0 and 10 using only addition and subtraction by 1 which are easy to learn as they are closely related to counting.

To add a to b where both a and b are between 0 and 10, first note that as a + b = b + a, we can assume that a >= b. Add 1 to a and subtract 1 from b to get a + b = a + 1 + b - 1.

For example: 5 + 6 = 6 + 5 = 6 + 1 + 5 - 1 = 7 + 4.

The process can then be repeated to add the new pair of numbers, continuing until either a = 10 or b = 0, both of which result in particularly easy sums, namely 10 + b = 1b or a + 0 = a.

To complete the example: 5 + 6 = 6 + 5 = 7 + 4 = 8 + 3 = 9 + 2 = 10 + 1 = 11.

Although in the worst case 5 + 5, 5 increments and decrements are required to complete the computation, this case only occurs once out of the 110 possible pairs to be summed and is easy to short cut by memorizing the fact that 2 × 5 = 10, a fact that is helpful in remembering the related sums 6 + 5, 5 + 6, 4 + 5, 5 + 4. Consider the proposed computation of a more difficult sum like 9 + 7 which produces the one step sequence: 9 + 7 = 10 + 6 = 16. Larger values of a and b result in shorter sequences of calculations on what might appear initially appear to be more difficult sums.