What are the basic facts?

Edward C. Rathmell
University of Northern Iowa

The basic facts for addition and multiplication are simply the facts that are a single digit plus (times) a single digit. For addition, the addends or parts are single digits even though the sum or whole might be a two-digit number. For multiplication, each of the factors must be a single digit. Accordingly, there are 100 possibilities combined with each of 10 possibilities for the 2 digits to be added (multiplied) or 100 basic facts for each of addition and multiplication.

For subtraction, the whole might be a two-digit number, but the parts must be single digits just as with addition, however, the whole can be a two-digit number. In fact, if 13 - 8 is treated as if it is a renaming problem, when you rename, you still get 13 - 8. When you rename for larger numbers, you change the subtraction in each place value column to a basic fact. When you rename a basic fact in subtraction, you get the same problem again.

Addition and subtraction facts can be matched in fact families. For two addition facts with the same parts, there are two subtraction facts with the same parts. For example, for 3 + 5 and 5 + 3, the subtraction facts are 8 - 5 and 8 - 3. Consequently, there are 100 subtraction facts.

Similarly, there are multiplication and division fact families. For example, 3 x 7 and 7 x 3 are paired with 21 ÷ 3 and 21 ÷ 7. But for division, there are only 90 facts. You cannot divide by zero. This is easily seen if you consider the following:

20 ÷ 4 = _?_ means the same as 4 x _?_ = 20

If you know the answer to one of these equations, you also know the answer to the other. Similarly,

3 ÷ 0 = _?_ means the same as 0 x _?_ = 3

However, what can you multiply by 0 to get 3? It is impossible. As an aside, what is 0 ÷ 0? In this case, if you write the multiplication sentence, any number can be used to solve the equation.

0 ÷ 0 = _?_ means the same as 0 x _?_ = 0

For this special case, any number solves the multiplication sentence above. But since we want one and only one answer for each division problem, and since every number satisfies the multiplication sentence, we cannot find a unique solution. Therefore, we cannot divide by zero in this case either.

Return to Questions