As another example of the possibility of confusing number classes, consider the difference between 5 apple trees containing 100 apples each, and 100 apple trees containing 5 apples each. The equality of the number of apples in each of these cases is asserted in the abstract expression 5×100=100×5. Although when considering soley the number of apples, this equality is certainly valid, it can become misleading to consider these two expressions “equivalent” in any sense more specific than that of the abstract counting of apples.
The two terms of a multiplication are formally labeled the multiplicand – the number of items to be multiplied – and the multiplier – the number of times the multiplicand is to be repeated. This distinction is critical in understanding the process of multiplication, and the proper meaning of the multiplication. Again the issue is in the class or type of numbers being combined. Here I will restrict the discussion to multiplication of positive, real numbers by natural numbers. The multiplicand is either a count of units, or a measure of a continuous quantity in units. The multiplier is a natural abstract number representing a (whole) number of repetitions. When separated, these numbers are not of the same class, and therefore cannot be interchanged in two expressions and considered equivalent. When considered in this manner, 5 sets of 100 units each is not equivalent to 100 sets of 5 units each. Only when the multiplication is “computed”, and the resulting total measure taken, can we state that the two products yield an equal measure of units.
The danger in confusing the two levels of meaning present in a simple multiplication, such as 5×100 and 100×5 becomes apparent when abstract substitutions are made, out of the context of the computation of units, in a lengthy computation. Unless the mathematician is taking great care to interpret the meaning of his expressions, a valid sequence of abstract mathematical expressions can lead to a meaningless result.