Monthly Archives: July 2006


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.

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Distinctions between Number Classes

In the discussion of signed integers, it was noted that the nature of arithmetic involved in their use was fundamentally different than arithmetic performed with natural numbers. This fact is a result of a deeper distinction between these numbering systems. A natural number is not a signed integer. The correspondance between positive integers and natural numbers is one of analogy, but not of identification or definition.

It is perhaps easier to understand the distinction between classes of numbersby considering the historical development of arithmetic and geometry. The oldest examples of arithmetic consist almost purely of addition, subtraction and multiplication. When the need to divide arose in real-world problems, the results were expressed (in Sumerian, Babylonian and Egyptian “texts”) as combinations of natural numbers and sums of ratios of natural numbers. Thus an answer would beƂexpressed as 3 + 1/6 + 3/8. In fact, in ancient Egypt a sophisticated system of reducing such expressions to a standard form N+1/a+1/b+1/c+… was devised to “simplify” this usage. It is to be noted here that the idea ofbeing able to “combine” the various fractions and natural numbers was missing from these earliest arithmetic concepts – the math was much more cumbersome, but more conceptually honest at the dawn of our history.

Thedevelopment ofgeometry wasan entirely separated form of study in ancient history. Geometric problems were worked out usingconstructions based upon assumed self-evident axioms. The use of numbers to represent geometric concepts is first seen in classical Greece, but did not fully reach its maturity until as recently as the time of Descartes. A principle concern in geometry is the concept of “length” of line segments, or “distance” in general. The measurement of length using an established unit leads immediately to the need to express partial units, as length is a continuous attribute, unlike the count of a number of discrete items. In this key extension of the meaning of “number” from counting to measuring continuous value – and in particular the length of a line – a new class of number was introduced, encompassing the concept of integral subdivisions of units, or fractions.

The Greeks (first Pythagoras, or perhaps one of his students) first noted the incompleteness of this new numbering system, when considering the length of the diagonal of a square with a side one unit in length. It is a fairly easy proof (when using today’s notation) to show that this length is not a fractional expression of the side length. Hence, there must exist additional “numbers” that arenot expressible as fractions. It was over 1000 years later when Dedekind completed the identification of this new numbering system with the “real numbers” that represent- and can be represented as – lengths on a line.

The point to be made here again, is that the natural number system is fundamentally different from the fractional numbering system. That the fractions 8/4 and 2/1 are equivalent is a true statement; however that these numbers and the natural number 2 are equivalent is not at all true. As to whether the fractional numbering system is of the same class as the real numbering system is a more difficult question, depending on the conceptual framework from which the fractional numbering system is constructed. If, as presented here, the fractional numbering system is meant to represent the possible magnitudes of a value of continuous extent, then the fractional numbering system is incomplete, and must be extended to the (positive) real numbers to form a meaningful concept. [There is circularity in that statement, through the inexact use of the term “continuous”, which requires further thought]. On the other hand, if the fractional numbering system was developed as, say, a monetary reckoning system for trade in tangible items, it may be considered as a separate number class.

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