Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.3.COUNTING

The numbers that can be used for counting are denoted by 1, 2, 3, 4, 5, and so on, and they are called natural numbers. Sometimes, zero is also included in the list of natural numbers, in order to be able to express the absence of things. On the other hand, for the Greek philosophers of antiquity, counting started with two objects, hence one was not regarded as a number. But no matter where we let them begin, the natural numbers are the basis for the understanding and mathematical construction of other types of numbers, such as negative numbers, rational numbers, and even real numbers—the numbers used for measuring quantities. German mathematician Leopold Kronecker (1823–1891) has best described the fundamental role of the natural numbers in his often-quoted dictum: “God made natural numbers; all else is the work of man.”

We typically make our first conscious acquaintance with natural numbers when we learn to count. Whether or not one likes mathematics, the ability to count has become second nature to us. As soon as we acquire this ability, we forget about this tedious learning process. Counting, then, appears to be a simple exercise, and we are usually not aware of its inherent complexities. But, in fact, counting is a rather delicate process, and it takes some maturity in abstract reasoning to describe it in more detail.

Can you estimate the number of pebbles in figure 1.2? If you want to know exactly, you will have to count them. By observing ourselves when counting, we find that this task consists of several steps:

1. We start with an arbitrary object in the collection and say “one.”

2. We mark this object as “already counted” (at least in our mind, in order to avoid counting it twice).

3. We select a new object (either by pointing with a finger or simply by looking at it).

4. We say the next “number word” (using number words always in the same strict order).

5. We go back to step 2 and repeat until there are no more uncounted objects. The last number word obtained in that way describes the number of objects.

Figure 1.2: Explicit counting of a set of pebbles.

Counting is a process of associating number words with objects in a collection. One of the more difficult tasks involved here is that one has to divide the collection of objects into those that have already been counted and those that still remain to be counted. This is fairly easy if we can put the objects in a row, but it could be impossible if the objects were moving and kept changing places.

When counting nonpermanent objects or events—for example, the chimes of a clock striking the hour—we typically say a number word as the event occurs. When the events are separated by long time intervals, we normally have to create a permanent record of that event—for example, tally marks on a sheet of paper—and finally determine the number of events from the record.