# Numbers and Numerals

Number is a mathematical concept that is used to develop various formal systems of mathematical reasoning. Generally, the concept rests at the tacit level of understanding in the sense that we assume that we know what it means, and that others have the same meaning in mind. But the concept becomes more complex as we play with it.

Consider the formal number systems.

• There are natural numbers, which is the set [1, 2, 3, …].
• There are whole numbers, which is the set of natural numbers with the added element 0.
• Within the set of whole numbers there is the subset : [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Each of the ten elements in this subset is called a digit. Digits are combined in various ways with other symbols to represent numbers from a variety of number sets.
• There are integers, which is based on the property of symmetry around 0: the set is […, -3, -2, -1, 0, 1, 2, 3, …]. We say that integers can be negative or positive, depending on their location with respect to 0.
• There are rational numbers, which is the set of integers plus all of the ratios of two integers, p/q, where q is not 0. (Thus, ½ is a rational number, as is 312/311.)
• Then there are numbers that cannot be expressed as the ratio of integers, such as the ratio of the circumference of a circle to its diameter, called pi. These are called irrational numbers.
• The set of rational numbers combined with the set of irrational numbers is called the set of real numbers.

There are larger sets of numbers, but the point is made that number is a mathematical concept that is a building block for formal systems of mathematical reasoning.

For most purposes, especially in the social sciences, in measurement discourse numerical references are to elements in the set of real numbers. What does that mean?

Usually we use the term “number” to refer to the result of counting specific units, or the result of performing some group of mathematical operation on that result. For example, six (6) cars went through the red traffic light at this intersection on Monday between 7:00 a.m. and 9:00 a.m. The unit is a car that has certain characteristics (passed through the intersection in the two-hour period when the traffic light was red). The number “6” represents the result of someone counting the defined units. (The same principle applies if the counting was done by a set of instruments set up at the intersection.)

Suppose the results of five days of counting were 6, 7, 8, 10, and 5. What does the statement mean: In the five days an average of 7 1/5 cars went through the red light. The number “7 1/5” is the result of the mathematical operations of adding the five numbers and dividing the total by the number 5 (which is the result of counting the number of days that cars were counted). This is a rational number (it can be expressed as the ratio of integers 36/5). The process for deriving the number 7 1/5 is clear, but what does it mean?

Mathematically speaking, it means that 7 1/5 cars going through the red light every day for five days is the same total number of cars that were actually observed going through a red light in that period. Without knowing the actual counts for each day, we usually assume that about 7 cars were observed each day, maybe a few more on a particular day and a few less on another day. Presumably the observer did not see a part of a car go through the intersection.

Numeral

A numeral is a symbol that represents a number. Numerals have been devised to facilitate mathematical reasoning. The result of counting six cars can be represented by the Arabic numeral “6” or the Roman numeral “vi”. Arabic numerals are much easier to work with.

In measurement discourse, it is essential that we clearly define the unit that is being counted, the counting process and the mathematical operations performed. That is, it is essential that we define the numbers that are represented by the numerals in a particular situation. Otherwise, confusion lurks in the background.

For fascinating accounts of some relationships between numerals and numbers see Isaac Asimov,  (1977). Asimov on numbers. Garden City, NY: Doubleday.