When is a Number, Not a Number?

 When Is a Number, Not a Number?

What numbers are not numbers? In some cases, as discussed below, information is represented with digits but does not represent a numeric value to be used in arithmetic operations. Basically, there are three types of numbers in general use: symbols, ordinals, and cardinals.

Symbols

Many of the numbers we use every day are not numbers at all. They are identifiers of some sort, not representing quantities or ranks. Here are the most common examples:

·        Phone Numbers: These are now almost entirely digital. Years ago, shorter numbers included an alphabetic prefix. For example, an old phone number might have been “LI5-7624”, where "LI" stood for "Lincoln." Now, area codes and country codes are required for calls between cities and countries, respectively.

·        Identification Numbers: These include Social Security numbers, student numbers, serial numbers, part numbers, etc. They are digital identifiers for which no mathematical operations can be performed.

·        Credit Card Numbers: These are among our most important “numbers though not numbers.” Typically, they have 16 digits, along with a special three-digit security code visible only to the physical cardholder.

·        Vehicle Identification Numbers (VINs): These are 17-character codes containing mostly digits and some letters. Similarly, router ID numbers are usually formed with fewer digits.

·        Postal (ZIP) Codes: These are five-digit codes used for mail delivery. For example, my ZIP code is 77840. ZIP codes start from 00501. The ZIP+4 system extends the standard five-digit ZIP code with four additional digits to improve mail delivery accuracy and speed.

·        Addresses: An address might be something like 2932. Doubling it or taking its square root has no meaning relative to the address. For apartment complexes, the address would include "Street address + Building number + Unit number," none of which make sense for arithmetic operations.

It may seem odd, though true, that many of the numbers in your life are not numbers at all, but merely symbols. Our next category, ordinals, are not truly numbers either, but they do have an "ordering," making them closer to actual numbers, the cardinals.

Ordinals (or Ranks)

These are rankings: first, second, third, and so forth, often abbreviated as 1st, 2nd, 3rd, etc. They are not suitable for arithmetic. For example, you cannot add 1st and 2nd to get anything meaningful. Consider tennis rankings: the rank is an ordinal number based on tournament points scored over a year. Arithmetic does not apply, but you can consider the number of those ranked between various rankings.

Ordinals are crucial when ranks are important, such as in sports, stocks, wealth, and all forms of measurable excellence where there is a "top dog." In the animal kingdom, the alpha leader (1st) of any pack makes decisions, eats first, and controls reproductive activities. The matriarch/patriarch of a family or herd is the de facto leader, controlling important decisions. Similarly, business operations, organized religions, and governments are based on an ordinal hierarchy. Military organizations and operations are strictly hierarchical (ordinal) by nature.

While ordinals are not truly numbers in the sense of arithmetic, they have supreme importance in organized society and all organizations. Let us now turn to the “real” numbers on which all mathematics and our daily lives depend.

Cardinals

Cardinals represent quantities, like apples, volts, dollars, etc. These are the only numbers suitable for arithmetic to obtain meaningful values. For instance, you can add two apples to three apples to get five apples, but you cannot add two apples to three oranges meaningfully. In school, math class typically deals with cardinals without tags, i.e., abstract numbers.

Cardinals can extend to fractions, decimals, and real numbers. Infinity (∞) has unique arithmetic properties such as ∞ + ∞ = ∞. Cardinals can be abstracted to new objects such as groups, rings, and fields, which have arithmetic-like properties. These are very useful in modern living, though we rarely see them. One particular number system, the binary system, is highly useful in computing machines.

Cardinals, rationals, and reals form the class of numbers that truly are the numbers we love – and need.