Last updated 1996 Sep 29
Numbers and Their Characteristics
Introductory Cautions
| Classification Schemes
| Other Characteristics
Numbers serve many purposes well; however, we must match the power of numbers with care in their use. Context determines much of the meaning of a number, just as for other words. The correct use of quantitative methods begins with an understanding of how numbers can be used. Be wary of implicit assumptions.
Much nonsense can be traced to misunderstanding the meaning of particular numbers. For example, the ancient Greeks used the letters of their alphabet to also represent numbers, alpha for 1, beta for 2, etc. In that way every word was also a number. People would add Tom to Dick and if the answer happened to be Harry, they thought they had discovered something. Today an equally silly computation would be to take the square root of one's Social Security Number. Before using some numbers to solve a problem, think about what the numbers represent.
Traditional
| Modern
| Special Scales
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Many languages have a system of numbers that applies to every type of thing: seven people, seven houses, seven miles, and so on. Other languages have distinct number systems for, say, people on the one hand and inanimate objects on the other. In a similar way, for counting certain things, we still use Roman numerals rather than the common Arabic. Examples are
- upper case Roman numerals for the year a motion picture was made
- upper case Roman numberals on cuckoo clocks
- lower case Roman numerals for the pages in the preface of a book
Europe used Roman numerals until the Renaissance, often in lower case. In England the pronoun I was capitalized to distinguish it from the number i.
We caution each other not to "add apples and oranges" because we use the same number words for what are really different things. Using one system of number symbols for many purposes does lead to economy in keeping records and computation.
Before we use numbers to "prove" something, we should think critically what sort of numbers we are dealing with. For example, what sort of numbers are "Social Security numbers", "income", "age", "number of employees", "inventory turnover", and so on?
European languages generally have two versions of their underlying words for numbers
- Cardinal: one, two, three, four, etc. for counting;
- Ordinal: first, second, third, fourth, etc. for ranking.
In modern times people have distinguished what are called "number scales" by how the numbers are used, namely,
- Nominal Number Scale:
- Numbers are simply unique identifiers. Order is incidental to meaning. Order is used for listing and searching. For example
- Social Security numbers
- telephone numbers
- customer numbers
- Uniform Product Codes
- players' numbers in sports.
- Ordinal Number Scale:
- The natural order of numbers assigned corresponds to some physical order of the objects. "Greater than" is meaningful; arithmetic is not. For example
- house numbers on a street
- Mohs scale of hardness:
- talc
- gypsum
- calcite
- flourite
- apatite
- orthoclase
- quartz
- topaz
- corundum
- diamond
- critics' movie rankings: 1 to 10 with 10 best
- service numbers given to customers as they arrive
- Interval Number Scale:
- Equal intervals between assigned numbers correspond to some equal physical measurements. Addition and subtraction are meaningful. For example
- calendar dates
- altitude above sea level
- longitude (from Greenwich)
- Fahrenheit and Celsius temperature
- serial numbers
- Rational Number Scale:
- Ratios as well as intervals have meaning, and the number zero represents a physical limit. For example
- prices
- weights
- times in which contestants complete a race
- latitude
- Kelvin temperature
- ph for acidity = - log( hydrogen ion concentration), calibrated so 7 is neutral
- Richter for earthquake magnitude = (log(energy in ergs) - 11.8)/1.5. Originally magnitude was log10( swing amplitude at standard distance) / (swing amplitude of standard earthquake)
Divisibility
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