Our first contact with numbers probably came from “counting” fingers, rocks, beans, toys at the behest of our mother. She needed the proof, or at least the appearance that we were intelligent as perceived by others in the reciting of numerals while pointing at successive objects. This became a basis for understanding numbers as an abstract property of any set of things.
The next level of difficulty is communicating numbers, by voice and by writing. We have names for each of our fingers in the counting system: one, two, three, … and symbols for each also: 1, 2, 3, … . Before we had these names and symbols, people sometimes counted things by marks on a stick, one mark for each item. These marks were also sometimes made on stone, sheepskin, and papyrus. This system led to Roman numerals where the marks resembled fingers and hands: I, II, III, IIII, V, … , VIIII, X.
In the first millennium A.D., in the far east, our current system based on groups of ten, the decimal system, was used, popularized and exported. By 1200 A.D. all of the European continent was using the decimal system for commerce. Very little of the Roman system remained, but it is still used in some places today, especially clocks and dates.
Next came adding, subtracting, multiplying, dividing. In the seventh century A.D. they were using the decimal system in China, India and the middle East. The next development was to include with their numbers a zero and the negative numbers. A few philosophers and teachers from BCE, like Euclid and Archimedes, had already begun to codify a system of numbers, their relations and their operations. In the nineteenth century, mathematics became a complete logical system where our decimal numerals were only an example of a number system. That’s when we began to prove things about numbers solely from a set of axioms about an abstract set of things.
In algebra, we use logical Inference to solve problems and discover other deducible properties.
The assumed properties (field axioms) of the operations on “real numbers” (add and multiply) are found in the document at: real numbers.
The assumptions about the ordering of real numbers are:
- If a and b are two real numbers (a, b ∈ R) exactly one of the following is true: a > b or a=b or b >a.
- If a,b,c∈R and a>b, b>c, then a>c.
- If a,b,c∈R and a>b, then a+c>b+c.
- If a,b,c∈R and a>b, c>0 then ac>bc
- a < b means b > a
- a ≥ b means a > b or a = b
- a ≤ b means a < b or a = b
The real number line is a graphic representation of our numbers as a horizontal line containing a point representing the number zero which divides the line into two parts: positive to the right and negative to the left. On this line, a > b indicates a is to the right of b. At this point it may serve to notice that a > b is equivalent to the statement a + (-b) > 0 which, by convention is written a – b > 0.
The axiom about completeness of the real numbers is:
This property comes up in discussing the “limit” of numeric series, sequences and infinite decimal representations. It is sometimes called the lub or “least upper bound” property.
In summary, the real numbers are a complete, Ordered Field.