Functions of Numbers

A function of numbers has a name, a rule and a domain.  The name can be any string of characters, but it is best if it is short and somehow memorable.  If all you know about the function is its rule in arithmetic form, then it is customary to use f, g, h etc. for the name.  Similarly, an element of the domain might have a name related to its units , if applicable.  For example, the function whose value is the distance associated with a time in hours traveled by a vehicle moving at 45 miles per hour:  Dist(hr)  or d(t).  As to our study of algebra, the names are insignificant, but sometimes convenient mnemonic devices.

The domain is sometimes specified, like “all the positive numbers” but more often it defaults to all real numbers for which the function is well-defined, i.e., valid.

The rule that associates a “value” in the domain with a real number is typically a numeric expression with a place-holder name for the argument (element of the domain).  The function of a specific argument is evaluated by substituting that argument for the place-holder.

Example:

F(x) = 2x² – 3x + 5

F(-2) = 2((-2)² – 3(-2) + 5  =  8 + 6 + 5 = 19

Numbers and Their Properties

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.

Field

The assumed properties (field axioms) of the operations on “real numbers” (add and multiply) are found in the document at: real numbers.

Ordered

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,cR and a>b, b>c, then a>c.
• If a,b,cR and a>b, then a+c>b+c.
• If a,b,cand a>b, c>then ac>bc

Notation:

• 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.

Complete

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.

What is and Why study Algebra

What: “Algebra (from Arabic ‘al-jabr’ meaning ‘reunion of broken parts’) is one of the broad parts of mathematics, together with number theory, geometry and analysis. ”  For more detail, click Algebra.

In today’s vernacular, “Using the rules of arithmetic to deduce a numeric value from given numeric properties.”

This process can be encapsulated in  “Finding the zeros of real functions” and is illustrated in textbooks by showing how to find the zeros of linear $(ax+b)$, quadratic $( a^2+ 3x - 8 )$ and other rational  (polynomial divided by polynomial) functions.  Other functions studied in an algebra course include:

1.  radical functions like $\sqrt x$
2. exponential functions like $e^x$