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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses variables, constants, and real numbers. By the end of the module students should be able to distinguish between variables and constants, be able to recognize a real number and particular subsets of the real numbers and understand the ordering of the real numbers.

Section overview

  • Variables and Constants
  • Real Numbers
  • Subsets of Real Numbers
  • Ordering Real Numbers

Variables and constants

A basic distinction between algebra and arithmetic is the use of symbols (usually letters) in algebra to represent numbers. So, algebra is a generalization of arithme­tic. Let us look at two examples of situations in which letters are substituted for numbers:

  1. Suppose that a student is taking four college classes, and each class can have at most 1 exam per week. In any 1-week period, the student may have 0, 1, 2, 3, or 4 exams. In algebra, we can let the letter x size 12{x} {} represent the number of exams this student may have in a 1-week period. The letter x size 12{x} {} may assume any of the various values 0, 1, 2, 3, 4.
  2. Suppose that in writing a term paper for a biology class a student needs to specify the average lifetime, in days, of a male housefly. If she does not know this number off the top of her head, she might represent it (at least temporarily) on her paper with the letter t size 12{t} {} (which reminds her of time ). Later, she could look up the average time in a reference book and find it to be 17 days. The letter t size 12{t} {} can assume only the one value, 17, and no other values. The value t size 12{t} {} is constant .

    Variable, constant

  1. A letter or symbol that represents any member of a collection of two or more numbers is called a variable .
  2. A letter or symbol that represents one specific number, known or unknown, is called a constant .

In example 1 , the letter x size 12{x} {} is a variable since it can represent any of the numbers 0, 1, 2, 3, 4. The letter t size 12{t} {} example 2 is a constant since it can only have the value 17.

Real numbers

Real number line

The study of mathematics requires the use of several collections of numbers. The real number line allows us to visually display (graph) the numbers in which we are interested.

A line is composed of infinitely many points. To each point we can associate a unique number, and with each number, we can associate a particular point.

Coordinate

The number associated with a point on the number line is called the coordinate of the point.

Graph

The point on a number line that is associated with a particular number is called the graph of that number.

Constructing a real number line

We construct a real number line as follows:
  1. Draw a horizontal line.

    A horizontal line with arrows on the end.

  2. Origin

    Choose any point on the line and label it 0. This point is called the origin .

    A horizontal line with arrows on the end. The center has a hash  mark labeled 0.

  3. Choose a convenient length. Starting at 0, mark this length off in both direc­tions, being careful to have the lengths look like they are about the same.

    A horizontal line with arrows on the end. The center has a hash mark labeled 0. There are numerous evenly-spaced hash marks on either side of the zero.

We now define a real number.

Real number

A real number is any number that is the coordinate of a point on the real number line.

Positive numbers, negative numbers

Real numbers whose graphs are to the right of 0 are called positive real numbers , or more simply, positive numbers . Real numbers whose graphs appear to the left of 0 are called negative real numbers , or more simply, negative numbers .

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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