Web Site:  www.waygate.com/bbjames

Section P.1: Algebraic Expressions and Real Numbers

Algebraic Expressions

We use letters to represent numbers. If the letter represents a number that can change, we call it a variable. Most often we use x as the variable.
           u, x, y, z - usually variables
           a, b, c, d - usually constants
By combining of letters and numbers using the operations of addition, subtraction, multiplication, division, powers or roots, we get algebraic expressions.
When adding, the expressions (and numbers) are called terms.
When multiplying, the expressions (and numbers) are called factors.

An example of an algabraic expression is:       (x + 3y + 2z - 4) . (4u - v - 2w + 7)
The expression has 2 factors of 4 terms each.

Remember that we take short cuts and leave out a lot of parentheses, ( ).
           a b = c     ==>     a . b = c     ==>      (a) . (b) = (c)
Also remember that "1" is a factor of every number and expression because of the multiplicative identity -
           a . 1  =  1 . a  =  a

Often, the algebraic expressions involve exponents.

Exponential Notation - An exponential expression, an - is some number a, called the base, raised to a power, b, called the exponent.
           x2  =  x . x                  and is read as x squared
           x3  =  x . x . x             and is read as x cubed
           x4  =  x . x . x . x        and is read as x to the 4th power
           x5  =  x . x . x . x . x   and is read as x to the 5th power
           et cetera.

           bn = b . b . . . . . b      { b is a factor n times

NOTE:   00   is indeterminant (similar to undefined).

Another shortcut that mathematicians use:  We don't write exponents of 1.

           a  =  a1              ab2c  =  a1b2c1

Another problem that students often have - the exponent only applies to the symbol immediately preceding it.

           - x2  =  - (x2)         not  (-x)2

Example 1.1

example 1

Evaluating Algebraic Expressions - The Order of Operations

To evaluate mathematical expressions , use the following order:
  1. Evaluate expressions in grouping symbols, ( ), [ ], { }.  The fraction bar is a grouping symbol.  If you have grouping symbols within other symbols (nested), evaluate the innermost and work your way out.
  2. Evaluate all terms containing exponents and roots.
  3. Evaluate all multiplications and divisions, from left to right.
  4. Evaluate all additions and subtractions, from left to right.
P E M D A S - may be used to help remember the order.
Example 1.2

example 1.2

Use the same order when evaluating expressions with variables.

Example 1.3

example 1.3
Example 1.4

Carbon dioxide, CO2, production for all countries except for the US, Canada, and western Europe, in millions of metric tons, is given by:
           CO2 = 0.078x2 - 0.39x + 0.55                       where x is each 10 year period since 1905.

           a)  Find CO2 produced in 1945:
                      First find x:  1905 = 0,  1915 = 1,  1925 = 2,  1935 = 3,  1945 = 4  
                CO2 = 0.078(4)2 - 0.39(4) + 0.55
                        = 0.158 million metric tons

           b)  Find the predicted CO2 production in 2005:
                      First find x:  1955 = 5,  1965 = 6,  1975 = 7,  1985 = 8,  1995 = 9  2005 = 10
                CO2 = 0.078(10)2 - 0.39(10) + 0.55
                        = 3.95 million metric tons


Formulas and Mathematical Models - Study the section in the book on your own.

Sets

A set is a collection of objects or elements.  A set may be defined in 2 ways:  1) the roster or list method, and 2) set builder notation.

In the roster method, you just name the items in the set -
           A = {ECSU, ECU, NCSU, UNC}
Elements only need to be named once.

The null set or the empty set is denoted by   { }   or   Ø.

The equality and inequality symbols are used in set builder notation. The symbols are:   eqn 1.1.  These symbols represent the relationship between numbers on a number line.

Set builder notation

           A = { x | x > a }

                 {                } - the braces mean "The set of"
                    x                - the first x means "all elements x"
                        |             - the vertical bar means "such that"
                            ^        - the expression after the vertical bar is the property that x must have to be in the set.

Union and Intersection

Union Intersection

Sets of Numbers
Subsets of the Number System
Natural or Counting Numbers N = {1, 2, 3,   . . . }
Whole Numbers W = {0, 1, 2, 3,   . . . }
Integers I = { . . . , -2, -1, 0, 1, 2,   . . . }
Rational Numbers
Irrational Numbers H = {x | x is a real number that is not rational}
Real Numbers R = {x | x is a point on the number line}

The decimal representation of a rational number will be a terminating or repeating decimal.
Example 1.5

Determine the elements of the set which are elements of the Number subsets:
a. natural 2, 4
b. whole 2, 4
c. integer 2, 4, -100, -7
d. rational 2, 4, -5.33, 9/2, -100, -7, 4.7
e. irrational
f. real all elements in the set

Ordering the Real Numbers

By convention, the numbers on the real number line increase from left to right. For any specific number, numbers to the left of it have a value less than the specific number while numbers to the right of it have values greater than it.