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MAT091 :: Lecture Note :: Week 01
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Overview
Handouts
Assignments

Syllabus Review

It is the student's responsibility to read and understand the MAT091 syllabus.

I became an instructor at SCC during fall of 1997. During my time at SCC I have learned that students who attend class do better than those who don't.

Faculty were told the following:

   "An instructor is paid to hold class from time A to time B.
    It is considered fraud if an instructor is consistently 
    letting students out early or coming to class late."

The Math Tutor Lab in CM-441A is available for help learning arithmetic and algebra.

MAT108 - Tutored Mathematics Course can be taken to help with respect to learning the MAT091 material.

Calculators are available for rent in the IT (Information Technology) building.

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What is Algebra?

Stand firm in your refusal to remain conscious during algebra.
In real life, I assure you, there is no such thing as algebra.

-- Fran Lebowitz (01950-) { American author; more... } [algebra]

I don't know why I should have to learn Algebra...
I'm never likely to go there.

-- Billy Connolly (01942-?????) {Scottish comedian; more...} [math]

As long as algebra is taught in school, there will be prayer in school.
-- Cokie Roberts (01943-?????) { American journalist and author; more... } [math]

Algebra is the metaphysics of arithmetic.
-- John Ray (01627-01705) { English naturalist; more... } [math]

To speak algebraically, Mr. M. is execrable, but Mr. G. is e(x+1)ecrable.
-- Edgar Alan Poe (01809-01849) { American poet and writer; more... } [math/algebra]

My mother is a mathematician, so she knows how to induce good behavior.
"If I've told you n times, I've told you n+1 times...."

-- Anonymous { She was only a mathematicians daughter, but she knew how to multiply. } [math]

Men are liars. We'll lie about lying if we have to. I'm
an algebra liar. I figure two good lies make a positive.

-- Tim Allen (01953-) { American comedian; more... } [life]

In a nutshell, Algebra is x-sighting!

BAB:: Joke... Terror Alert -- Al-gebra Movement

So... What is Algebra? Algebra is a branch of mathematics that provides a foundation for other branches of mathematics, science, engineering, computing, and so on. Via Wikipedia.org: "In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics." Algebra involves the evaluating of expressions and the solving of equations that contain one or more variables.

YouTube.com::Introduction to Algebra via KahnAcademy.org.

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Exponents (integer exponents greater than or equal to 0)

Math.com tells us that exponents are a "shorthand way to show how many times a number, called the base, is multiplied times itself." In other words, just like multiplication is a form of "repeated addition," exponents are a form of "repeated multiplication."

   5 + 5 + 5 = 3 * 5              (repeated addition)
   5 * 5 * 5 = 53      (repeated multiplication)

Exponents imply the operation of "raising to a power." For example, 105 is read "10 raised to the 5th power," with 10 being the base and 5 the exponent (or power).

By mathematical law (definition), any non-zero number raised to the power of zero is one.

   50 equals 1
   99990 equals 1
   820 equals 1
   -420 equals -1
   (-42)0 equals 1
   00 equals ??? [Google says 1]

Any number raised to the power of one is that number.

   31 equals 3
   99991 equals 9999
   -421 equals -42

Any number raised to the m-th power (where m > 1) is that number multiplied by itself 'm' times.

   54 = 5 * 5 * 5 * 5
   37 = 3 * 3 * 3 * 3 * 3 * 3 * 3
   23 = 2 * 2 * 2
   -52 = -(5 * 5)
   (-5)2 = -5 * -5

[special exponent values] A number squared is a number raised to the power of 2 and a number cubed is a number raised the power of 3.

   8 squared is 82 which equals 8 * 8
   4 cubed is 43 which equals 4 * 4 * 4
BAB:: Square Number Playing Results in Square Number Discovery [24 May 2007]

Recall, the base-10 (i.e. decimal) number system has the ones, tens, hundreds, thousands and so one. These positional values are based upon 10 being raised to the whole numbers 0, 1, 2, and so on.

100 equals 1 (one)
101 equals 10 (ten [deka-])
102 equals 100 (hundred [hecto-])
103 equals 1000 (thousand [kilo-])
104 equals 10,000 (10 thousand)
105 equals 100,000 (100 thousand)
106 equals 1,000,000 (million [mega-] [1,000 thousand])
107 equals 10,000,000 (10 million)
108 equals 100,000,000 (100 million)
109 equals 1,000,000,000 (billion [giga-] [1,000 million])
...
1012 equals 1,000,000,000,000 (trillion [tera-] [1,000 billion])
...
1015 equals 1,000,000,000,000,000 (quadrillion [peta-] [1,000 trillion])
More... On Notation

The caret ^ symbol is sometimes used to imply exponents. GDT calls this "calculator notation."

   2^4 = 24 = 2 * 2 * 2 * 2 = 16
   8^3 = 83 = 8 * 8 * 8 = 512
   1^2 = 12 = 1 * 1 = 1
More... One Expression That Contains Lots of Math

Let's take a peek at the following expression reading it from left-to-right.

   a^n ⋅ 1 = a^n = a^(n + 0) = a^n ⋅ a^0
More... Something From a Math Reading Group

GDT enjoyed seeing pictures drawn to scale that turned the earth into a pixel followed by reducing the sun into pixel. GDT also liked how powers of ten were used to demonstrate the base-10 number system.

   5555 = 5 * 10^3 + 5 * 10^2 + 5 * 10^1 + 5 * 10^0
   209 = 2 * 10^2 + 0 * 10^1 + 9 * 10^0
   790016 = ???
   ddd,ddd,ddd = 3 * 10^8 + 5 * 10^5 + 2 * 10^3 + 0 * 10^2 + 10^0
External Hyperlink(s)

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Exponents (integer exponents less than 0)

A non-zero number (base) raised to a negative exponent is equal to one divided-by the number raised to the absolute value of the exponent.

                                        1
   ab where 'b' is less than 0 equals  ---
                                        ab

   7-2  equals  1/72  equals  1/49

   2-5  equals  1/25  equals  1/32

   10-2  equals  1/102  equals  1/100
   10^-1   1/10     tenth          deci-
   10^-2   1/100    hundreth       centi-
   10^-3   1/1000   thousandth     milli-
   10^-6   1/10^6   millionth      micro-
   10^-9   1/10^9   billionth      nano-
   10^-12  1/10^12  trillionth     pico-
   10^-15  1/10^15  quadrillionth  femto-
What's an absolute value?

The absolute value of a number is its distance from zero on the number line. For example, -5 is five ones away from zero; therefore, its absolute value is 5.

Symbolically, two vertical bars with a number (or mathematical expression) between them represents an absolute value.

 
   |7| equals 7
   |-7| equals 7
   |3 - 4| equals |-1| equals 1
   -|-5| equals -5
   3 * |-2| = 3 * 2 = 6

Function notation: Sometimes abs(n) is the absolute value function. The function outputs the absolute value of input n.

External Hyperlinks

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Math Terminology: Part 0

This is a quickie terminology review that is intended to be augmented by lecture.

A mathematical expression is a combination of one or more terms that can be evaluated.

Expressions are evaluated resulting in a value. In other words, expressions "express" a value.

A term is either a number, variable (i.e. unknown number), function, or a combination of these objects.

In an expression terms are separated by addition and subtraction operators.

Expressions can contain grouping symbols (parenthesis, brackets, fraction bars) to alter order of operations when expressions are evaluated.

Numbers or variables that are multiplied together are called factors. A term can consist of multiple factors.

   5 + 3 * 2

   the expression has two terms:  5 and 3 * 2
   the 3 * 2 term has two factors (3 and 2)
   + and * are arithmetic operators
   * has a higher precedence than +; therefore,
     it is evaluated first:  5 + 3 * 2 = 5 + 6 = 11
     if + needs to be evaluated before *, then the
     expression is:  (5 + 3) * 2 = 8 * 2 = 16

An equation is a relationship between two expressions. Relational operators are used to relate expressions.

   3 + 5 = 4 * 2

   3 + 5 evaluates to 8
   4 * 2 evaluates to 8
   
   3 + 5 = 4 * 2 
     8   =   8

Equations have a left-side and a right-side. The left-side is left of the relational operator. The right-side of an equation is to the right of the relational operator.

General syntax for a mathematical equation:

   expression  relational-operator  expression

   where relational-operator is equal, not equal, greater than,
                                greater than or equal, less than,
                                less than or equal, approximate

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Like Terms

Evaluating expressions and solving equations can sometimes be made "easier" if they are "simplified." One step in simplifying expressions/equations is the "combining" of like-terms. Like-terms are terms having the same variables raised to the same exponents.

   5x + 2x = 7x
   2x - 5x = 2x + (-5x) = -3x
   x2 + 4x2 = 5x2 [x2 = 1x2]
   9x10 + 9y10 are not like-terms [different variables]
   5x2 and 2x are not like-terms [different exponents]

wyzant.com:: Simplifying Expressions Calculator

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Algebraic Expressions

An algebraic expression is a mathematical expresssion that contains at least one variable along with zero or more numbers (i.e. constants) and zero or more arithmetic operators.

The following are algebraic expressions.

   n
   3n
   5n + 10
   2x^2 - 2y^3
   9a + 2b - 3c
   11(x + y^-3 + 2w)

Variables are unknown values that are represented by single-character letters. For example, 3n is 3 times 'n', where 'n' is a variable. If a value is assigned to 'n', then the expression can be evaluated.

   3n + 8

   if n = 3, then 3(3) + 8 = 100
   if n = -5, then 3(-5) + 8 = -7
   if n = 0, then 3(0) + 8 = 8

The replacement (or swapping) of variables with values can be described in a variety of ways: Values can be "plugged into" or "assigned to" variables; or variables can be "replaced with" or "substituted with" values; or we "let" variables be certain values.

   5x + 1  assign 5 to x ........... 5(5) + 1 = 25 + 1 = 26
   5x + 1  replace x with 5 ........ 5(5) + 1 = 25 + 1 = 26
   5x + 1  plug 5 into x ........... 5(5) + 1 = 25 + 1 = 26
   5x + 1  substitute x with 5 ..... 5(5) + 1 = 25 + 1 = 26
   5x + 1  let x be 5 .............. 5(5) + 1 = 25 + 1 = 26

Again, expressions are typically evaluated after variables have been assigned (i.e. given) values.

If a variable is multiplied by a constant, then the constant is called a coefficient.

   3n ...... 'n' is a variable and 3 is a coefficient
   x ....... 'x' is a variable and 1 is the coefficient
   -1y ..... 'y' is a variable and -1 is the coefficient

The combination of a coefficent and variable is called a factor. Factors are numbers, variables, and expressions that are multiplied together to produce a product.

Factors separated by addition and subtraction operators are called terms.

   3n + 1 .............. 1 factor, 2 terms
   4a + 2b - 10 ........ 2 factors (4a and 2b), 3 terms
   7x - 5 + 8y - 11 .... 2 factors (7x and 8y), 4 terms
   2x^3 + 10^2.......... 1 factor (2x^3), 2 terms
   3(4) - 2(1 + 3)...... 2 factors, 2 terms

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Simplifying Expressions

   a.  Simplify:  4 - 2(x + 3)
       4 + -2(x + 3)     # rewrite a - b as a + -b
       4 + -2x + -6      # distribute -2
       -2x - 2           # combine like terms (the constants)
       -2x + -2          # rewrite as an addition problem
       -2(x + 1)         # factor out a -2

       check work:  4 - 2(x + 3)               -2(x + 1)

       let x = 0... 4 - 2(0 + 3)               -2(0 + 1)
                    4 - 2(3) = 4 - 6 = -2      -2(1) = -2

       let x = 1... 4 - 2(1 + 3)               -2(1 + 1)
                    4 - 2(4) = 4 - 8 = -4      -2(2) = -4

       let x = -1... 4 - 2(-1 + 3)             -2(-1 + 1)
                     4 - 2(2) = 4 - 4 = 0      -2(0) = 0

       check work using the calculator:
         press Y=
         enter 4-2(x+3) into y1
         enter -2(x+1) into y2
         press GRAPH to see one line
         press 2nd GRAPH to see table values
         press Y=
         change y2 to -2(x+2)
         press GRAPH to see two lines
         press 2nd GRAPH and compare table values

   b.  Simplify:  (x^2 + x - 5) - (2x^2 - x + 3)
       # convert subtractions into additions
       (x^2 + x + -5) + (-1)(2x^2 + -x + 3)   

       x^2 + x + -5 + -2x^2 + x + -3      # drop ()s and distribute -1
       x^2 + -2x^2 + x + x + -5 + -3      # combine like terms (step 1)
       -x^2 + 2x + -8                     # combine like terms (step 2)

       check work:  let x = 0
       (x^2 + x - 5) - (2x^2 - x + 3)
       (0^2 + 0 - 5) - (2(0^2) - 0 + 3) = -5 - 3 = -8

       -x^2 + 2x + -8
       -0^2 + 2(0) + -8 = 0 + 0 + -8 = -8

       check work:  let x = 1
       (x^2 + x - 5) - (2x^2 - x + 3)
       (1^2 + 1 - 5) - (2(1^2) - 1 + 3) 
       (     -3    ) - (      4       ) = -7

       -x^2 + 2x + -8
       -1^2 + 2(1) + -8 
        -1  + 2    + -8 = -7

       check work:  let x = -1
       (x^2 + x - 5) - (2x^2 - x + 3)
       ((-1)^2 + -1 - 5) - (2(-1^2) - -1 + 3)
       (   1   + -1 - 5) - (2       - -1 + 3)
       (       -5      ) - (     6          ) = -11

       -x^2 + 2x + -8
       -(-1)^2 + 2(-1) + -8 = -1 + -2 + -8 = -11

       check work using the calculator:
          press Y=
          enter (x^2 + x - 5) - (2x^2 - x + 3) into y1
          enter -x^2 + 2x + -8  into y2
          on the y2 line... left arrow to the \ and press ENTER
          press GRAPH to see only one parabola
          press 2nd GRAPH to compare table values

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Define: horizontal and vertical

The horizon is the "apparent junction of earth and sky."

The following is a number line written horizontally.

   <--+---+---+---+---+---+---+---+---+---+---+-->
         -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5

The vertical is "perpendicular to the plane of the horizon or to a primary axis." [Perpendicular is "being at right angles to a given line or plane."]

The following is a number line written vertically.

      ^
      |
   +5 +
      |
   +4 +
      |
   +3 +
      |
   +2 +
      |
   +1 +
      |
    0 +
      |
   -1 +
      |
   -2 +
      |
   -3 +
      |
   -4 +
      |
   -5 +
      |
      v
Right Angle

Perpendicular lines intersect and their point of intersection is called the origin. Perpendicular lines form four right angles at the origin. A right angle measures ninety degrees (90°).

With respect the Cartesian coordinate system, the horizonal line is the x-axis and the vertical line is the y-axis.

           | <-- vertical-axis (y-axis)
           |
           |
           |
   --------o-------- <-- horizonal-axis (x-axis)
           |\
           | \
           |  o is the origin
           |   

Note: The Cartesian coordinate system was developed by French mathematician and philosopher Rene Descartes in approximately 1637.

RoadHacker:: Horizon/Vertical Slideshow [opens new window]

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Inequalities

The following are some inequalities graphed on the number line. An end-point marked with 'x' (or ']') implies that value is included, while an end-point marked with 'o' (or ')') is excluded.

   x > 2
   
    ---------------------o=========>
   -5 -4 -3 -2 -1  0  1  2  3  4  5

   x ≥ -4

    ---x===========================>
   -5 -4 -3 -2 -1  0  1  2  3  4  5

   x ≤ 3

  <=========================x------
   -5 -4 -3 -2 -1  0  1  2  3  4  5

   x < -2

  <==========o---------------------
   -5 -4 -3 -2 -1  0  1  2  3  4  5

Linear inequalities are solved just like linear equations with one exception: the inequality sign is "flipped" whenever there is a multiply (or divide) by a negative. The term "flipped" is used to imply less-than becomes greater-than and vice versa; and that less-than-or-equal-to becomes greater-than-or-equal-to and vice versa.

   Example 1:  8x + 1 < 2x - 5

   8x - 2x + 1 < -5
   6x + 1 - 1 < -5 - 1
   6x < -6
   x < -1

   try x = 0...
      8(0) + 1 = 9; 2(0) - 5 = -5
      9 is not less than -5

   try x = -1...
      8(-1) + 1 = -7; 2(-1) - 5 = -7
      -7 is not less than -7

   try x = -2...
      8(-2) + 1 = -15; 2(-2) - 5 = -9
      -15 is less than -9

   Example 2:  -2x < 5

   -2x / -2 > 5 / -2      # < flipped to > (divide-by negative)
   x > -2.5

   try x = 5...
      -2(5) < 5
      -10 < 5 is true

   try x = -5...
      -2(-5) < 5
      10 < 10 is false

The following uses numbers to help understand when inequalities are flipped (switched).

   10 > 5  is true
   multiply both sides by -1
   10(-1) = -10  and  5(-1) = -5
   -10 > -5 is false
External Hyperlink(s)

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