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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 room IT-100.

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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

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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 = 5 x 3   (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)

{PurpleMath} {MathPapa} {KhanAcademy} {UdacityMOOC} {OpenAlgebra} {InteractiveMathematics} {TIcalculatorHelp}
{SCC math resources... MathAS::id 4689, key 4689 | OER::textbook | playbook | MathGym::drop-in tutoring hours}
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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

{PurpleMath} {MathPapa} {KhanAcademy} {UdacityMOOC} {OpenAlgebra} {InteractiveMathematics} {TIcalculatorHelp}
{SCC math resources... MathAS::id 4689, key 4689 | OER::textbook | playbook | MathGym::drop-in tutoring hours}
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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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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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Introduction to Functions

A function is a "relationship" that maps each input into one and only one output.

"Relationship" is quoted because functions can be defined in a variety of ways. The following was copied from the Wikipedia.

   "The mathematical concept of a function expresses the 
    intuitive idea of deterministic dependence between two 
    quantities, one of which is viewed as primary (the independent 
    variable, argument of the function, or its 'input') and the 
    other as secondary (the value of the function, or 'output'). 

In the Wikipedia's function definition, the word deterministic implies that every input to a function always results in the same output.

Terminology

The valid inputs to a function represents the domain of the function.

The range of a function are all the possible outputs.

Inputs are independent of the outputs, but the outputs are dependent on the inputs.

Functions are typically given one letter names. In many instances, inputs are labeled (named) 'x' and outputs are labeled (named) 'y'.

              +----------+
   input ---> | function | ---> output
     x        +----------+         y

   y = f(x)   or    f(x) = y

   y is the output, x is the input, f() is the function name

The following is a function with an implementation.

   f(x) = 2x                   [each input is multiplied by 2]

   f(-2) outputs -4            [input is -2]
   f(-1) outputs -2            [input is -1]
   f(0) outputs 0              [input is 0]
   f(1) outputs 2              [input is 1]
   f(2) outputs 4              [input is 2]

   f() is a function because each input produces only one output.
   Notice how each output value depends on the input value.

   The domain of f() is all real numbers.
   The range of f() is all real numbers.

   Function f() could be named doubler() because the output
   is always double (or 2 times) the input.  

   doubler(5) outputs 10
   doubler(-10) outputs -20
   doubler(5 * 3) outputs 30
   doubler(1 - 2 - 3) outputs -8

Public domain dot-png from Wikipedia.org...

Are They Functions?
   Is f() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  f()  | ---->  -1          f(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  f()  | ---->  -2          f(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  f()  | ---->  -3          f(3) = -3            (3, -3)
           +-------+


   Is g() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  g()  | ---->  -1          g(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  g()  | ---->  -2          g(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  g()  | ---->  -2          g(3) = -2            (3, -2)
           +-------+

   Is h() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  h()  | ---->  -1          h(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  h()  | ---->  -2          h(2) = -2            (2, -2)
           +-------+
           +-------+
   2 ----> |  h()  | ---->  -3          h(2) = -3            (2, -3)
           +-------+
           +-------+
   3 ----> |  h()  | ---->  -4          h(3) = -4            (3, -4)
           +-------+

   Is Q() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  Q()  | ---->  -1          Q(1) = -1            (1, -1)
           +-------+
           +-------+ 
   2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
           +-------+
           +-------+
   2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  Q()  | ---->  -3          Q(3) = -3            (3, -3)
           +-------+

If a function receives an input that is not in its domain, then the output of the function is "undefined." (i.e. A function does not work on inputs that are not in its domain.) If the domain of a function is not stated, then the its domain is all real numbers.

Related BABs and External Hyperlink(s)

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{SCC math resources... MathAS::id 4689, key 4689 | OER::textbook | playbook | MathGym::drop-in tutoring hours}
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