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MAT081 :: Lecture Note :: Week 01
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Overview
Handouts
Assignments

Syllabus Review

It is the student's responsibility to read and understand the MAT081 Syllabus.

Prior to the start of classes, 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.

If your funds and schedule permit, then MAT108--Tutored Mathematics can be taken pass/fail for additional help with this course.

KhanAcademy.org::Arithmetic is a learning resource that is provided for free.

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Introduction to Arithmetic

Let's start learning about arithmetic with some quotes.

Arithmetic is being able to count up to twenty
without taking off your shoes.

-- Mickey Mouse (01928-????) { famous Walt Disney cartoon character; more... } [arithmetic]

The hardest arithmetic to master is that
which enables us to count our blessings.

-- Eric Hoffer (01902-01983) {American social philosopher; more...} [arithmetic]

The most unsuccessful four years in the education of a cost-estimator is fifth grade arithmetic.
-- Norman R. Augustine (01935-?????) { U.S. aircraft businessman; more... } [arithmetic]

If it's green, it's biology. If it stinks, it's chemistry.
If it has numbers, it's math. If it doesn't work, it's technology.

-- unknown source {21st century = computing + math + biology + science} [future]

I am not a number -- I am a free man!
-- Number Six (01967-01968) {The Prisoner [01967 UK science fiction television series]; more...} [life]

A teacher's day is half bureaucracy, half crisis, half monotony and one-eighth epiphany. Never mind the arithmetic.
-- Susan Ohanian { public school teacher and freelance writer } [teaching]

Half of Americans don't understand math and the other three-fourths don't care.
-- Marshall Trimble { Arizona's historian; more... } [arizona history/math]

Let's write the whole number 61 on the whiteboard and ask the question: What does the number 61 mean? {answer}

Without some form of unit, a number is a number is a number. Many of us prefix numbers with a dollar sign ($) when dealing with money. Roadtrippers and commuters suffix numbers with the unit "miles" to represent distances between two locations. Numbers are sometimes suffixed with the degree (°) symbol to represent temperatures when talking about the weather and cooking.

Arithmetic involves executing four basic operations on numbers: addition (+), subtraction (-), multiplication (x) and division (÷).

Definitions

A whole number is zero (0) and the counting (or natural) numbers 1, 2, 3 and so on. Integers are whole numbers that also include their opposites. For example, the opposite of number 5 is number -5 and the opposite of -7 is +7. Numbers prefixed with dash are called negative numbers.

Negative numbers are less than (<) zero, while positive numbers are greater than (>) zero. Zero is neither positive nor negative.

Two numbers added together result in a sum. Two numbers multiplied together result in a product. One number subtracted from another number results in a difference. One number divided by another number results in a quotient (with potentially a remainder).

PurpleMath.com:: Number Types

Wikipedia.org

Wikipedia.org is "the free encyclopedia" and the GDT website uses it frequently as a resource.

The Wikipedia is a community-based project that belongs to every user of the WWW. It is full of information, mis-information and dis-information; however, many parts of the Wikipedia has usable information.

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Introduction to the Number Line

Technically, a number line is a one-dimensional graph. [Give examples of two- and three-dimensional graphs.]

   ... ---+---+---+---+---+---+---+---+---+---+---+--- ...
         -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5

The ... notation implies continuation; in other words, it goes on and on and on. Infinity is when something goes on forever. { example}

Numbers to the right of zero are positive and those to the left are negative.

Negative numbers are prefixed (started) with a dash - character. Positive numbers can be prefixed with a plus + character, but usually the plus sign is not used.

   negative seven:  -7
   positive seven:  7 or +7

Zero is neither positive, nor negative; however, sometimes -0 and +0 may be used to represent real numbers that are close to zero. For example, nano represents one billionth which is the value 0.000000001.

YouTube.com::Numberphile::Number Line

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Some Basic Arithmetic Terminology

Addition Terminology

Addition is the combination of two or more numbers. The result of an addition is called a sum.

   general equation:  a + b = c

'a' is called the augend, 'b' is called the addend, while 'c' is called the sum.

   5 + 3 = 8

   5  is the augend 
   3  is the addend 
   8  is 5 added with 3
   +  is the arithmetic addition operator
   =  is the equals operator
Subtraction Terminology

Subtraction is the arithmetic operation in which the difference between two numbers is calculated.

   general equation:  a - b = c

'a' is called the minuend, 'b' is called the subtraend, while 'c' is called the difference.

   5 - 3 = 2

   5  is the minuend 
   3  is the subtrend 
   2  is the difference 5 and 3
   -  is the arithmetic subtraction operator
   =  is the equals operator

Subtraction is related to addition as follows. If a + b = c, then c - b = a and c - a = b.

If you subtract a larger number from a smaller number, then the difference is a negative number (i.e. it is less than zero). Negative numbers are prefixed with a dash - character.

   3 - 5 = -2

   3 minus 5 equals a negative 2 (i.e. -2)
   5 subtracted from 3 is a difference of -2
   the difference between 3 and 5 is -2
Multiplication Terminology

Multiplication is a technique for adding identical numbers.

   4 * 5  is equal to  4 + 4 + 4 + 4 + 4
   9 * 2  is equal to  9 + 9
   3 * 8  is equal to  3 + 3 + 3 + 3 + 3 + 3 + 3 + 3
   general equation:  a * b = c

'a' is called the multiplicand, 'b' is called the multiplicator, and 'c' is called the product.

For example, it is 11 miles between my home in Tempe and the SCC campus. The type (i.e. unit of measurement) is miles and 11 is a value. Round-trip my commute is 22 miles (2 times 11). Two times eleven can be written in the following ways.

   2 × 11
   2(11)
   2 ⋅ 11
   2 * 11

It is a good idea to memorize the 12-by-12 multiplication table.

Division Terminology

Division is the reverse operation of multiplication.

   general equation:  a / b = c

   'a' and 'b' and 'c' are "variables"
   variables are assigned "values"

'a' is called the dividend, 'b' is called the divisor, and 'c' is called the quotient.

Exercise: If fourteen (14) Artie Artichoke dolls are to be split evenly between two (2) people, how many dolls will each person receive?

   14 / 2
   14 ÷ 2
    _______
   2)14

   14 divided by 2 equals 7

   Given the general equation:  a / b = c
   In this specific problem:  'a' has the
   value 14, 'b' has the value 2, and 'c'
   has the value 7.

[side-bar] 14 is an even number because when divided by 2 there is zero remainder. In other words, 14 is evenly divisable by 2; therefore, 14 is an even number.

   7 / 2 = 3 with a reminder of 1

Division is the reverse operation of multiplication. If a * b = c and b is not zero, then the equation is equal to a = c / b.

   a * b = c
   ---------  let 'a' equal 4 and 'b' equal 2
   4 * 2 = 8

   a = c / b
   ---------
   4 = 8 / 2

Division by zero is undefined (i.e not allowed). {MathForum.org:: Ask Dr. Math: Dividing by Zero}

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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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Introduction to Statistics

Wikipedia.org offers the following definition for statistics.

   "Statistics is a type of data analysis which practice 
    includes the planning, summarizing, and interpreting 
    of observations of a system possibly possibly mathematical 
    model of future events based on a forecasting followed by 
    predicting or of the system being observed."

According to a t-shirt sold by Computer Gear, "53.7% of all statistics are made up." In other words, caution must be exercised when presented with statistics.

Definitions

A set is a collection of zero or more things. The things in a set are called elements. A set with no elements is the empty set.

In many math worlds, the things in a set are numbers. For example, a set might contain a collection of temperatures or a collection of test scores or a collection of counters and so on.

Set notation uses braces {} to enclose the elements of a set.

A set is typically populated using data obtained from either observation or experimentation.

The following are some of the basic operations performed on a set.

    range: difference between largest and smallest
           values in a set
     mean: the average of a set of numbers
   median: the number found in the middle of sorted set
     mode: most commonly occuring value in a set of numbers;
           a set can have zero or more modes

      Sample data set S:  { 5  10  10  7   3  }
   S sorted (ascending):  { 3  5   7   10  10 }

   range  = 10 - 3 = 7
   mean   = 5 + 10 + 10 + 7 + 3 = 35; 35/5 = 7
   median = 7
   mode   = 10 

   In this example, the range, mean, and median
   are all the same value; however, this is not
   always true.

He uses statistics as a drunken man uses lamp-posts... for support rather than illumination.
-- Andrew Lang (01844-01912) { Scottish poet, novelist, literary critic; more... } [statistics]

External Hyperlink(s)
BAB::Statistics 0.1

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