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MAT081 :: Lecture Note :: Week 01
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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 CM441A is available for help learning arithmetic and algebra.
If your funds and schedule permit, then MAT108Tutored 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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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 (0190201983) {American social philosopher; more...} [arithmetic]The most unsuccessful four years in the education of a costestimator 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 (0196701968) {The Prisoner [01967 UK science fiction television series]; more...} [life]A teacher's day is half bureaucracy, half crisis, half monotony and oneeighth epiphany. Never mind the arithmetic.
 Susan Ohanian { public school teacher and freelance writer } [teaching]Half of Americans don't understand math and the other threefourths 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) numbers1, 2, 3
and so on. Integers are whole numbers that also include their opposites. For example, the opposite ofnumber 5
isnumber 5
and the opposite of7
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 communitybased project that belongs to every user of the WWW. It is full of information, misinformation and disinformation; however, many parts of the Wikipedia has usable information.
 Wikipedia.org:: Arithmetic ... Number Line ... Number
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Technically, a number line is a onedimensional graph. [Give examples of two and threedimensional graphs.]
... +++++++++++ ... 5 4 3 2 1 0 +1 +2 +3 +4 +5The
...
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 +7Zero 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 value0.000000001
.YouTube.com::Numberphile::Number Line
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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 operatorSubtraction 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 operatorSubtraction is related to addition as follows. If
a + b = c
, thenc  b = a
andc  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 2Multiplication 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 + 3general 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) ismiles
and11
is a value. Roundtrip my commute is22
miles (2
times11
). Two times eleven can be written in the following ways.2 × 11 2(11) 2 ⋅ 11 2 * 11It is a good idea to memorize the 12by12 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.[sidebar]
14
is an even number because when divided by2
there is zero remainder. In other words,14
is evenly divisable by2
; therefore,14
is an even number.7 / 2 = 3 with a reminder of 1Division is the reverse operation of multiplication. If
a * b = c
andb
is not zero, then the equation is equal toa = c / b
.a * b = c  let 'a' equal 4 and 'b' equal 2 4 * 2 = 8 a = c / b  4 = 8 / 2Division by zero is undefined (i.e not allowed). {MathForum.org:: Ask Dr. Math: Dividing by Zero}
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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 = 5^{3} (repeated multiplication)Exponents imply the operation of "raising to a power." For example,
10^{5}
is read "10 raised to the 5th power," with10
being the base and5
the exponent (or power).By mathematical law (definition), any nonzero number raised to the power of zero is one.
5^{0} equals 1 9999^{0} equals 1 82^{0} equals 1 42^{0} equals 1 (42)^{0} equals 1 0^{0} equals ??? [Google says 1]Any number raised to the power of one is that number.
3^{1} equals 3 9999^{1} equals 9999 42^{1} equals 42Any number raised to the mth power (where
m > 1
) is that number multiplied by itself 'm' times.5^{4} = 5 * 5 * 5 * 5 3^{7} = 3 * 3 * 3 * 3 * 3 * 3 * 3 2^{3} = 2 * 2 * 2 5^{2} = (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 of3
.8 squared is 8^{2} which equals 8 * 8 4 cubed is 4^{3} which equals 4 * 4 * 4BAB:: Square Number Playing Results in Square Number Discovery [24 May 2007]Recall, the base10 (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.
10^{0}
equals1
(one)
10^{1}
equals10
(ten [deka])
10^{2}
equals100
(hundred [hecto])
10^{3}
equals1000
(thousand [kilo])
10^{4}
equals10,000
(10 thousand)
10^{5}
equals100,000
(100 thousand)
10^{6}
equals1,000,000
(million [mega] [1,000 thousand])
10^{7}
equals10,000,000
(10 million)
10^{8}
equals100,000,000
(100 million)
10^{9}
equals1,000,000,000
(billion [giga] [1,000 million])
...
10^{12}
equals1,000,000,000,000
(trillion [tera] [1,000 billion])
...
10^{15}
equals1,000,000,000,000,000
(quadrillion [peta] [1,000 trillion])More... On Notation
The caret
^
symbol is sometimes used to imply exponents. GDT calls this "calculator notation."2^4 = 2^{4} = 2 * 2 * 2 * 2 = 16 8^3 = 8^{3} = 8 * 8 * 8 = 512 1^2 = 1^{2} = 1 * 1 = 1More... One Expression That Contains Lots of Math
Let's take a peek at the following expression reading it from lefttoright.
a^n ⋅ 1 = a^n = a^(n + 0) = a^n ⋅ a^0More... 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 base10 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 + 10^0External Hyperlink(s)
 Wikipedia.org:: Exponent (Exponentiation) [opens new window]
 PurpleMath.com:: Exponents: Basic Rules [opens new window]
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A nonzero number (base) raised to a negative exponent is equal to one dividedby the number raised to the absolute value of the exponent.
1 a^{b} where 'b' is less than 0 equals  a^{b} 7^{2} equals 1/7^{2} equals 1/49 2^{5} equals 1/2^{5} equals 1/32 10^{2} equals 1/10^{2} equals 1/10010^1 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 femtoWhat'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 is5
.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 = 6Function notation: Sometimes
abs(n)
is the absolute value function. The function outputs the absolute value of inputn
.External Hyperlinks
 PurpleMath.com:: Negative Exponents
 YouTube.com::Power of Ten  Are We Alone In The Universe
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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 tshirt 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 lampposts... for support rather than illumination.
 Andrew Lang (0184401912) { Scottish poet, novelist, literary critic; more... } [statistics]External Hyperlink(s)
 KhanAcademy.org::Statistics intro: mean, median and mode [opens new window]
 YouTube.com:: Mean Median and Mode [opens new window]
 ComputerGear.com::Tshirt:: 53.7% of all statistics are made up [opens new window]
BAB::Statistics 0.1
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