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MAT081 :: Lecture Note :: Week 02
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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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The place values for whole numbers in the decimal (base10) number system are defined using positive integer powers of
10
.10^{0} is 1 ... ones 10^{1} is 10 ... tens 10^{2} is 100 ... hundreds 10^{3} is 1,000 ... thousands 10^{4} is 10,000 ... ten thousands 10^{5} is 100,000 ... hundred thousands 10^{6} is 1,000,000 ... millions ... 10^9 is billions; 10^12 is trillions; 10^15 is quadrillions 10^100 is googol; 10^googol is googolplex caret ^ is calculator notation for exponents (raising a number to a power) n^m = n^{m} (e.g. 5^3 = 5^{3})Example: The integer number
4,096
has four units of one thousand, zero units of one hundred, nine units of one ten, and six units of one. The digit '4' is in the "thousands place," the digit '0' is in the "hundreds place," the digit '9' is in the "tens place" and the digit '6' is in the "ones place."4 x 1000 = 4000 0 x 100 = 0 9 x 10 = 90 6 x 1 = 6 4000 + 0 + 90 + 6 = 4096The place values for decimal digits in the base10 number system are defined using negative integer powers of ten.
10^{1} equals 1/10 equals 0.1 10^{2} equals 1/100 equals 0.01 10^{3} equals 1/1000 equals 0.001 10^{4} equals 1/10,000 equals 0.0001 ... 10^{9} equals 1/1,000,000,000 equals 0.000000001To the left of the decimal point are the ones, tens, hundreds, thousands, and so on. On the fractional side of decimal number are the tenths, hundreths, thousandths, and so on. There are no oneths.
The number
0.5214
is less than one and greater than zero. The digit '5' is in the tenths place, the digit '2' is in the hundreths place, the digit '1' is in the thousandths place and the digit '4' is in the ten thousandths place.5 x 0.1 = 0.5 2 x 0.01 = 0.02 1 x 0.001 = 0.001 4 x 0.0001 = 0.0004 0.5 + 0.02 + 0.001 + 0.0004 = 0.5214PurpleMath.com:: Number Bases
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Wikipedia says..."
0
is a number and a numerical digit."Zero represents nothing (or null or nil or void or absence of value).
let 'a' represent any whole number a + 0 = a 55 + 0 = 55 a  0 = a 18  0 = 18 a * 0 = 0 33 * 0 = 0 a / 0 = not defined cannot divide by zero 0 / a = 0 0 / 13 = 0 0 is neither positive nor negative 0 is not a prime numberMost definitions indicate that 0 is an even number; however, some people believe 0 is neither even nor odd. Even numbers are integers that are evenly divisable by 2; thus, according to this definition, 0 is an even number.
GDT::BAB:: Google's Calculator Has Stopped DividingByZero [28 July 2005]
Wikipedia.org:: 0 (number)
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The number one often represents a unit (i.e. "a single undivided whole).
Let 'a' represent any whole number. a + 1 = (a incremented by 1) 7 + 1 = 8 a  1 = (a decremented by 1) 9  1 = 8 a * 1 = a 55 * 1 = 55 a / 1 = a 33 / 1 = 33 1 is an odd number 1 is not a prime number 1 is the first whole number?A fraction with one as its numerator is called a unit fraction.
Wikipedia.org:: 1 (number)
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Precedence means "priority of importance."
Given an expression such as
3 + 5 x 8
we need to concern ourselves about the order of evaluation. For example, if we add 3 and 5 and multiply the sum by 8, then we get 64 for an answer; however, if we multiply 5 by 8 and add 3 to the product, then we get an answer of 43.order of operations =================== groupings () []  exponents multiply, divide [equal precedence; lefttoright] addition, subtraction [equal precedence; lefttoright] 3 + 5 * 8 Multiply has a higher precedence than addition; therefore, do it first followed by the addition. 5 * 8 is 40, add 3 gives 43 (3 + 5) * 8 Grouping has the highest precedence; therefore, do it first and then mulitply. 3 + 5 is 8, times 8 gives 64 8 + 3  2 Addition and subraction have the same precedence; therefore, evaluate lefttoright. 8 + 3 is 11, subtract 2 gives 9 (4 + 1)^{2} Grouping has highest precedence; therefore, do it first. 4 + 1 is 5, 5 squared is 25 4 + 1^{2} Exponent has higher precedence than addition. 1 squared is 1, plus 4 gives 5 3 + 2  2 * 5 Separately evaluate the numerator (top) and denominator (bottom) and then divide the denominator into the numerator. 3 + 2 = 5 ... 2 * 5 = 10 ... 5 dividedby 10 = 0.5 Note: (3 + 2) / (2 * 5) does not equal 3 + 2 / 2 * 5What's PEMDAS?
The following was from a Fall 2004 student.
Please excuse my dear aunt sally. P  parenthesis E  exponent M  multiply D  divide A  add S  subtractThere are some special cases when using PEMDAS.
 brackets
[]
go ahead of parenthesis()
 multiply and divide are at the same level of precedence and they are evaluated lefttoright
 add and subtract are at the same level of precedence and they are evaluated lefttoright
B  brackets E  exponents D  divide M  multiply A  add S  subtractExternal Hyperlinks
 Wikipedia.org:: Order of operations
 PurpleMath.com:: Order of Operations: PEMDAS
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