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MAT081 :: Lecture Note :: Week 02
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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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Base-10 Place Values

The place values for whole numbers in the decimal (base-10) number system are defined using positive integer powers of 10.

   100 is         1 ... ones
   101 is        10 ... tens
   102 is       100 ... hundreds
   103 is     1,000 ... thousands
   104 is    10,000 ... ten thousands
   105 is   100,000 ... hundred thousands
   106 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 = nm (e.g. 5^3 = 53)

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 = 4096

The place values for decimal digits in the base-10 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   0.000000001

To 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 = 0.5214

PurpleMath.com:: Number Bases

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Some Properties of Zero

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 number

Most 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 Dividing-By-Zero [28 July 2005]

Wikipedia.org:: 0 (number)

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Some Properties of One

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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Operator Precedence (Order of Operations)

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; left-to-right]
   addition, subtraction     [equal precedence; left-to-right]

   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 left-to-right.
      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 + 12

      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 divided-by 10 = 0.5
      Note: (3 + 2) / (2 * 5) does not equal 3 + 2 / 2 * 5
What'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 - subtract

There are some special cases when using PEMDAS.

   B - brackets
   E - exponents
   D - divide
   M - multiply
   A - add
   S - subtract
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