
SECTION 1.1: THE NUMBER TREE
AND DEFINITIONS

Maria Gaetana Agnesi
Maria Gaetana Agnesi was born in Milan on May 16, 1718, to a wealthy and literate family. She was the oldest of 21 children. Her father was a professor of mathematics and provided her a profound education. She was recognized as a child prodigy very early; spoke French by the age of five; and had mastered Latin, Greek, Hebrew, and several modern languages by the age of nine. At her teens, Maria mastered mathematics.
By the age of twenty, she began working on her most important work, Analytical Institutions, dealing with differential and integral calculus. It is said that she started writing Analytical Institutions as a textbook for her brothers, which then grew into a more serious effort. When her work was published in 1748, it caused a sensation in the academic world. It was one of the first and most complete works on finite and infinitesimal analysis. Maria's great contribution to mathematics with this book was that it brought the works of various mathematicians together in a very systematic way with her own interpretations. The book became a model of clarity, it was widely translated and used as a textbook.
In Accelerated PreCalculus and Physics you will be reading math and physics
textbooks that will be using the language of physics and mathematics. In order
to fully appreciate the concepts that you will be encountering in Accelerated
PreCalculus and Physics(i.e. TEAMS) course, you MUST learn this language.
In TEAMS it is assumed that you have some basic familiarity with the ideas about to be expressed. Above is the number tree with branches spreading out to many types of numbers. In TEAMS you will be
dealing with the REAL NUMBER system. You will seldom touch on COMPLEX
NUMBERS. Now on to the REAL NUMBER system.
The real number system uses the digits 0,1,2,3,4,5,6,7,8, and 9 to construct
all real numbers. The ten digits build what is called a "base
ten" number system.

SETS AND ELEMENTS

A "set" is a collection of objects. Those objects are generally called "members or elements of the set."
"Roster form,", pronounced "rawstir," is a way of writing a set. If you put a list of elements between two "set brackets" you are using "roster form."
Here are some examples of sets in roster form:
ex. 1) {January, June, July}
The elements of this set are January, June and July.
ex. 2) {1, 2, 3, 4, 5}
The elements of this set are 1, 2, 3, 4 and 5.
ex.3) {1, 3, 5...}
The elements of this set are all the positive odd numbers.
This set of odd numbers goes on "forever." The "ellipsis," ... , indicates the fact that the odd numbers go on forever. When a set of elements go on "forever" we say the set is "infinite" which means there are "infinitely many members in the set." Thus, the set of positive even numbers can be written as {2, 4, 6, 8, ... } and is also infinite.
A MIND BENDER  RUSSELL'S PARADOX

IS THERE A SET OF ALL THE SETS THAT DO
NOT CONTAIN THEMSELVES AS MEMBERS?
Teacher Comment: Language is an odd tool. Humans can construct sentences that that are grammatically correct but produce interesting contradicions. 
See http://plato.stanford.edu/entries/russellparadox


THE REAL NUMBERS

Natural Numbers or Counting Numbers: {1, 2, 3, 4,
...}
The set of all natural numbers is often denoted by the symbol .
= {1, 2, 3, 4,
...}
Whole Numbers: {0, 1, 2, 3, 4,...}
The whole numbers include the
Natural Numbers. The set of all whole numbers is not named by any symbol.
Integers: {...,4, 3,2, 1,0, 1, 2, 3, 4, 5,...}
These include Whole and Natural numbers. The set of all integers is often denoted by the symbol .
= {...,4, 3, 2, 1, 0, 1, 2, 3, 4, 5,...}
Rational Numbers: numbers that can be expressed as a
ratio(a fraction) of two integers, r/s,
and s ≠ 0 and r/s is in "lowest terms." The set of all rational numbers is often named by the symbol .
Comment: Many students have a difficult time thinking about rational numbers so lets consider some examples. Let's create some definitions first. These definitions will help us understand rational numbers.
terminating decimal number_{Definition }: a number with a decimal
point and there are a finite number of digits.
ex 1) .5612345
ex
2) .239765534356575
repeating decimal number_{Definition }: a number with a decimal
point, and there are an infinite number of digits that have a repeating
pattern.
ex. 1)
.
ex
2)
.
Comment: a line above a sequence of digits is an indicator that those digits will
continually repeat.
Any terminating or
repeating decimal number can be expressed as a ratio of two
integers.
Below are 5 examples of rational numbers:
ex 1) 10 = 20/2 note: 10 can be rewritten as a fraction of two integers.
ex. 2) .333... = 1/3 note: .333... is a repeating decimal number and can be rewritten as a fraction of two integers.
ex. 3).25 = 1/4 note: .25 is a terminating decimal number and can be rewritten as a fraction of two integers.
ex. 4) 5 = 50/10 note: 5 can be rewritten as a fraction of two integers.
ex. 5) .75 = 3/4 note: .75 is a terminating decimal number and can be rewritten as a fraction of two integers.
Irrational number: numbers that cannot be expressed as
the ratio of two integers.
The set of all irrational numbers is often denoted by a boldface I.
note: Any nonterminating and nonrepeating decimal
number will be irrational.
Examples of irrational numbers:
ex 1)with no repetition of a pattern occurring.
ex.
2) with no repetition of a pattern occurring.
EVEN, ODD, PRIME AND COMPOSITE NUMBERS
The numbers ...6, 4, 2, 0, 2, 4, 6, 8, 10... are even
numbers and the numbers ...7, 5, 3, 2, 1, 1, 3, 5, 7 ... are odd numbers. The words even and odd only apply to integers.
even number_{definition }: a number that can be written in the
form 2n, where n is an integer.
examples:
a) 6 is even because it can be written as 2 x 3 where n = 3 and 3 is an integer.
b) 50 is even because it can be written as 2 x 25 where n = 25 and 25 is an integer.
c) 100 is even because it can be written as 2 x 50 where n = 50 and 50 is an integer.
odd number_{definition }: a number that can be written in the
form 2n + 1, where n is an integer.
examples:
a) 5 is odd because 5 can be written as 2 x 2 + 1 where n = 2 and 2 is an integer.
b) 11 is odd because 11 can be written as 2 x 5 + 1 where n = 5 and 5 is an integer.
c) 13 is odd because 13 can be written as 2 x (7) + 1 where n = 7 and 7 is an integer.
prime number_{definition }: an integer greater
than 1 which can only be divided by 1 and itself with a remainder of 0.
comment: Some students have a hard time with this idea so make sure you think about this!
examples:
a) 2 is prime since it can be divided by 1 and 2 having a remainder of 0: 2/1 = 2 remainder 0 and 2/2 = 1 remainder 0.
b) 5 is prime since it can be divided by 1 and
5 having a remainder of 0: 5/1 = 5 remainder 0 and 5/5 = 1 remainder 0
composite number_{definition }: a natural
number( also called a counting number) that is not prime.
examples:

RADICALS AND RADICANDS

The following
symbol,,
is called a radical symbol. Often you will see a number
inside the small "v" portion of the radical symbol like this: . The
number 3 in the radical symbol is called the index. Also, you will see a number inside the radical symbol like this: . The number inside the radical symbol, in this instance 8, is called the radicand. Lets look at another example. In ,
5 is the index and 32 is the radicand. Expressions like and represents
a number. Now how would you determine the number? Please see Chapter 9 for an explanation of .

SECTION 1.2: IRRATIONAL NUMBERS

What is an "irrational number"? An irrational number
is a number that cannot be expressed as a fraction r/s ,
where r and s are integers and s ≠ 0. Also, irrational numbers have decimal expansions that neither terminate nor
repeat. Hmmm, very hard to think about this idea, isn't it?
Math teachers usually do not explain to you how one knows a number is
irrational. This is due to the fact that one needs some sophisticated logical
skills and a deep understanding of the characteristics of numbers.
For example, you have been told that
pi(π) is an irrational number. But has anyone ever explained to you how a
mathematician determined the mathematical fact that pi cannot be expressed as
a fraction(ratio) of two integers?
is a classic example of an irrational number. is asking you "what number raised to the power of 2 will produce 2?" Go ahead,
start guessing!
Let's try to convince you that the is irrational. We will use a classical proof technique called "Proof by Contradiction." The "Proof by Contradiction" is also known as reductio ad absurdum, which is Latin for "reduce it to something absurd". Here's the idea:
Assume what you are trying to prove is untrue.
Based on that assumption hope to reach one or more conclusions that that you know in fact are not true.
If the assumption produces these obvious conclusions that are not true then the assumption must be false
THE PROOF THAT THE IS
IRRATIONAL
Our goal is to prove that the is not a rational number by a Proof By Contradiction. So here goes!
Let's assume is a rational number. Since I am claiming that is a rational number then it must be able to be rewritten as a quotient of two integers.Right? Right!
Therefore, I could write where "a", "b" are integers, and b ≠ 0 ,
and a/b is simplified to the lowest terms.
So lets follow out this assumption with our knowledge of algebra and logic.
if
I am squaring both sides of the equation. This is a rule of Algebra.
→ ( eq. 1) Here is what results from squaring both sides. Note that when one squares a square root one gets the radicand. This equation will be used later in this proof.
→ b^{2} () I multiplied both sides of the above equation by b^{2}.
→ 2b^{ 2 }= a^{ 2} or a^{ 2} = 2b^{ 2} Therefore, a^{2 } is
an even number since it is equal to two times an integer.
Comment: Please note that ANY integer mutiplied by 2 ALWAYS results in an even number.
Therefore,
"a"
also is an even number. (see comment below)
comment: Why is "a" and even number? Because of the "mathematical fact" that if a number squared results in an even number then the number being squared will also be an even number. Please think
about this for a moment, and use examples to verify this "mathematical
fact."
Let's continue with our proof.
If
"a"
itself is an even number, then
"a"
is 2 times some other integer, or a = 2k where k is this integer.
If we substitute 2k for a into the equation ( see eq. 1 above ),
this is what we get:
I squared 2k to get 4k^{2}
b^{ 2 }() I'm multiplying both sides of the equation by b^{ 2}.
Here is the result from the the previous step
This means b^{2 } is
even. Therefore, b itself is an even number!!!
HENCE A CONTRADICTION! Now why is this a contradiction? Because we started the whole
process saying that is simplified to the lowest terms. It turns out that our proof concluded that "a" and "b" are both even, and must have a common factor, therefore they are NOT reduced to lowest
terms which is a contradiction.
QED: So the cannot
be rational!

SUMMING UP: EXAMPLES OF RATIONAL AND IRRATIONAL
NUMBERS
ex. 1) Is "0.83333333333333333333333333333333..." a rational
or an irrational number? In this decimal the digit "3" is infinitely repeated.
Therefore it can be transformed into a fraction. In fact it is equal to the
fraction 5/6. Therefore "0.83333333333333333333333333333333..." is a rational
number.
ex. 2) Is "3.60555127546398929311922126747050..." a rational
or an irrational number? This is an infinite decimal, and it does not appear
to be an infinite repeating decimal. It is therefore cannot be transformed in
a fraction, and is considered to be an irrational number.
ex. 3) Is
"3.605551275463989293119221267470507050705070507050..." a rational or an
irrational number? This number is obscenely long, but eventually the digits
"7050" start repeating over and over again. Therefore this is a repeating
decimal, which makes it a rational number.
ex. 4) Is "3.1415926535897932384626433832795..." a rational
or an irrational number? This is an infinite decimal, and it does not appear
to be a infinite repeating decimal. It is therefore an irrational number. In
fact this is a very important irrational number known as "pi".
ex. 5) Is "0.31415926535897932384" a rational or an
irrational number? This gargantuan number is obscenely long, but it does end.
Therefore since it is a finite decimal this number can be transformed into the
fraction .
Therefore 0.31415926535897932384 is a rational number.
ex. 6) Is 5.6 a rational or irrational number? 5.6 is a
finite decimal. It can be changed into the fraction 28/5.
Therefore 5.6 is considered to be rational.

SECTION 1.3: IMAGINARY AND
COMPLEX NUMBERS

What is an "imaginary number"?
imaginary number_{definition } : a radical with a positive even integer index and a radicand that is a negative
real number.
Let me give you a few examples.
ex. 1)
ex. 2)
ex. 3)
Now we will use the definition of an imaginary number for the definition of a
"complex number."
complex number_{definition} : an imaginary
number added to or subtracted from a real number.
Let me give you a
few examples.
ex. 1)
ex. 2)
ex. 3)



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
