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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 1: THE NUMBER SYSTEM

 

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SECTION 1.1: THE NUMBER TREE AND DEFINITIONS

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Figure


 

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 aagnesi 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 Pre-Calculus 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 Pre-Calculus 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.

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SETS AND ELEMENTS
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A "set" is a collection of objects. Those objects are generally called "members or elements of the set."

"Roster form,", pronounced "raw-stir," 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
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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/russell-paradox
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THE REAL NUMBERS

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Natural Numbers or Counting Numbers: {1, 2, 3, 4, ...}

The set of all natural numbers is often denoted by the symbol natnum.
natnum = {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 integer.
interger synbol = {...,-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 rational 1.

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 numberDefinition : a number with a decimal point and there are a finite number of digits.

ex 1) .5612345             

ex 2) .239765534356575

repeating decimal numberDefinition : a number with a decimal point, and there are an infinite number of digits that have a repeating pattern.

ex. 1) .MATH  

ex 2) .MATH

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 non-terminating and non-repeating decimal number will be irrational.

Examples of irrational numbers:

ex 1)MATHwith no repetition of a pattern occurring.

ex. 2) MATHwith 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 numberdefinition : 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 numberdefinition : 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 numberdefinition : 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 numberdefinition : a natural number( also called a counting number) that is not prime.

examples:

a) 6 b) 8 c) 12 d) 30

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RADICALS AND RADICANDS
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The following symbol,$\sqrt{}$, is called a radical symbol. Often you will see a number inside the small "v" portion of the radical symbol like this: $\root{3} \of{}.$. The number 3 in the radical symbol $\root{3} \of{}.$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 $\root{5} \of{32}$ , 5 is the index and 32 is the radicand.  Expressions like $\root{3} \of{8}$ and $\root{5} \of{32}$ represents a number. Now how would you determine the number? Please see Chapter 9 for an explanation of MATH.

 

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SECTION 1.2: IRRATIONAL NUMBERS

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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?

$\sqrt{2}$ is a classic example of an irrational number. $\sqrt{2}$ 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 $\sqrt{2}$ 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 $\sqrt{2}$ IS IRRATIONAL

    Our goal is to prove that the $\sqrt{2}$ is not a rational number by a Proof By Contradiction. So here goes!

    Let's assume $\sqrt{2}$ is a rational number. Since I am claiming that $\sqrt{2}$ is a rational number then it must be able to be rewritten as a quotient of two integers.Right? Right!

    Therefore, I could write $\sqrt{2}=$ $\dfrac{a}{b},$ 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 $\sqrt{2}=$ $\dfrac{a}{b}$

    MATH  I am squaring both sides of the equation. This is a rule of Algebra.

    MATH     ( 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.

    b2 (MATH)         I multiplied both sides of the above equation by b2.

    2b 2 =   a 2  or  a 2  = 2b 2      Therefore, a2 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 MATH( see eq. 1 above ), this is what we get:

    MATH

    MATH      I squared 2k to get 4k2

    b 2 (MATH)    I'm multiplying both sides of the equation by b 2.

    $2b^{2}=4k^{2}$     Here is the result from the the previous step

    $b^{2}=2k^{2}$

    This means b2 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 $\dfrac{a}{b}$ 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 $\sqrt{2}$ 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 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.

     

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    SECTION 1.3: IMAGINARY AND COMPLEX NUMBERS

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    What is an "imaginary number"?

    imaginary numberdefinition : 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) $\sqrt{-4}$

    ex. 2) $\root{4} \of{-16}$

    ex. 3) MATH

     

    Now we will use the definition of an imaginary number for the definition of a "complex number."

    complex numberdefinition : an imaginary number added to or subtracted from a real number.

    Let me give you a few examples.

     

    ex. 1) $-5+\sqrt{-4}$      

    ex. 2) $7-\root{4}\of{-16}$    

    ex. 3) MATH


     

    -----------------------------
    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: X-INTERCEPT(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