.5 Solving Equations Containing **Rational** Expressions Integrated Review-Summary on **Rational** Expressions Integrated Review-Summary on **Rational** Expressions 7.6 Proportion and Problem Solving with **Rational** Equations 7.6 Proportion and Problem Solving with **Rational** Equations 7.7 Simplifying Complex Fractions 7.7 Simplifying Complex Fractions 10. Radicals, **Rational** Exponents, and Complex **Numbers** 10. Radicals, **Rational** Exponents, and Complex **Numbers** 10.2 **Rational** Exponents 10.2 **Rational** Exponents 10.3 Simplifying/

Realizations; Design **Classes** Introduction to **Rational** Rose 2000 v6.5 Copyright © 1999 **Rational** Software, all rights reserved **9** Where are we – Subsystem Design Realizations between the subsystem’s interfaces and contained **classes** are defined. Detailed/**for** the remaining messages. Remember, you must add them in order.. Note: What kind of **class** is RegisterForCoursesForm? Note the **number** of messages from the actor (external) to this **class**. Introduction to **Rational** Rose 2000 v6.5 Copyright © 1999 **Rational**/

A decide. Whatever A gets, B gets also. B knows people fall into two **classes**, those choosing ONE and those choosing BOTH. B would very much want A to / mathematical derivations). P gets an answer or two **For** X =1, P gains **9** **For** all other X, P gains ~.2X (the larger the **number** of simulation trials, the closer is the convergence /answer is 1/3 (and have written journal articles saying so). What then is the **rational** answer? The Absent-Minded Driver Perhaps Sleeping Beauty is not such a paradox (social consensus /

**for** housekeeping before Python reclaims object memory **9** **Class** Attributes – One copy of attribute shared by all objects of a **class** – Represents “**class**-wide” information Property of the **class**, not of an object of the **class** – Initialized once in the **class**/15 **Rational** **numbers** in Python Strange rounding-off errors: Create **class** **for** representing **rational** **numbers** by storing numerator and denominator >>> 2/3.0 0.6666666666666663 >>> 2/6.0 0.3333333333333331 >>> 5/6.0 0.8333333333333337 16 **Rational** **numbers** **rational**./

**class** **Rational** { friend ostream& operator<< (ostream&, const **Rational**&); friend istream& operator>> (istream&, **Rational**&); public: **Rational** (int n = 0, int d = 1); private: int numerator; int denominator; }; Example (Cont ..) ostream& operator<< (ostream &out, const **Rational** &r) { out << r.numerator << "/" << r.denominator; return out; } istream& operator>> (istream &in, **Rational** &r) // complete the code main() **Rational** rObj; cout << "Enter a **rational** **number** like 2/**9**: "; cin >> rObj; cout << "The **rational**/

**9**.00001; 0.75 Irrational **numbers** A **number** that is not **rational** is called irrational. A decimal like 3.5655655565555655556… is not **rational** because although there is a pattern, it does not repeat. It is an irrational **number**. Compare this to 3.556556556556556556… It is **rational** because 556 repeats. It is a **rational** **number**/to find the product. Standard Algorithm **for** Multiplying Decimals Why do we do what we do? Multiply 2.1 × 1.3 Explain the algorithm. Dividing Decimal **Numbers** What model should we use? Dividing/

] Display the data Sample of program import java.awt.Container; import javax.swing.*; public **class** lab91 extends JApplet { double[] **numbers**; String output=""; public void init() { **numbers** = new double[**9**]; **for** (int i = 0; i < **9**; i++) { **numbers**[i] = Double.parseDouble(JOptionPane.showInputDialog( "Enter a floating-point value" )); } **for** (int i = 0; i < **9**; i++) { output += "**number**" + (i + 1) +": "+**numbers**[i]+" "; } JTextArea outputArea = new JTextArea(150, 300); outputArea.setText( output); Container container/

.6666…; 0.727272… Terminating decimals: 4.8; **9**.00001; 0.75 Irrational **numbers** A **number** that is not **rational** is called irrational. A decimal like 3.5655655565555655556… is not **rational** because although there is a pattern, it does not repeat. It is an irrational **number**. Compare this to 3.556556556556556556… It is **rational** because 556 repeats. It is a **rational** **number**. Comparing Decimals When are decimals equal? 3.56/

<<(long); ostream& operator<<(double);... 6 Output of User-Defined Types #include **class** **rational** { int num, den; public: **rational**(int n=0, int d=1) { num=n; den = d/**number** of characters read by the last input operation 32 #include void main() {char v[20]; int n; cin.read( v, 10); n = cin.gcount() ; cout < >dec >>n; cin >>hex >>m; cout< > ws >> x; cout << tab <<"x = "<< tab << x << ENDL; }//ex10-**9**.cpp 41 How Does it Work? **class**/iostream.h u strstream.h//strstrea.h **for** VC++ 80 Types of ostrstream Objects/

.6666…; 0.727272… Terminating decimals: 4.8; **9**.00001; 0.75 Irrational **numbers** A **number** that is not **rational** is called irrational. A decimal like 3.5655655565555655556… is not **rational** because although there is a pattern, it does not repeat. It is an irrational **number**. Compare this to 3.556556556556556556… It is **rational** because 556 repeats. It is a **rational** **number**. Comparing Decimals When are decimals equal? 3.56/

**Rational** Software Corporation Multiple Systems Visual Modeling Promotes Reuse Reusable Components Page **9** R Copyright © 1997 by **Rational** Software Corporation What is the UML? UML stands **for** /**Rational** Software Corporation Multiplicity and Navigation Multiplicity defines how many objects participate in a relationships – Multiplicity is the **number** of instances of one **class** related to ONE instance of the other **class** – **For** each association and aggregation, there are two multiplicity decisions to make: one **for**/

– some of these terms are ‘used’ interchangeably… E.g. Design model contains **class** diagrams, sequence diagrams, and a **number** of others…. Diagrams: means by which we view of the system. Different building blocks (model elements) **for** different types. E.g.: **classes**, interfaces, collaborations, components, nodes, dependencies, generalizations, and associations. Introduction to **Rational** Rose 17 Views Just as there are many views of a house under/

whole **numbers** with no common factor. **For** example, express 36 + 8 as 4 (**9** + 2). (6.NS.B.4) (DOK 1) Apply and extend previous understandings of **numbers** to the system of **rational** **numbers**. (6.NS.C) 5.Understand that positive and negative **numbers** are/ Energy transformations Conservation of energy “Conservation of energy and increase in disorder” is an essential concept of a world-**class** secondary science curriculum. Included in “conservation of energy and increase in disorder” is the following content: The total /

Unlike Fractions Lesson 2-7Lesson 2-7Solving Equations with **Rational** **Numbers** Lesson 2-8Lesson 2-8Problem-Solving Investigation: Look **for** a Pattern Lesson 2-9Lesson 2-9Powers and Exponents /0.182 D.0.252 SCHOOL In Mrs. Townley’s eighth grade science **class**, 4 out of 22 students did not turn in their homework. Find the/ Lesson **9** Ex3 Evaluate Powers Evaluate **9** 5. **9** 5 = **9** ● **9** ● **9** ● **9** ● 9Definition of exponents Answer: 59,049 = 59,049Simplify. Check using a calculator. **9** 5 59049 ENTER 1.A 2.B 3.C 4.D Lesson **9** CYP3/

{j, k} ≠ {l, m}. Proof: Consider n = 1729, j = **9**, k = 10, l = 1, m = 12. Now just check that the equalities hold. Another Constructive Existence Proof Theorem: **For** any integer n>0, there exists a sequence of n consecutive composite integers. - Same statement/intelligent student in the **class**, and Y≠X. Therefore, the premises do not imply the conclusion. Therefore, the premises do not imply the conclusion. Example : Q3 Q3: Prove that the sum of a **rational** **number** and an irrational **number** is always irrational. /

our next **class** meeting Averaging Real **Numbers** To average a set of real **numbers** we add all the **numbers** and then divide by the **number** of **numbers** in the set Find the average of the following set of **numbers**:. Divisibility A real **number** is divisible by another if the division has no remainder On the following slides are tests **for** divisibility by all the **numbers** between 2 and **9**, except **for** 7/

in **class**, the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will be able to use properties of **rational** and irrational **numbers** to /properties of **rational** and irrational **numbers** to write, simplify, and interpret expressions based on contextual situations. Like Radicals… Two radical expressions are like radicals if they have the same radicand. **For** instance, /5) 2√6 + 10√18 2√6 + 10√**9**∙2 2√6 + 30√2 3√24 - 15√8 3√(46) - 15√(42) (32)√6 - 152√2 6√/

C 4.D Example 2 A.5.1 B.5.15 C.5.2 D.5.**9** Example 3 AGRICULTURE A farmer lost the fruit on 8 of 15 orange trees because /148 C.0.182 D.0.252 SCHOOL In Mrs. Townley’s eighth grade science **class**, 4 out of 22 students did not turn in their homework. Find the fraction /3 CHEMISTRY The values **for** the approximate densities of various substances are shown in the table. Order the densities from least to greatest. Order **Rational** **Numbers** Write each fraction as a decimal. Example 3 Order **Rational** **Numbers** Answer: From the/

Page **9** Copyright © 1997 by **Rational** Software Corporation Putting the UML to Work The ESU University wants to computerize their registration system – The Registrar sets up the curriculum **for** a/**Rational** Software Corporation Multiplicity and Navigation Multiplicity defines how many objects participate in a relationships – Multiplicity is the **number** of instances of one **class** related to ONE instance of the other **class** – **For** each association and aggregation, there are two multiplicity decisions to make: one **for**/

(StudentInfo) name tenureStatus ScheduleAlgorithm Page 32 R Copyright © 1998 by **Rational** Software Corporation Multiplicity and Navigation Multiplicity defines how many objects participate in a relationships – Multiplicity is the **number** of instances of one **class** related to ONE instance of the other **class** – **For** each association and aggregation, there are two multiplicity decisions to make: one **for** each end of the relationship Although associations and aggregations are/

(define (angle z) (cdr z)) **9** Tagged Data n In order to support both types of complex **numbers** we need to be able to guarantee that,/**number** scheme-**number**) (lambda (x y) (tag (* x y)))) 40 (put ’div ’(scheme-**number** scheme-**number**) (lambda (x y) (tag (/ x y)))) (put ’make ’scheme-**number** (lambda (x) (tag x))) ’done) ;; External constructor **for** ordinary **numbers**: ;; Sch-Num --> **Number** (define (make-scheme-**number** n) ((get ’make ’scheme-**number**) n)) 41 **Rational** **Number** - generic **number** arith. (define (install-**rational**/

**9**: Do the right thing A new **class** should be consistent with the rules of the language. It should respond to standard messages, it should behave properly with typical functions (assuming the type allows that kind of call). "The Practice of Computing Using Python", Punch & Enbody, Copyright © 2013 Pearson Education, Inc. An example Consider a **Rational** **number** **class**/ Education, Inc. __radd__ vs __add__ 1 + r1 try 1.__add__(r1), failure look **for** an __radd__ if it exists, remap 1 + r1 r1.__radd__(1) "The Practice /

**Rational** Software, all rights reserved **9** VOPC – more on last slide… The generalization relationship has been shown because it is important to understand the relationship between the two new **classes**, ScheduleOfferingInfo and PrimaryScheduleOfferingInfo. The PrimaryScheduleOfferingInfo **class** contains information about a primary CourseOffering that is schedule-specific. **For**/ 1998-1999 **Rational** Software, all rights reserved 14 ClassName > Attribute : Type = InitValue CourseOffering > **number** :String=“100” /

Perfect Square The square of an integer **For** this **class**, you are required to memorize all perfect squares from /**number** is -4.5669? **rational** #4 Give an example of a whole **number** that isn’t positive. REAL **NUMBERS** **Rational** **numbers** and irrational **numbers** form the set of real **numbers**. Subsets of real **numbers**: Natural Whole Integer **Rational** Irrational Inequalities An inequality is a mathematical sentence comparing the values of two expressions using an inequality symbol. 012345678-8-7-6-5-4-3-2-**9**/

moment and review our preferred practices. Introduction to **Rational** Rose 2000 v6.5 Copyright © 1999 **Rational** Software, all rights reserved 23 Preferred Practices **Class** Design Relationships Generalization Use **for** “is a” relationships. Don’t use simply to make definitions visible, except **for** > **classes**. Aggregation Use **for** “part of” relationships. Don’t use in place of dependency between loosely coupled **classes**. Navigability Reduce bi-directional to unidirectional relationships, where/

Points 7. Relationships 8. Use-Case Diagrams **9**. Other Diagrams/enclosures Use-Case Report: Template The Use-Case / **Number** and name sections to simplify reviewing Remember the description is **for** people to read, not **for** computers **Rational** Requirements Management with Use Cases v5.5 Copyright © 1998-2000 **Rational** Software/ **Rational** Requirements Management with Use Cases v5.5 Copyright © 1998-2000 **Rational** Software, all rights reserved 51 Exercise: Non-Functional Requirements **For** your **class** /

**Rational** Software Corporation Rose **for** Apex, v. 4.0, p. 8 Creating Subsystems & Views R Copyright © 1996 by **Rational** Software Corporation Rose **for** Apex, v. 4.0, p. **9** Creating New View R Copyright © 1996 by **Rational** Software Corporation Rose **for** Apex, v. 4.0, p. 10 Step 3: Select the **Classes**/ The default **number** of errors is set to 30 R Copyright © 1996 by **Rational** Software Corporation Rose **for** Apex, v. 4.0, p. 12 Error Limit (Model Property) R Copyright © 1996 by **Rational** Software Corporation Rose **for** Apex, v/

that must interact with the system under development StudentRegistrarProfessorBilling System Page **9** R Copyright © 1997 by **Rational** Software Corporation Use Cases A use case is a pattern of /**Rational** Software Corporation Multiplicity and Navigation Multiplicity defines how many objects participate in a relationships – Multiplicity is the **number** of instances of one **class** related to ONE instance of the other **class** – **For** each association and aggregation, there are two multiplicity decisions to make: one **for**/

**Class** Standard 8.NS.1 Menu **RATIONAL** NUMBERSIRRATIONAL **NUMBERS** MULTIPLYING IRRATIONAL **NUMBERS** TERMINATING VS. REPEATING TERMINATING DECIMAL QUIZ **Rational** **Numbers** A **rational** **number** is any real **number** that can be expressed as the quotient or fraction of two integer Example 1.5 is a **rational** **number** because 1.5= 3/2 It can be written as a fraction More Examples NumberAs a Fraction **Rational**/ 2 x 11. The **number** of digits that repeat are referred to as the period of the repeating decimal. **For** example, thirds have a /

a matter of “**rational** choice?” In order **for** behavior to be **rational** - must this behavior/is not limited by **class** or other social variables. STRENGTHS/**number** of motivated criminals. The strength of this approach is that it can explain fluctuations in crime and delinquency rates and shows how victim behavior can influence criminal choices. Routine activities provides a macro view of crime, **rational**/**9**. Target removal 10. Identifying property 11. Reducing temptation 12. Denying benefits Reducing Anticipated Rewards **9**/

, 45, 47, 49, 50, 61, 70, 72, 78. HOME LEARNING: Review **for** test on Friday 11/20/15. Go to review section of your textbook. AGENDA: PRE-CALCULUS HONORS. DAY 60; FRI. NOV.20, 2015; (2 nd **9**- WEEKS) CHAPTER TEST: POLYNOMIAL AND **RATIONAL** FUNCTIONS LEARNING OBJECTIVES; SWBAT:- Solve complex **numbers**, quadratic functions, **rational** functions and direct, inverse, joint and combination variation problems. ACTIVITIES: Chapter/

() addStudent(StudentInfo) name tenureStatus ScheduleAlgorithm Page 15 Copyright © 1997 by **Rational** Software Corporation Multiplicity and Navigation Multiplicity defines how many objects participate in a relationships – Multiplicity is the **number** of instances of one **class** related to ONE instance of the other **class** – **For** each association and aggregation, there are two multiplicity decisions to make: one **for** each end of the relationship Although associations and aggregations are/

**For** each use-case realization Find **Classes** from Use-Case Behavior Distribute Use-Case Behavior to **Classes** **For** each resulting analysis **class** Describe Responsibilities Describe Attributes and Associations Qualify Analysis Mechanisms Unify Analysis **Classes** Checkpoints Object Oriented Analysis and Design Using the UML v2000 Copyright © 2000 **Rational** Software, all rights reserved **9**/ Message **Numbering** Object Oriented Analysis and Design Using the UML v2000 Copyright © 2000 **Rational** Software,/

stimuli Define a separate server thread of control **for** each service Reduce **number** of threads to what can be supported OOAD Using the UML - Describe Concurrency, v 4.0 Copyright 1994-1998 **Rational** Software, all rights reserved 23 R **Class** diagrams Active **classes** as processes Composition relationships from processes to **classes** Composition relationships from processes to subsystem interfaces **Class** Name Process Name > Interface Name Process Name > Process/

4.2 Copyright 1998-1999 **Rational** Software, all rights reserved **9** Course Catalog SystemRegister **for** CoursesStudent > RegistrationController Example: Finding Control **Classes** Recommend: Identify one control **class** per use case Each control **class** is responsible **for** orchestrating/controlling the processing that implements /incorrect user input) Examples of optional flows include the following: The actor decides-from a **number** of options-what the system is to do next The subsequent flow of events depends on /

resources **for** both: –RDz –COBOL **9** Welcome to Enterprise COBOL and **Rational** Developer **for** System z COBOL runs the worlds enterprises RDz is the way we make COBOL development easy IBM z/10 **Rational** Developer **for** System z 10 Why COBOL? Lets look at some **numbers**… /) specific product features to install. These are the defaults … and they’ll work fine **for** the Distance Learning **class** Note that CA Endevor is optional **for** this **class** Next > Click Next > 28 Installation Process – 10 of 12 You can choose/

the on-demand business logo, **Rational**, the **Rational** logo, and other IBM **Rational** products and services are trademarks or/And Datasets TransferSourceFiles 137 Workshop – Workshop – Allocate PDS (Libraries) **for** **Class** Using RDz it is simple to allocate a new PDS: Right/ –Picture string –Non-numeric literal –Comment –Numeric literal –Sequence **number** OK Click OK If you dont like your changes/ Find all variables that end in "data" PIC.**9**|PIC **9**( Find all numeric variable declarations Expression See Notes/

**numbers** except zero The graph has two symmetrical parts, called branches. **For** each point (x, y) on one branch, there is a corresponding point (-x, -y) on the other branch. The graph of On Your Calculator… Graph Describe what happens when… a > 1 in a < 0 in All **rational**/through the points, using the asymptotes as guidelines. Example 2 Graph: Real Life Uses The Junior **class** is sponsoring a dinner. The cost of catering the dinner is $**9**.95 per person plus a delivery charge of $18. (a)Write a model that gives the/

post this power point to my FIS **class** files so you can use it to study #1 – Simplifying Complex **Numbers** (a+bi form) Simplify the following /**Rational** Expressions Multiply and then simplify the following **rational** expression. You may want to factor first. Solution #7 #8 – Adding/Subtracting **Rational** Expressions Add and subtract the following expressions Find the LCD (least common denominator 1 st ) Solution #8 #**9** – Simplifying **Rational** Expressions Anyone can shoot **for** this problem. DOUBLE points! Solution #**9**/

the same **class** as a parameter **For** example, the concat method of the String **class** takes as a parameter another String object str3 = str1.concat(str2); This drives home the idea that the service is being requested from a particular object Java Foundations, 3rd Edition, Lewis/DePasquale/Chase5 - 75 Dependency The following example defines a **class** called RationalNumber to represent a **rational** **number** A **rational** **number** is a/

600 rounded to the nearest tenth is 24.5. The math **class** drops a small ball from the top of a stairwell. They/ to fall. Exploring Square Roots and Irrational **Numbers** LESSON 1-2 48 = 16t 2 Substitute 48 **for** d. d = 16t 2 Use the /**number** as **rational** or irrational. Explain. Exploring Square Roots and Irrational **Numbers** LESSON 1-2 d. 6.36366366636666... Irrational; the decimal does not terminate or repeat. **Rational**; the decimal repeats.a. –**9**.3333 b. 4 7979 **Rational**; the **number** can be written as the ratio. 43 **9**/

4:30-6:30pm Also see this page (**for** large **classes** like ComS 207): http://www.iastate.edu/~registrar/exams/ Addison-Wesley. All rights reserved Rule #1: 1..**9** must be in each row © 2004 Pearson Addison-Wesley/**class** called **Rational** to represent a **rational** **number** A **rational** **number** is a value that can be represented as the ratio of two integers Some methods of the **Rational** **class** accept another **Rational** object as a parameter See RationalTester.java (page 297)RationalTester.java See **Rational**.java (page 299)**Rational**/

digits: 0 to **9**, followed by the letters A through F to indicate the values 10 to 15. The following diagrams show how the **number** forty-two appears in/G, T (binary/decimal) Initialization to default value **for** **Class** var’s Instance var’s Array components Not **for** local var’s! static data 0000 stack FFFF heap/ calculation. 1/2 + 1/3 + 1/6 = 1 public void run() { **Rational** a = new **Rational**(1, 2); **Rational** b = new **Rational**(1, 3); **Rational** c = new **Rational**(1, 6); **Rational** sum = a.add(b).add(c); println(a + " + " + b +/

10 to 50. Unification of ideas began in mid 90’s. Rumbaugh joins Booch at **Rational** ’94 v0.8 draft Unified Method ’95 Jacobson joins **Rational** ’95 UML v0.**9** in June ’96 UML 1.0 offered to OMG in January ’97 UML 1.1/attribute that are actions Identifying attributes Look **for** information that must be maintained about each **class** Several nouns rejected as **classes**, may now become attributes An attribute should generally contain a simple value E.g. string, **number** Tips about identifying and specifying valid attributes /

are related due to the controller’s logic. Only the CourseCatalogSystem (a boundary **class**) **class** interacts with the external legacy Course Catalog System. Note inclusion of the actors. /**Rational** Software, all rights reserved 20/20 More on Sequence Diagrams Sequence diagrams show the explicit sequence of messages and are better **for** real-time specifications and **for** complex scenarios. A sequence diagram includes chronological sequences, but does not include object relationships. Sequence **numbers**/

LEARNING: Complete review worksheet **for** TEST on THURSDAY (11/06/14); AGENDA: PRE-CALCULUS HONORS. THUR. DAY 55; 11/06/14 (2 nd **9**-WEEK CHAPTER 2 TEST ON POLYNOMIALS & **RATIONAL** FUNCTIONS. LEARNING OBJECTIVES; SWBAT: Perform operations with square roots of negative **numbers**, solve problems involving quadratic/Discuss Pg. 501 # 74 – 83. C/W: Pg. 513 # 1 – 22 (all) HOME LEARNING: Complete **class** work. Go to http://killian.dadeschools.net and print & continue to work on Chapter 4.2 – 4.4.http://killian.dadeschools.net [Due/

Software Use-Case Analysis - Steps Supplement the Use-Case Description **For** each use-case realization Find **classes** from use-case behavior Distribute use-case behavior to **classes** **For** each resulting analysis **class** Describe responsibilities Describe attributes and associations Qualify architectural analysis mechanisms Unify analysis **classes** Checkpoints Next J. Scott Hawker 2004-09-23p. **9** Material © IBM **Rational** Software Analysis **Classes**: A First Step Towards Executables Use-CasesAnalysis/

the IBM logo, the on-demand business logo, **Rational**, the **Rational** logo, and other IBM **Rational** products and services are trademarks or registered trademarks of/**classes**, comprising **class** object definitions and object definitions. The ability to define data encapsulated inside **class** objects and objects. The ability to define methods **for** **class**/special) **9** Only numeric data - digits (0 **9**) V Part of numeric definition. Assumed decimal point S Part of numeric definition. Sign – allows **for** negative **numbers** COBOL /

& its **class** are separated by a colon & underlined. Reflexive Message Object Lifeline 1.1: PerformAnother Responsibility The Anatomy of Sequence Diagrams Hierarchical Message **Numbering** A Sequence / 1998-1999 **Rational** Software, all rights reserved **9** Example: Sequence Diagram (cont.) :CourseOffering : Student : RegisterForCoursesForm : RegistrationController : Schedule : Student : PrimaryScheduleOfferingInfo Repeat **for** all primary course offerings. An attempt is made to register the student **for** all selected/

enough detail. OOAD Using the UML - Use-Case Analysis, v 4.2 Copyright 1998-1999 **Rational** Software, all rights reserved **9** Supplement the Use-Case Description **For** each use-case realization Find **Classes** from Use-Case Behavior May be easier now to identify our candidate analysis **classes** **for** the system (candidate means first cut; possible **classes**) We should be able to identify a set of candidate analysis/

Ads by Google