discussed a cute example of existential proof—is it possible **for** an **irrational** **number** power another **irrational** **number** to be a **rational** **number**—we proved it is possible, without actually giving an example). 11/**9** The absence of evidence is not the evidence of /– a. Develop “tractable subclasses” of languages **and** require the expert to write all their knowlede in the procrustean beds of those sub-**classes** (so we can claim “complete **and** tractable inference” **for** that **class**) OR –Let users write their knowledge in the/

Proof by Contradiction: Example 2 Theorem: **For** all real **numbers** x **and** y, if x is a **rational** **number**, **and** y is an **irrational** **number**, then x+y is **irrational**. Proof Assume x is any **rational** **number**, y is any **irrational** **number** **and** that x+y is a **rational** **number**. Then x+y = a / b **for** some a Z **and** some b Z + Since x is **rational**, x = c /d **for** some c Z **and** some d Z + Then (c /d/

2 1 In addition to level 3.0 **and** above **and** beyond what was taught 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 write, simplify, **and** interpret expressions on contextual situations. - justify the sums **and** products of **rational** **and** **irrational** **numbers** -interpret expressions within the context of a problem/

**numbers** that are not **rational**, **and** approximate them with **rational** **numbers**. - Convert a decimal expansion that repeats into a fraction. - Approximate the square root of a **number** to the hundredth & explain the process. - **For** all items listed as a 2, students can explain their process. Students will know the subset of real **numbers**. - Know that all **numbers** have a decimal expansion. - Compare the size of **irrational** **number**. - Locate approximately **irrational** **numbers** on a **number**/

notebook paper divided into **9** sections. You will turn in your work at the end of the **class**. In order **for** your team to /**Numbers** look **for** key words to determine the operation. how many times really means DIVIDE how many more really means SUBTRACT combined means to ADD Time to play: STATION 1 Each team is getting a board with two sections labeled **Rational** **and** **Irrational**. Your team will get an envelope with several **numbers** in it. Your team must separate the **numbers** into two groups, **Rational** **and** **Irrational**/

following **numbers** as **Irrational**, **Rational**, Whole Natural, **and**/or Integers: 0909 0 **Rational**, integer, whole, natural 0 3.666666666 0 **rational** 0 -5 0 **Rational**, Integer 0 4.75 0 **rational** 0 7.154976352118568763215489736 0 **irrational** 7 Worksheet 0 Complete Worksheet in Groups 0 Complete Part I then stop 0 Complete Part II then stop 0 Complete Part III then stop 0 Review Time!! 0 Complete Page 1-12 **for**/

Section 1-3 Explore Real **Numbers** SPI 12A: Order a given set of **rational** **numbers** TPI 12F: Explore various representations of Absolute Value Objectives: Investigate the Real **Number** System Classify **and** Compare **Numbers** Natural **Numbers** Real **Number** System Whole **Numbers** Integers **Rational** **Numbers** **Irrational** **Numbers** X y Real **number** line Coordinate Plane Natural **Numbers** 1, 2, 3, … 1 2 3 4 5 6 7 8 **9** 10 11 Whole **Numbers** 0, 1, 2, 3, … 0 1 2 3/

8-Polynomials **and** **Rational** Expressions Dividing polynomials; Simplifying **rational** expressions; Exponent rules; Adding, subtracting, multiplying, **and** dividing **rational** expressions; Complex fractions Chapter **9**- **Rational** Expressions in Open Sentences Percent problems; Fractional equations; Word problems: Work, motion; Ratio **and** proportion; Direct, inverse, joint, combined variation Chapter 10-**Irrational** **Numbers** **and** Radicals **Rational** **numbers**; Decimals **and** fractions; **Rational** square roots; **Irrational** square/

**for** any Θ **and** integer n (modulus action suspended) i.e. ∴ Adjust Ω’’ to get f is monotonic with Ω → Ω’’ [Ω, Ω’] Analytic Approach Continued Fraction n th order approximation: Approximation of **irrational** **number** by **rational** fractions **for** i = 1,2,3,… Golden Ratio → Fibonacci **numbers**/ increasing K pass 1 → Chaos Ω= 0.606661 Ω= 0.5 Universality **Classes** of Circle maps Criteria: functional form near inflection point (Θ = 0, /0 Fractal dim of q.p. regions: see chap **9**. Breaking up of the torus in a Benard experiment/

Manipulating propositions –not, **and**, or, implication, biconditional Truth tables Propositional equivalences Table 6 (page 27) 3 Last **class**: quick recap – contd/x y z (x y) (x < z < y) (y < z < x) Is it true? **9** Nested Quantifiers - 2 x y (x + y = 0) is true over the integers Proof: Assume an arbitrary integer x./**irrational** Suppose 2 is **rational**. Then 2 = p/q, such that p, q have no common factors. Squaring **and** transposing, p 2 = 2q 2 (even **number**) So, p is even (previous slide) Or p = 2x **for**/

in our **class** This means /**RATIONAL** **NUMBERS**: Definition 1 – A **rational** **number** is a **number** that can be expressed as the quotient of two integers Examples: Definition # 2 – a **rational** **number** is a **number** that can be written as either a terminating or repeating decimal. Examples:.8, -1.85, **IRRATIONAL** **NUMBERS**: Definition # 1 – An **irrational** **number** is a **number** that cannot be written as the quotient of two integers Definition # 2 – An **irrational** **number** is a **number** that is a nonterminating **and**/ **and** then come back **for** /

to science, 2 Foundations of Research 42 Are we **rational**? Are people “**rational**”? Are our beliefs generally scientific? **Irrational** beliefs have increased in the U.S. in the 21 /**9**/15 EXAMPLE Foundations of Research 49 3 rd variables in spurious correlations Spurious correlations …often a 3 rd variable actually causes both terms in the correlation. Shoe size **and** reading performance **for** elementary school children Age: Older children have larger shoe sizes **and** read better. **Number** of police officers **and** **number**/

they will get an A in this **class** (q). What can you say about a student / write x = y + 3, **for** some y 1. Thus, x 2 = (y+3) 2 = y 2 + 6y + **9** > 6y + **9** 6 + **9** = 15. Indirect Proof Since p/**rational** **number** **and** an **irrational** **number** is **irrational**. Proof: Assume that the product of a **rational** **and** an **irrational** **number** is **rational** (the negation of what we want to prove.) Then we can express this as xw=y, where x **and** y are **rational**, **and** w is **irrational**. Thus, we can write x=a/b **and** y=c/d, **for** some integers a, b, c, **and**/

**number**/ 29 (**9**) 28 (11) 3 /**Class** discussion paper: Masulis (1980)- Highly significant Announcement effects: +7.6% **for** leverage increasing exchange offers. -5.4% **for** leverage decreasing exchange offers. Practical Methods employed by Companies. Trade off models: PV of debt **and** equity. Pecking order. Benchmarking. Life Cycle. Increasing Debt? time Introduction to Behavioural Finance. Standard Finance assumes that agents are **rational** **and** self-interested. Behavioural finance: agents **irrational**. **Irrational**/

it can be written as the ratio 1/3 Practice **Irrational** **Numbers**! Which of these is an **irrational** **number**? A -2 B √56 C √64 D 3.14 Which of these is an **irrational** **number**? A √3 B -13.5 C 7 11 D 1 √**9** Converting **Rational** **Numbers**! Fraction Decimal Percent Place **number** over its place value **and** reduce Divide by 100 Multiply by 100 75 = 3 100 4 0/

3/5 Example 3: Adding **and** Subtracting Fractions with Like Denominators Add or Subtract. 6/11 + **9**/11 -3/8 – 5/8 Try these on your own… -2/**9** – 5/**9** -7/**9** 6/7 + (-3/7) 3/7 Example 4: Evaluating Expressions with **Rational** **Numbers** Try these on your own… 12.1 – x **for** x = -0.1 12.2 7/10 + m **for** m = 3 1/10 3/

**Rational** Portfolios **for** **Irrational** Markets Global Strategic Allocation We believe a global approach to investing offers potential benefits **for** all investors, whether their primary objective is capital growth, income, capital preservation or a balance between them. Portfolio Construction We believe both active management **and** broad asset **class**/50 is expansionary. A **number** close to 60 is indicative of rapid economic growth. Economy The Global Purchasing Managers Index **for** manufacturing includes countries that /

7 §12-3 De Moivre Student Discussion. 8 §12-3 De Moivre’s Error Function **9** §12-3 Sterling’s Formula **for** n large. nActualSterlingAccuracy 13622702080061872394750.6% 202.432902008 10 18 2.422786847 10 18 0.4% 503./**number**! The five basic constants in mathematics in one neat formula. Hence, God must exist! 18 §12-5 Euler **For** any polyhedron the following holds: v + f = e + 2 19 §12 - 6 Clairaut, d’Alembert **and** Lambert Student Comment 20 §12 - 6 is **irrational** If x 0 is **rational**, then tan x is **irrational**/

the two solutions will be **rational** **numbers**. If not, they’re both **irrational**.) b2 – 4ac > 0 Graph of y = ax2 + bx + c Kinds of solutions to ax2 + bx + c = 0 Discriminant b2 – 4ac Example Use the discriminant to determine the **number** **and** type of solutions **for** the following equation. 5 / formula instead (**and** you will find it to be easier **and** quicker than factoring.) Reminder: This homework assignment on Section 8.2 is due at the start of next **class** period. You may now OPEN your LAPTOPS **and** begin working on/

Equivalences –Proof by contradiction Strategies Summary Problems **and** Discussion Next Lecture Notes 2 Learning Goals: “Pre-**Class**” Be able **for** each proof strategy below to: –Identify /in my computer’s machine language. Problem: prove the theorem. **9** You pick: Java or Racket. WLOG Example: My Machine Speaks /**Irrational** Theorem: The 2 is an **irrational** **number**. Opening steps: (1) Assume **for** contradiction that 2 is **rational**. (2) Using our knowledge of **rationals**, we know 2 = a/b, where a Z, b Z+, **and** a **and**/

Worked Problem: 2 is **Irrational** Theorem: The 2 is an **irrational** **number**. Opening steps: (1) Assume **for** contradiction that 2 is **rational**. (2) Using our knowledge of **rationals**, we know 2 = a/b, where a Z, b Z+, **and** a **and** b have no common factor except/the “Sequential Circuits” readings at the bottom of the “Textbook **and** References” area of the website.Textbook **and** References Complete the open-book, untimed quiz on Vista that was due before **class**. snick snack More problems to solve... (on your own/

only looks neater but it is more accurate. **Irrational** **numbers** involving square roots are often written using surds. See N2.5 Surds. Examples of **irrational** **numbers** include: π √3 **and** sin 50° **Rational** or **irrational**? Decide whether the **number** shown is **rational** of **irrational**. N4.3 Calculating with decimals Contents N4 Decimals **and** rounding A N4.1 Decimals **and** place value A N4.2 Terminating **and** recurring decimals A N4.3 Calculating with decimals/

My e-mail address (**for** teacher purposes only) is: Day 2: January 31st Objective: Review expectations **for** **class** **and** homework. Work together to share mathematical ideas **and** to justify strategies as you//7, 22, **and** so on) a **number** that can NOT be expressed as an integer fraction (π, √2, **and** so on) **Irrational** **Numbers**: NONE Symbols **for** **Number** Set Real **Numbers**: The set of all **rational** **and** **irrational** **numbers** **Rational** **Numbers** Integers **Irrational** **Numbers** Real **Number** Venn Diagram: Natural **Numbers** less than or /

**9**)(3x + 8) = 0, which leads to the two solutions of **9**/4 **and** -8/3. The Quadratic Formula The quadratic formula is another technique we can use **for**/**and** a perfect square, then you will get two real solutions that are real **and** **rational** **numbers**. If the discriminant is positive but not a perfect square, then you know **for** sure that your polynomial is prime (can’t be factored without using radicals), but there will be two **irrational**/ is due at the start of the next **class** session. Lab hours in 203: Mondays through /

; -77/3 **9** The **Irrational** **Numbers** Real **number** that is not **rational** Can not be written as a ratio of two integers Example: √2, √3,π In Computation with **irrational** **numbers** we use **rational** approximation **for** them Example: √2≈1.4141; π≈3.14 : the ratio of the circumference **and** diameter of every circle Note: not all square roots are **irrational** Example: √**9** = 3 10 The **Number** Line Every real **number** corresponds to one **and** only one/

Graphs **9** X Y f y = f (x) f : X → Y if **for** each x ∊ X ∃ a unique y ∊ Y such that y = f(x). Domain of f Range of f Co-domain of f f[X] = {f(x) : x ∊ X} Set function **Class** of sets Functions **and** its Graphs 10 X Y f y = f (x) If f : X → Y then ∃ two set/

designers. 4) The lecturer took up the **class** after a short break. Idioms **and** Phrases 8. **for** (so) long **for** a long time 很長一段時間 I haven’t seen Betty **for** so long that I doubt if she can still recognize me. 8. **for** (so) long 很長一段時間 I have been waiting here **for** my friend **for** so long that I fell asleep. Idioms **and** Phrases **9**. catch up on something to do something/

“**Rational** “**Rational** **and** **Irrational** Adventures in the Direct Loan Wonderland” Presented by Sherry Proper Director of Financial Aid & Enrollment Support Allegheny College 2 “Chapter 1 – Down the Uncertain Chasm” Liquidity injected into FFELP by Government purchasing of loans in 2008-09 & 2009-10 –Very unlikely to happen **for** 2010-11 July 1, 2010 not that far away Will FFELP $$$ be available? Will lenders remain in FFELP/

designers. 4) The lecturer took up the **class** after a short break. Idioms **and** Phrases 8. **for** (so) long **for** a long time 很長一段時間 I haven’t seen Betty **for** so long that I doubt if she can still recognize me. 8. **for** (so) long 很長一段時間 I have been waiting here **for** my friend **for** so long that I fell asleep. Idioms **and** Phrases **9**. catch up on something to do something/

**9** D. 15 7a A **rational** **number** can be written in the form: or a/b (where b can equal 1 or any other **number** except zero: b ≠ 0) The **rational** **number** is like a ratio of a to b, hence the term “**rational**.” Which of the following is not a **rational** **number**? 7b The square root of a non-perfect square is called an **irrational**/C.85 D.240 The table below records Lester’s results **for** 3 of his **classes**. 25c. Lester predicted that the average score **for** his ELA, Math **and** Science scores would be higher than 70%. How accurate was/

repeats. This is the decimal equivalent of 1 over 3. Yes this is a **rational** **number**. Slide 1- 15 Copyright © 2011 Pearson Education, Inc. **Irrational** **number**: Any real **number** that is not **rational**. Examples: Real **numbers**: The union of the **rational** **and** **irrational** **numbers**. Slide 1- 16 Copyright © 2011 Pearson Education, Inc. Objective 3 Graph **rational** **numbers** on a **number** line. Slide 1- 17 Copyright © 2011 Pearson Education, Inc. Example 3 Graph on a/

COMMUNICATOR Be prepared to share your work with the **class**. Prepare **for** my Unit 5 Test By completing my practice test **and** making a 3x5 card of notes. Write down 5 **Rational** **Numbers** **and** 5 **Irrational** **Numbers** Defend your choices to your partner (justify your answers/ the pattern. Do you see a pattern? **RATIONAL** **NUMBERS** Decimals that terminate or repeat. Examples: 0.356 **and** 0.555… **IRRATIONAL** **NUMBERS** Real **numbers** that are NOT **RATIONAL** The CRAZY Ones! Perfect Squares MEMORIZE THEM 1 4 **9** 16 25 36 49 64 81 100 121/

**for**. Example 1:4 kilometers = 4000 meters Example 2:36 millimeters = 3.6 centimeters COACH LESSON 11 65 Kilo - Hecto - Deka - Meter Liter Gram Deci - Centi- Milli - 66 PRACTICE UNIT CONVERSIONS! The students in a math **class** measured **and**/**rational**, because it can be written as the ratio 1/3 Practice **Irrational** **Numbers**! 73 Which of these is an **irrational** **number**? A -2 B √56 C √64 D 3.14 Which of these is an **irrational** **number**? A √3 B -13.5 C 7 11 D 1 √**9** FractionDecimalPercent Place **number** over its place value **and**/

, 4, **9**, 16, … Perfect Squares These **numbers** are called the Perfect Squares. Their square roots are integers. Real **Numbers** real **numbers** The real **numbers** (are there unreal **numbers**?) can be divided into two infinite sets: **Rational** **and** **Irrational**. **rational** numbersAll **rational** **numbers** can be written as a ratio of integers. (Ex: 4, 2/3, -6/25, etc.) Real **Numbers** real **numbers** The real **numbers** (are there unreal **numbers**?) can be divided into two infinite sets: **Rational** **and** **Irrational**. **irrational** numbersNo **irrational**/

5.5%10.3%8,660-74.**9**% 2.0%0..2%10.3%/**rational** **and** **irrational**. The market continually **and** automatically weighs all these factors. (A random walker would have no qualms about this assumption either. He would point out that any **irrational** factors are just as likely to be one side of the market as on the other.) (3) Disregarding minor fluctuations in the market, stock prices tend to move in trends which persist **for**/market Measure: This is a measure of the **number** of stocks in the market which have advanced relative/

**rational** **and** **irrational**. Unjust acts inspire **rational** hate. Hatred of a person based on race, religion, sexual orientation, ethnicity, or national origin constitutes **irrational** hate. Therefore: hate crimes=**irrational** hate Both **irrational** **and** **rational**/invitation, or any **number** of other actions to demean **and** isolate. The haters even may adopt a name **for** their group (Stage/**9**.5 percent) **and** anti-Islamic (**9**.0 percent) hate crimes. Disability: In 2007, 62 hate crimes against individuals with mental disabilities **and**/

**irrational** use Discuss strategies **and** interventions to promote **rational** use of medicines Some questions to ponder Could there have been a better term than "**Rational**" ? The **rational** use of drugs requires that patients receive medications appropriate to their clinical needs, in doses that meet their own individual requirements **for** an adequate period of time, **and**/project. How many LMICs can provide this data? This provides antibiotics by **class** **and** total; how many of your countries can provide even the total? /

Logic – Model Checkers – Hoare Logic **9** Learning the tools is not easy… / implications (essentially how an attacker would break our policy) We have two basic **classes** of rules: – Network topology – Attack vulnerability Example rules (network topology): / 6.Flee country 17 Process **for** applying theory to practice 1./**irrational**. Now consider the **number** y = x x. By law of excluded middle, we know that either – y is **rational**: in this case a = b = x – y is **irrational**: in this case observe that y x = 2. Thus a = y **and**/

**class** of linear functions. –What is the slope? –What is the intercept? –Solutions to systems of linear equations. Review the basic differential calculus skills required to successfully solve homework, quiz, **and** exam problems. 3 Types of **Numbers** Integers –…,-3, -2, -1, 0, 1, 2, 3,… **Rational** **Numbers** –An Integer Divided By An Integer (e.g. ½ = 0.5 & -4/3 = -1.333…) –All Integers Are **Rational** **Numbers** **Irrational** **Numbers** –Not **Rational**/

to a proposition 10/31 Midterm returned Make-up **class** on Friday 11/**9** (morning—usual **class** time) Herbrand Interpretations Herbrand Universe –All constants Rao,Pat –/entails every instantiation of it) –Existential instantiation (an existentially quantified statement holds **for** some term (not currently appearing in the KB). Can we combine these / **and** home once. so x is either home or office Existential proofs.. Are there **irrational** **numbers** p **and** q such that p q is **rational**? **Rational** **Irrational** This **and** the/

**and** effectiveness Psychological Disorders- Etiology Neurotic disorder (term seldom used now) *usually distressing but that allows one to think **rationally** **and** function socially *Freud saw the neurotic disorders as ways of dealing with anxiety Psychotic disorder *person loses contact with reality *experiences **irrational** ideas **and**/Continue reading below... **For** example, they develop/the **number** 13 (triskaidekaphobia) e) Fear of water (aquaphobia) **9**) /j) Autism _____ A **class** of disorders including depersonalization /

**for** the **class**, but … You must master the material **and** pass the tests/quizzes to continue into Geometry in the fall. Notes **and** /**irrational** **numbers**:Special **numbers** like Roots that are not whole **numbers** likeDecimals that don’t repeat the exact same thing like.34334433344433334444… Real NumbersALL **numbers** you know so farBoth **rational** **and** **irrational** **numbers** together Tell which **numbers** in this set are …NaturalWholeIntegersRational numbersIrrational numbersReal **numbers** Place, or = between each pair of **numbers**/

**irrational** use Discuss strategies **and** interventions to promote **rational** use of medicines Some questions to ponder Department of Essential Medicines **and** Health Products TBS 2012 The **rational** use of drugs requires that patients receive medications appropriate to their clinical needs, in doses that meet their own individual requirements **for** an adequate period of time, **and**/of Essential Medicines **and** Health Products TBS 2012 How many LMICs can provide this data? This provides antibiotics by **class** **and** total; how /

400 – 350 BCE) The theory was designed to deal with (**irrational**) lengths using only **rational** **numbers** Length λ is determined by **rational** lengths less than **and** greater than λ Then λ 1 = λ 2 if **for** any **rational** r λ 1 ) Note: the theory of proportions can be used to define **irrational** **numbers**: Dedekind (1872) defined √2 as the pair of two sets of positive **rationals** L √2 = {r: r 2 2} (Dedekind cut) The/

(aka source code) **and** need to rewrite code 1.Backups essential 2.Use RAID **for** critical disks; tape **for** all; off-site backups **for** disaster The rest of the story: About a week after the last person left, there was a system crash that caused major problems. [See note page **for** details.] May 25, 2015SE 477: Lecture **9** 3 of 93 SE 477 – **Class** **9** Topics: Miscellaneous: »Agile/

fist date of two party goers DUE NEXT **CLASS**! Anxiety Disorders Disorders in which excessive anxiety leads to personal distress **and** atypical, maladaptive **and** **irrational** behavior Specific Phobia Social Phobia & Agoraphobia Panic Disorder Generalized Anxiety Disorder Obsessive-Compulsive Disorder Phobia: **irrational** fear A strong **and** persistent fear of specific objects/situations that is excessive or unreasonable (**irrational**) Specific Phobia **For** example, there was woman with a specific phobia/

QualPro Appendix 2 ACT Math Concepts **and** Problems Math Vocabulary area of a circle chord circumference collinear complex **number** congruent consecutive diagonal directly proportional endpoints function y = R (x) hypotenuse integer intersect **irrational** **number** least common denominator logarithm matrix mean median obtuse perimeter perpendicular pi polygon prime **number** quadrant quadratic equation quadrilateral quotient radian radii radius **rational** **number** real **number** slope standard coordinate plane transversal/

does, A can give B 3. Similarly **for** **9**, **and** all multiples of 3. Hence if A/STOP, **and** end the game with both players getting nothing. This may seem as **irrational** to /B knows people fall into two **classes**, those choosing ONE **and** those choosing BOTH. B would very/**for** the **rationality** of one or another decision are heavily weighted by problem framing, **and** by examples. The finite exchange paradox also reveals a difference that a **number** of researchers have noted between ‘uncertainty’ **and** ‘vagueness’. The highest **number**/

: The 2 is an **irrational** **number**. Opening steps: (1) Assume 2 is **rational**. (2) Using our knowledge of **rationals**, we know 2 = a/b, where a Z, b Z+, **and** a **and** b have no common factor except 1. [But, we know nothing more about a **and** b!] Next, play around with the formula 2 = a/b **and** see where it takes you! Strategies **for** Predicate Logic Proofs Have/

**and** taking the limit as. Does not require that we commit to a fixed – or even finite – **number** of **classes**. Effective **number** of **classes** can grow with **number**/3 Species 4 Species 5 Species 6 Species 7 Species 8 Species **9** Species 10 FeaturesNew property Structure S (85 features from Osherson et al., e.g., **for** Elephant: ‘gray’, ‘hairless’, ‘toughskin’, ‘big’, ‘bulbous’,/ Snow **and** the cause of cholera (1854) **Rational** analysis of cognition Often can show that apparently **irrational** behavior is actually **rational**. Which/

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