PPT-Direct Proof and Counterexample II: Rational Numbers

The word rational contains the word ratio which is another word for quotient A rational number can be written as a ratio of integers Example 1 Determining Whether Numbers Are Rational or Irrational. Pearson . Pre-AP Unit 1. Topic . 2: Reasoning and Proof. 2-1. : . Patterns and Conjectures. Pearson Texas Geometry ©2016 . Holt Geometry Texas ©2007 . TEKS Focus:. (4)(C) Verify that a conjectures is false using a counterexample.. This Lecture. Now we have learnt the basics in logic.. We are going to apply the logical rules in proving mathematical theorems.. Direct proof. Contrapositive. Proof by contradiction. Proof by cases. Dr Frost. Objectives: . Appreciate the difference between a rational and irrational number, and how surds can be manipulating both within brackets and fractions.. Learning Objectives. By the end of this topic, you’ll be able to answer the following types of questions:. Inquiries into the Philosophy of Religion. A Concise Introduction. Chapter 5. God And Morality. By . Glenn Rogers, Ph.D.. Copyright. ©. 2012 . Glenn Rogers. Proof of God?. God and Morality. Aristotle referred to man (humankind) as the rational animal, emphasizing that it is human rationality that sets humans apart from animals. . Fill-in-the-blank:. Rational. or . Irrational. ?. 1) The sum of two rational numbers is ________.. 2) The product of two rational numbers is ______.. 3) The sum of a rational and an irrational is ____.. 1.1 Propositional Logic. 1.2 Propositional Equivalences. 1.3 Predicates and Quantifiers. 1.4 Nested Quantifiers. 1.5 Rules of Inference. 1.6 Introduction to Proofs. 1.7 Proof Methods and Strategy. We wish to establish the truth of. The notion of divisibility is the central concept of one of the most beautiful subjects in advanced mathematics: . number theory. , the study of properties of integers.. Example 1 – . Divisibility. Fractions and Icebergs!. Since 1912. when the Titanic sank after striking an iceberg, ships and planes from the INTERTATIONAL ICE PATROL have searched the waters of the North Atlantic to report icebergs that have broken off Greenland’s Ice Sheet and drifted into shipping. . grade Math – Numeration Unit. 7. π. -√36. ¾. 0. 2.12122….. Number Types. Whole . Integers. Rational Numbers. Irrational Numbers. Which is which? . How can you tell them apart?. Whole Numbers. beyond negligible. Yevgeniy. . Dodis. , . Abhishek. Jain, Tal Moran, . Daniel . Wichs. Hardness Amplification. Go from . “weak” security . to. . “strong” security. .. Weakly Secure. Strongly Secure. Investigation1: Extending the Number System. Students will demonstrate understanding of the number line and how to use it to compare numbers, add and subtract numbers.. Homework for this section. Due Thursday. Investigation2: Adding and Subtracting Rational Numbers. Students will demonstrate how to combine numbers with the same sign and different signs. Homework. Pg. 44. 1-11 odd. 19-29 odd. 30, 48,54,66. Rational and Learning TargetStudents can identify numbers as being rational or irrationalAll REALnumbers are made up of RATIONAL IRRATIONAL numbersRational NumbersA rational number is a number that c Now we have learnt the basics in logic.. We are going to apply the logical rules in proving mathematical theorems.. Direct proof. Contrapositive. Proof by contradiction. Proof by cases. Basic Definitions.