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. to. Hardness Amplification. beyond negligible. Yevgeniy. . Dodis. , . Abhishek. Jain, Tal Moran, . Daniel . Wichs. Hardness Amplification. Go from . “weak” security . to. . “strong” security. Remember to Silence Your Cell Phone and Put It In Your Bag!. Comparing Rational Numbers (in fraction form). Models. For , where b>0, iff a<c.. For , where b>0 and d>0, . 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. Rational Numbers. The . real number system. consists of rational and irrational numbers.. . Rational numbers. can be expressed in fractional form, , where a (the numerator) and b (the denominator) are both integers and b = 0.. 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 ____.. 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. Subtitle. Finding a Common Denominator. Adding and subtracting rational numbers with variables works much the same way as constants. The only variations we need to worry about is how to handle multiplication by variable factor and how that factor affects the numerator’s sum or difference.. 3 ½. ½. 3.45. 123.456…. Mixed Number. Fraction. Terminating Decimal. Repeating Decimal. Non-Terminating or Non-Repeating. Real Numbers. The set of . real numbers. is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of . Objective. TSW identify the parts of the Real Number System. TSW define rational and irrational numbers. TSW classify numbers as rational or irrational. Real Numbers. Real Numbers are every number.. Therefore, any number that you can find on the number line.. 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. 3 ½. ½. 3.45. 123.456…. Mixed Number. Fraction. Terminating Decimal. Repeating Decimal. Non-Terminating or Non-Repeating. Real Numbers. The set of . real numbers. is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of . Investigation 3: Multiplying and Dividing Rational Numbers. Students will demonstrate how to combine numbers with the same sign and different signs. Homework. Pg. 67. 25,38-41. Warm-up. 3.1 Multiplication Patterns with Integers. 3 ½. ½. 3.45. 123.456…. Mixed Number. Fraction. Terminating Decimal. Repeating Decimal. Non-Terminating or Non-Repeating. Real Numbers. The set of . real numbers. is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of . 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.