1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 2 Review of the Real Number System Chapter 1

3. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 3 1.1 Basic Concepts

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 4 1.1 Basic Concepts Objectives 1.Write sets using set notation. 2.Use number lines. 3.Know the common sets of numbers. 4.Find additive inverses. 5.Use absolute value. 6.Use inequality symbols.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 5 1.1 Basic Concepts Write Sets Using Set Notation A set is a collection of objects called the elements, or members, of the set. Set braces, { }, are used to enclose the elements. For example, 4 is an element of the set, {3, 4, 11, 19}. {3, 4, 11, 19} is an example of a finite set since we can count the number of elements in the set. A set containing no elements is called the empty set, or the null set, denoted by Ø.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 6 Examples of Sets Natural numbers Whole numbers Empty set N = {1,2,3,4,5,6,...} W = {0,1,2,3,4,5,6,…} Ø (a set with no elements) 1.1 Basic Concepts Certain sets of numbers have names: Caution: Ø is the empty set; { Ø } is the set with one element, Ø. Note: N and W are infinite sets. The three dots, called an ellipsis, mean “continue on in the pattern that has been established.”

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 7 Set-Builder Notation 1.1 Basic Concepts Sometimes instead of listing the elements of a set, we use a notation called set-builder notation. {x | x has property P }

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 8 Listing the Elements in Sets 1.1 Basic Concepts a){x | x is a whole number less than 3} The whole numbers less than 3 are 0, 1, and 2. This is the set {0, 1, 2}. b){x | x is one of the first five odd whole numbers} = {1, 3, 5, 7, 9}. c){z | z is a whole number greater than 11} This is an infinite set written with three dots as {12, 13, 14, 15, … }.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 9 Using Set-Builder Notation to Describe Sets a){ 0, 1, 2, 3 } can be described as {m | m is one of the first four whole numbers}. b){ 7, 14, 21, 28, … } can be described as {s | s is a multiple of 7 greater than 0}. 1.1 Basic Concepts

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 10 Using Number Lines A number line is a way to picture a set of numbers: 1.1 Basic Concepts 05–5 – 4 – 3 – 2 – 1 1 2 3 4 0 is neither positive nor negative Negative numbersPositive numbers

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 11 Using Number Lines The set of numbers identified on this number line is the set of integers: I = {…,–3, –2, –1, 0, 1, 2, 3, …} 1.1 Basic Concepts 05–5 – 4 – 3 – 2 – 1 1 2 3 4 Each number on the number line is called a coordinate of the point it labels. Graph of 3

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 12 Rational Numbers 1.1 Basic Concepts Rational numbers can be expressed as the quotient of two integers, with a denominator that is not 0. The set of all rational numbers is written: Rational numbers

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 13 Rational Numbers 1.1 Basic Concepts Rational numbers can be written in decimal form as: Terminating decimals: Repeating decimals: Bar means repeating digits.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 14 Irrational Numbers 1.1 Basic Concepts Irrational numbers have decimals that neither terminate nor repeat:

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 15 Graphs of Rational and Irrational Numbers 1.1 Basic Concepts 05–5 – 4 – 3 – 2 – 1 1 2 3 4 Irrational Numbers Rational Numbers

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 16 Real Numbers 1.1 Basic Concepts Rational numbers Integers Whole numbers Natural numbers Irrational numbers

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 17 Relationships Between Sets of Numbers 1.1 Basic Concepts Real numbers Irrational numbers Rational numbers Integers Noninteger rational numbers Positive integers Zero Negative integers

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 18 Sets of Numbers Natural numbers or Whole numbers Integers Rational numbers Irrational numbers Real numbers {1, 2, 3, 4, 5, 6, … } {0, 1, 2, 3, 4, 5, 6, … } {…,–3, –2, –1, 0, 1, 2, 3, … } 1.1 Basic Concepts

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 19 Relationships Between Sets of Numbers Decide whether each statement is true or false. a)All natural numbers are integers. b)Zero is an irrational number. c)Every integer is a rational number. d)The square root of 9 is an irrational number. e) is an irrational number. 1.1 Basic Concepts True False True False

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 20 Additive Inverse For any real number a, the number –a is the additive inverse of a. 1.1 Basic Concepts –4 units from zero4 units from zero 05–5 – 4 – 3 – 2 – 1 1 2 3 4 The number –4 is the additive inverse of 4, and the number 4 is the additive inverse of –4.

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 21 The Minus Sign The symbol “−” can be used to indicate any of the following: 1.a negative number, such as –13 or –121; 2.the additive inverse of a number, as in “ –7 is the additive inverse of 7”. 3.subtraction, as in 19 – 7. 1.1 Basic Concepts

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 22 Signed Numbers / Additive Inverses The sum of a number and its additive inverse is always zero. 4 + (–4) = 0 or –16 + 16 = 0 For any real number a, –(–a) = a. 1.1 Basic Concepts

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 23 Absolute Value Geometrically, the absolute value of a number, a, written |a| is the distance on the number line from 0 to a. 1.1 Basic Concepts Distance is 4, so |–4| = 4. Distance is 4, so |4| = 4. 05–5 – 4 – 3 – 2 – 1 1 2 3 4 Absolute value is always positive.

24. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 24 Formal Definition of Absolute Value 1.1 Basic Concepts Evaluate the following absolute value expressions: |–14||0|–|9|–|–13| |14| + |–7| –|6–3| –|8–8|

25. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 25 Equality vs. Inequality  An equation is a statement that two quantities are equal. 8 + 3 = 11 19 – 12 = 7  An inequality is a statement that two quantities arenot equal. One must be less than the other. 9 < 12 This means that 9 is less than 12. –7 > – 10 This means that –7 is greater than –10. 1.1 Basic Concepts

26. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 26 Inequalities on the Number Line On the number line, a < b if a is to the left of b; a > b if a is to the right of b. 05–5 – 4 – 3 – 2 – 1 1 2 3 4 1.1 Basic Concepts –2 < 3 1 > –4 The inequality symbol always points to the smaller number.

27. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 27 Inequality Symbols 1.1 Basic Concepts SymbolMeaningExample is not equal to –6 10 is less than –9 –3 is greater than 8 –2 is less than or equal to –8 is greater than or equal to –2 –7