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

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

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

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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.”

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 }

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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, … }.

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 – Each number on the number line is called a coordinate of the point it labels. Graph of 3

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

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

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

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

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

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

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 – The number –4 is the additive inverse of 4, and the number 4 is the additive inverse of –4.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 – Basic Concepts

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 – Absolute value is always positive.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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|

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Equality vs. Inequality  An equation is a statement that two quantities are equal = – 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 – Basic Concepts

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 – Basic Concepts –2 < 3 1 > –4 The inequality symbol always points to the smaller number.

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