# Algebra I Notes Review Real Numbers and Closure Unit 00a

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition, subtraction, multiplication, division, raising to powers, and taking roots. A binary operation requires two numbers. Addition is a binary operation, because "5 +" doesn't mean anything by itself. Multiplication is another binary operation. Are all operations binary? A.APR.A.1 Perform operations on polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. N.RN.B.3 Use properties of rational and irrational numbers. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Skills: Identify and apply real number properties using variables, including distributive, commutative, associative, identity, inverse, and absolute value to expressions or equations. Identify, understand, and explain the sets of numbers that are closed under given operations Math Background PREREQUISITE KNOWLEDGE/SKILLS: properties of real numbers. properties of closure for sets of real numbers. In this unit you will extend closure properties to polynomials. review or extend closure properties to irrational numbers. You can use the skills in this unit to understand and predict the nature of possible solutions. Essential Questions Why are all sets not closed under all operations? How can closure help understand the type of solution you might expect with operations? Is there a pattern for closure in sets? Note: A review of the skills is included at the end of these notes. Sample problems are provided to give teachers examples that guide instruction. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 1 of 20 8/20/2013

2 Sample Questions* (A.APR.A.1 and N.RN.B.3) 1. Determine if the set of polynomials is closed under division. Explain why or why not. a. The set of polynomials is closed under division. Just as multiplication is repeated addition, division is repeated subtraction. Since polynomials are closed under subtraction, they are also closed under division. b. The set of polynomials is not closed under division. Let f(x) and g(x) be polynomial expressions where g(x) is not equal to zero. f Then ( x ) f( x) is undefined if g(x) = 0. In this case, gx ( ) gx ( ) the set of polynomials is not closed under division. is not a rational expression, so c. The set of polynomials is closed under division. Since the set of polynomials is closed under multiplication, and division is the inverse operation for multiplication, the set of polynomials is also closed under division. d. The set of polynomials is not closed under division. Let f(x) and g(x) be polynomial expressions where g(x) is not equal to zero. By the definition of polynomial expressions, f( x) is not a polynomial expression, so gx ( ) the set of polynomials is not closed under division. (The quotient of two polynomial expressions is a rational expression.) Solution: D DOK Which number in the set, 6.1, 15, a c. is irrational? b. d. Solution: D DOK 1 3. Which number in the set 16, 3,,5 3 is rational? a. 16 c. 3 b. 5 3 d. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 2 of 20 8/20/2013 Solution: A DOK 1

3 4. Which word best describes the product of 7 and 11? a. Rational b. Natural c. Imaginary d. irrational Solution: D DOK 1 5. Which word best describes the sum of 19 and 3? a. Irrational b. Rational c. Natural d. imaginary Solution: A DOK 1 6. What is the sum of 3 5 and 17 5? Is the sum rational or irrational? a ; irrational c. 25; rational b ; rational d. 25; irrational Solution: A DOK 2 7. What is the product of 4 11 and? Is the product rational or irrational? a. 112 ; irrational c ; rational b ; irrational d ; rational Solution: C DOK 2 8. What is the product of and 2 7? Is the product rational or irrational? a ; rational c. 22.4; rational b ; irrational d. 22.4; irrational Solution: B DOK 2 Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 3 of 20 8/20/2013

4 9. What is the product of 5 and 2? Is the product rational or irrational? a. 10; rational c. 7; rational b. 10 ; irrational d. 2 5; irrational Solution: B DOK 2 SHORT ANSWER 10. Prove that the quotient of two nonzero rational numbers is rational. Let x and y be two nonzero rational numbers. Solution: DOK 4 By the definition of rational numbers, x can be written as a b and y can be written as c d, where a, b, c, and d are nonzero integers. a x b Divide x by y. y c d x a c ad To divide, multiply by the reciprocal. y b d bc Because the product of two nonzero integers is also a nonzero integer, the expressions ad and bc are both nonzero integers. Therefore, the quotient rational number. is the ratio of two integers. So, by definition, the quotient is a 11. Use an indirect proof to prove that the difference of a rational number and an irrational number is irrational. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 4 of 20 8/20/2013 Solution: DOK 4 Let a be a rational number and b be an irrational number. Let c be the difference of b and a. Assume that c is rational. Then: b a = c b = c + a Add a to both sides.

5 Because the sum of two rational numbers is rational, c + a is a rational number. But it is equal to b, which is an irrational number. Therefore, our assumption is wrong, and the difference of a rational number and an irrational number is irrational. 12. How can you determine whether the product of and is rational or irrational without multiplying? Solution: DOK 2 Both and are rational numbers. The product of two rational numbers is rational, so the product of and is rational. ESSAY 13. Let q 9 p, where p is a rational number. What type of number is q? Explain your reasoning. Solution: The number represented by q is a rational number. DOK 3 Sample explanation: 9 can be simplified to 3. Since we know p is a rational number, we know it can be written as a, where a and b are integers and b is not zero. So: b q 9 p 3p 3a b Since a is an integer, 3a is also an integer, so q is a rational number. 14. What kind of number is 5 9 m if m is a rational number? Explain your reasoning. Solution: A rational number DOK 3 Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 5 of 20 8/20/2013

6 Sample explanation: 5 is a rational number. m is also rational number, so it can be 9 written as a, where a and b are integers and b is not zero. So: b 5 m 9 5 a 9 b 5a 9b Since a and b are integers, 5a and 9b are also integers and 9b is not equal to zero, so 5 9 m is a rational number. 15. Consider the equation m k 17. k is a non-zero rational number. What type of number is m? Explain your reasoning. Solution: The number m must be irrational DOK 3 Sample explanation: Assume that m is rational. The number k is a non-zero rational number and is an irrational number. Therefore: m k m k If m and k are both rational, then their quotient is also rational. However, 17 is not a rational number. The assumption that m is a rational number is incorrect, and therefore it must be irrational. 16. Let t s 4.7, where s is an irrational number. What type of number must t be? Explain your reasoning. Solution: t is an irrational number. DOK 3 Sample explanation: Assume that t is rational. That means it can be written as a b, where a and b are integers and b does not equal zero. So: Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 6 of 20 8/20/2013

7 a s 4.7 b a 4.7 s b a 47 s b 10 10a 47b s 10b However, since a and b are integers, then 10a - 47b and 10b are also integers. This means that s can be written as the quotient of two integers, which contradicts the fact that s is irrational. Therefore t must also be irrational. 17. Let, where h is an irrational number. What type of number must g be? Explain your reasoning. Solution: g is an irrational number. : DOK 3 Sample explanation: Assume that g is rational. That means it can be written as, where a and b are integers and b does not equal zero. So: However, since a and b are integers, then and 5b are also integers. This means that h can be written as the quotient of two integers, which contradicts the fact that h is irrational. Therefore g must also be irrational. 18. Let p be a rational number that is not zero, and q be an irrational number. If, could r be a rational number? Explain your reasoning. Solution: No, r must be irrational. DOK 4 Sample explanation: Assume that r is rational. p is a rational number, so it can be written as, where a and b are integers and b does not equal zero. If r is also a rational number, so it can be written as, where c and d are integers and d does not equal zero. So: Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 7 of 20 8/20/2013

8 So q must also be rational, but this contradicts what we are given. Therefore, r must be irrational. 19. Let x and y be rational numbers. Prove that the sum of x and y is also a rational number. Solution: DOK 4 Because x and y are rational, then they can be written as, where a, b, c, and d are integers and b and d are not zero. Their sum is. Because a, b, c, and d are integers, ad + bc and bd are also integers, and bd is not zero because b and d are not zero. Therefore, the sum of x and y is also a rational number. 20. Let m and n be rational numbers. Prove that their product, mn, is also a rational number. Solution: DOK 4 Because m and n are rational numbers,, where a, b, c, and d are integers and b and d are not zero. The product of m and n is. Because a, b, c, and d are integers, ac and bd are also integers, and bd is not zero because b and d are not zero. Therefore, the product of m and n is a rational number. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 8 of 20 8/20/2013

9 For Algebra I, this section may be a review as needed. A set is a collection of objects. Each object in a set is called an element of the set. A set may have not elements, a finite number of elements, or an infinite number of elements. For example N = {1,2,3, } describes the set of natural numbers. A subset is a set contained entirely within another set. For example A= {2,6,1,50} is a subset of N above. When you first start dealing with numbers, you learn about the four main sets, or groups, of numbers, which nest inside one another: Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4,... and going on infinitely Integers: The set of counting numbers, zero, and negative counting numbers Rational numbers: The set of integers and fractions Real numbers: The set of rational and irrational numbers (which can't be written as simple fractions) The sets of counting numbers, integers, rational, and real numbers are nested, one inside another, similar to the way that a city is inside a state, which is inside a country, which is inside a continent. The set of counting numbers is inside the set of integers, which is inside the set of rational numbers, which is inside the set of real numbers. The diagram below shows other subsets of the set of real numbers. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 9 of 20 8/20/2013

10 Counting, or natural, numbers The set of counting numbers is the set of numbers you first count with, starting with 1. (so 0 is not a counting number.) Because they seem to arise naturally from observing the world, they're also called the natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9,... When you add two counting numbers, the answer is always another counting number. Similarly, when you multiply two counting numbers, the answer is always a counting number. Another way of saying this is that the set of counting numbers is closed under both addition and multiplication. Whole numbers The set of counting numbers and the number 0 make up the whole numbers. So you see the natural numbers are a subset of the whole numbers. Are whole numbers closed under addition? Are whole numbers closed under subtraction? No, because 1 3 2, which is not a whole number. Integers The set of integers arises when you try to subtract a larger counting number from a smaller one. For example, 4 6 = 2. The set of integers includes the following: The counting numbers Zero The negative counting numbers Here's a partial list of the integers:... 4, 3, 2, 1, 0, 1, 2, 3, 4,... Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction. Rational numbers The set of rational numbers includes all integers and all fractions. Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is that the rational numbers are closed under division. Real numbers Even if you filled in all the rational numbers on the number line, you'd still have points left unlabeled. These points are the irrational numbers. An irrational number is neither a whole number nor a fraction. In fact, an irrational number can only be approximated as a non-repeating decimal. In other words, no matter how many decimal places you write down, you can always write down more; furthermore, the digits in this decimal never become repetitive or fall into any pattern. The most famous irrational number is π, Together, the rational and irrational numbers make up the real numbers, which comprise every point on the number line. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 10 of 20 8/20/2013

11 CLOSURE A set of numbers is closed, or has closure, under a given operation if the result of the operation on any two numbers in the set is also in the set. For example, the set of even numbers is closed under addition,since the sum of two even numbers is also an even number. Note: A method to show that a set is NOT closed is to find a counterexample. There are many examples that can be used to demonstrate closure. Fill in the table with other examples. It is important to allow students to express the properties in their terms (as long as they are correct). Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 11 of 20 8/20/2013

12 Closure project Identify if the set is closed under the operations. Give examples or counterexamples. Add Subtract Multiply Divide Real numbers Rational numbers Irrational numbers Integers Whole numbers Polynomials Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 12 of 20 8/20/2013

13 Sum of a rational number and an irrational number? Special cases What type of number is the... Product of rational number and an irrational number? Difference of a rational number and an irrational number? Quotient of rational number and an irrational number? Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 13 of 20 8/20/2013

14 Real Number Properties Commutative Property of Addition: a b b a Real Number Example: Commutative Property of Multiplication: a b b a Real Number Example: Associative Property of Addition: a b c a b c Real Number Example: Associative Property of Multiplication: a bc a b c Real Number Example: Identity Property of Addition: a0 a Note: The additive identity is 0. Real Number Example: Identity Property of Multiplication: a1 a Note: The multiplicative identity is 1. Real Number Example: Inverse Property of Addition: a a 0 Note: The opposite of a number is the additive inverse. Real Number Example: Inverse Property of Multiplication: 1 a a 1 Note: The reciprocal of a number is the multiplicative inverse. Real Number Example: Zero Product Property: If ab 0, then either a 0 or b 0. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 14 of 20 8/20/2013

15 Distributive Property: a b c a b b c Real Number Example: Use the distributive property to multiply We can rewrite 32 as in order to use the distributive property so that we can perform the multiplication in our heads Absolute Value Absolute Value: the distance a number is away from the origin on the number line Ex: What two numbers have an absolute value of 4? Solution: There are two numbers that are 4 units away from the origin, as shown in the number line. Notation: x They are 4 and 4. x if x 0 and x xif x 0 Ex: Evaluate the expression Note: Absolute value bars are like grouping symbols in the order of operations. Evaluate the expression inside the absolute value bars. 2 3 Evaluate the absolute value and simplify Ex: Evaluate the variable expression Substitute x 1 and x x 34 x when x 1 and y y into the expression y. 3 Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 15 of 20 8/20/2013

16 Use the order of operations to simplify You Try: 1. Identify the property shown. c d e d c e 2. Show that the distributive property does not work over absolute value bars by showing that QOD: Determine if the statement is true or false. Explain your answer. The reciprocal of any number is greater than zero and less than 1. Sample Nevada High School Proficiency Exam Questions (taken from 2009 released version H): 1. Which equation illustrates the commutative property of addition? A 4 x 3x 3x 4 x B 3x4 x 0 3x4 x C 4 x 3x 4 x 3x D 3x4 x 3x4 3xx 2. The equation below illustrates a property of real numbers. 3 7x 9 21x 27 Which property is illustrated by the equation? A associative property B commutative property Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 16 of 20 8/20/2013

17 C distributive property D identity property 3. Which equation shows the use of the inverse property of multiplication? A 1 7x 1 7x B 7x1 7x C 7x 0 0 D 1 1 7x 7x 7 7 Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 17 of 20 8/20/2013

18 Skills: simplify algebraic expressions by adding and subtracting like terms. Terms: algebraic expressions separated by a + or a sign Constant Term: a term that is a real number (no variable part) Coefficient: the numerical factor of a term Like Terms: terms that have the exact same variable part (must be the same letter raised to the same exponent) Note: Only like terms can be added or subtracted (combined). To add or subtract like terms, add or subtract the coefficients and keep the variable part. Ex: Simplify This means we are adding 3 hearts plus 4 hearts. Therefore, we have a total of 7 hearts, which we can write as Ex: Use the expression 1 3x 4xy x 5y to answer the following questions. 1. How many terms are in the expression? Solution: There are 5 terms. 2. Name the constant term(s). Solution: There is one constant term, How many like terms are in the expression? Solution: There are 2 like terms. 3 x and x What is the coefficient of the xy term? What is the coefficient of the fourth term? Solution: The xy term has a coefficient of 4. coefficient of 1. The third term has a 5. Simplify the algebraic expression. Solution: Combine like terms. 1 3x 1x 4xy 5y 1 2x 4xy 5y How many terms are in the simplified form of the algebraic expression? Solution: There are 4 terms. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 18 of 20 8/20/2013

19 Simplifying an Algebraic Expression Ex: Simplify the expression x 4 8 5x 1. Step One: Eliminate the parentheses using the distributive property. 4x 32 5x 1 Step Two: Combine like terms. 4x 5x x 33 x n 3n 8 4n Ex: Simplify the expression 2 Step One: Eliminate the parentheses using the distributive property. 5 6n 16n 4n 2 2 Note: The expression in parentheses is being multiplied by 2n, which is what was distributed. Step Two: Combine like terms. Note: The answer may be written as n 4n 16n n n 16n 5 by the commutative property. n Ex: Write a simplified expression for the perimeter of a rectangle with length x 7 and width x 2. Note: The formula for the perimeter of a rectangle is P 2l 2w. Substitute the length and width into the formula. 2x 7 2x 2 Simplify using the distributive property and combining like terms. 2x14 2x 4 4x 10 You Try: 1. Simplify the expression x x Write a simplified expression for the area of a triangle with a base of 4x and a height of 2x 1. QOD: Explain in your own words why only like terms can be added or subtracted. Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 19 of 20 8/20/2013

20 Sample Practice Question(s): 1. Simplify the expression: 5 3 x 4 x A. 7x 4 B. 7x 32 C. 2x + 1 D. 2x 7 2. Simplify the expression: 2 2 5x 2x 6 4x 6x 1 A. B. C. D. 2 x x x 10x x 2x x 6x 5 3. Write an expression for the perimeter of the rectangle: y 2x A. 2xy B. 6xy C. 4x + 2y D. 4x 2y 2 2 Alg I Unit 00a Notes Real Numbers and Closurerev3 Page 20 of 20 8/20/2013

### 1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

### Chapter 1.1 Rational and Irrational Numbers

Chapter 1.1 Rational and Irrational Numbers A rational number is a number that can be written as a ratio or the quotient of two integers a and b written a/b where b 0. Integers, fractions and mixed numbers,

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### 1.3. Properties of Real Numbers Properties by the Pound. My Notes ACTIVITY

Properties of Real Numbers SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall The local girls track team is strength training by

### MAT Mathematical Concepts and Applications

MAT.1180 - Mathematical Concepts and Applications Chapter (Aug, 7) Number Theory: Prime and Composite Numbers. The set of Natural numbers, aka, Counting numbers, denoted by N, is N = {1,,, 4,, 6,...} If

### USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS

5 (1 5) Chapter 1 Real Numbers and Their Properties 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS In this section The properties of the real numbers can be helpful when we are doing computations. In

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Sect Properties of Real Numbers and Simplifying Expressions

Sect 1.6 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a.34 + 2.5 Ex. 1b 2.5 + (.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a).34 + 2.5 = 6.84

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Algebra 1: Topic 1 Notes

Algebra 1: Topic 1 Notes Review: Order of Operations Please Parentheses Excuse Exponents My Multiplication Dear Division Aunt Addition Sally Subtraction Table of Contents 1. Order of Operations & Evaluating

### Placement Test Review Materials for

Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the

### OPERATIONS AND PROPERTIES

CHAPTER OPERATIONS AND PROPERTIES Jesse is fascinated by number relationships and often tries to find special mathematical properties of the five-digit number displayed on the odometer of his car. Today

### MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

### MATH Fundamental Mathematics II.

MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### MATH REFRESHER I. Notation, Solving Basic Equations, and Fractions. Rockefeller College MPA Welcome Week 2016

MATH REFRESHER I Notation, Solving Basic Equations, and Fractions Rockefeller College MPA Welcome Week 2016 Lucy C. Sorensen Assistant Professor of Public Administration and Policy 1 2 Agenda Why Are You

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### ( ) 4, how many factors of 3 5

Exponents and Division LAUNCH (9 MIN) Before Why would you want more than one way to express the same value? During Should you begin by multiplying the factors in the numerator and the factors in the denominator?

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

### ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### is a real number. Since it is a nonterminating decimal it is irrational.

Name the sets of numbers to which each number belongs. 19. The number -8.13 is a real number. Since -8.13 can be expressed as a ratio where a and b are integers and b is not 0 it is also a rational number.

### troduction to Algebra

Chapter Three Solving Equations and Problem Solving Section 3.1 Simplifying Algebraic Expressions In algebra letters called variables represent numbers. The addends of an algebraic expression are called

### Logic and Proof Solutions. Question 1

Logic and Proof Solutions Question 1 Which of the following are true, and which are false? For the ones which are true, give a proof. For the ones which are false, give a counterexample. (a) x is an odd

### Section 1.9 Algebraic Expressions: The Distributive Property

Section 1.9 Algebraic Expressions: The Distributive Property Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Apply the Distributive Property.

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Square Roots. Learning Objectives. Pre-Activity

Section 1. Pre-Activity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of non-perfect

### MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

### Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### 1-2 Properties of Real Numbers

1-2 Properties of Real Numbers Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Simplify. 1. 5+5 0 2. 1 3. 1.81 4. Find 10% of \$61.70. \$6.17 5. Find the reciprocal of 4. Objective Identify and use properties

### Using the Properties in Computation. a) 347 35 65 b) 3 435 c) 6 28 4 28

(1-) Chapter 1 Real Numbers and Their Properties In this section 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS The properties of the real numbers can be helpful when we are doing computations. In this

### Sum of Rational and Irrational Is Irrational

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

### Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010

Section 7.1: Radicals and Rational Exponents Definition of nth root of a number Let a and b be real numbers and let n be an integer n 2. If a = b n, then b is an nth root of a. If n = 2, the root is called

### STRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.

STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary

### SIMPLIFYING ALGEBRAIC FRACTIONS

Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

### Algebra Revision Sheet Questions 2 and 3 of Paper 1

Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the

### A rational number is a number that can be written as where a and b are integers and b 0.

S E L S O N Rational Numbers Goal: Perform operations on rational numbers. Vocabulary Rational number: Additive inverse: A rational number is a number that can be a written as where a and b are integers

### Lesson 4.2 Irrational Numbers Exercises (pages )

Lesson. Irrational Numbers Exercises (pages 11 1) A. a) 1 is irrational because 1 is not a perfect square. The decimal form of 1 neither terminates nor repeats. b) c) d) 16 is rational because 16 is a

### Computing with Signed Numbers and Combining Like Terms

Section. Pre-Activity Preparation Computing with Signed Numbers and Combining Like Terms One of the distinctions between arithmetic and algebra is that arithmetic problems use concrete knowledge; you know

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

### MAT 0950 Course Objectives

MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify

### Clifton High School Mathematics Summer Workbook Algebra 1

1 Clifton High School Mathematics Summer Workbook Algebra 1 Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature: Parent Signature:

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

### The Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,

1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Example 1 Example 2 Example 3. The set of the ages of the children in my family { 27, 24, 21, 19 } The set of Counting Numbers

Section 0 1A: The Real Number System We often look at a set as a collection of objects with a common connection. We use brackets like { } to show the set and we put the objects in the set inside the brackets

### 1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS

Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS The following is a list of terms and properties which are necessary for success in Math Concepts and College Prep math. You will

### Rules for Exponents and the Reasons for Them

Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication

### Chapter 4 -- Decimals

Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Algebra 1A and 1B Summer Packet

Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the

### To Evaluate an Algebraic Expression

1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Math 1111 Journal Entries Unit I (Sections , )

Math 1111 Journal Entries Unit I (Sections 1.1-1.2, 1.4-1.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Higher Education Math Placement

Higher Education Math Placement 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry (arith050) Subtraction with borrowing (arith006) Multiplication with carry

### 4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

### Roots of Real Numbers

Roots of Real Numbers Math 97 Supplement LEARNING OBJECTIVES. Calculate the exact and approximate value of the square root of a real number.. Calculate the exact and approximate value of the cube root

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers

Section 1.8 The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers Each is said to be a subset of the real numbers.

### Prep for College Algebra

Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

### CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

### HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)

HFCC Math Lab Intermediate Algebra - 7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example

### Introduction to the Practice Exams

Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra

In order to simplifying radical expression, it s important to understand a few essential properties. Product Property of Like Bases a a = a Multiplication of like bases is equal to the base raised to the

### Pre-Algebra Curriculum Map 8 th Grade Unit 1 Integers, Equations, and Inequalities

Key Skills and Concepts Common Core Math Standards Unit 1 Integers, Equations, and Inequalities Chapter 1 Variables, Expressions, and Integers 12 days Add, subtract, multiply, and divide integers. Make

### USING THE PROPERTIES

8 (-8) Chapter The Real Numbers 6. (xy) (x)y Associative property of multiplication 66. (x ) x Distributive property 67. 4(0.) Multiplicative inverse property 68. 0. 9 9 0. Commutative property of addition

### , Seventh Grade Math, Quarter 2

2016.17, Seventh Grade Math, Quarter 2 The following practice standards will be used throughout the quarter: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.

### STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

### Sect Exponents: Multiplying and Dividing Common Bases

40 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says

### Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER

Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the

### Domain Essential Question Common Core Standards Resources

Middle School Math 2016-2017 Domain Essential Question Common Core Standards First Ratios and Proportional Relationships How can you use mathematics to describe change and model real world solutions? How

Table of Contents Sequence List 368-102215 Level 1 Level 5 1 A1 Numbers 0-10 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 0-10 64 H2 Operations and Properties 3 A3 Addition 0-10 65 H3 Evaluating

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### 7 th Grade Pre Algebra A Vocabulary Chronological List

SUM sum the answer to an addition problem Ex. 4 + 5 = 9 The sum is 9. DIFFERENCE difference the answer to a subtraction problem Ex. 8 2 = 6 The difference is 6. PRODUCT product the answer to a multiplication

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,