MATH 10021 Core Mathematics I

1 math 10021 . core Mathematics I. Department of Mathematical Sciences Kent State University July 23, 2010. 2. Contents 1 Real Numbers and Their Operations 5. Introduction .. 5. Integers, Absolute Values and Opposites .. 19. Integer Addition and Subtraction .. 36. Integer Multiplication and Division .. 55. Order of Operations .. 62. Primes, GCF, & LCM .. 77. Fractions and Mixed Numbers .. 91. Fraction Addition and Subtraction .. 125. Fraction Multiplication and Division .. 146. Decimals and Percents .. 162. Decimal Operations .. 189. Introduction to Radicals .. 207. Properties of Real Numbers .. 216. 2 Algebra 229. Variables and Algebraic Expressions .. 229. Linear Equations .. 241. Problem Solving .. 260. Proportions and Conversion Factors .. 270. Linear Inequalities .. 285. 3 Graphing and Lines 299.

2 Cartesian Coordinate System .. 299. Graphing Linear Equations .. 312. Function Notation .. 323. Slope .. 334. Equations of Lines .. 348. 3. 4 CONTENTS. A Real Number Operations 361. B Multiplying or Dividing Mixed Numbers 365. C Answers to Exercises 369. Chapter 1. Real Numbers and Their Operations Introduction Welcome to core Mathematics I ( math 10021 ) or core Mathematics I and II ( math 10006)! Beginnings are always difficult, as you and your fellow students may have a wide range of backgrounds in previous math courses, and may differ greatly in how long it has been since you last did math . So, to start, we want to suggest some ground rules, refresh your memory on some basics, and demonstrate what you are expected to know already. Let's start with the ground rules. First, this course will be easier if you keep up with the material.

3 math often builds upon itself, so if you don't understand one section, the next section may be impossible. To avoid this, practice, practice, practice, and get help from a tutor or the instructor immediately if you are having trouble. This book has many problems after each section with answers to all in the back, so feel free to do more than what is assigned to be collected! The student who shows up an hour prior to the exam and tells the instructor that they need to go over everything is going to be sorely disappointed. Second, please do NOT ask your instructor When am I ever going to use this? , as this may lead him or her to develop nervous tics or to begin muttering under his or her breath. This is college, and the student's role at college is to acquire knowledge. Some of the material in this book may have obvious applications in your future job, some may have surprising applications, and some you may never use again.

4 ALL OF IT is required to pass this course! 5. 6 CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONS. Finally, you may already know, or will soon learn, that you will be expected to do the first exam without a calculator. The idea is that the material covered will just be basic operations on real numbers, and you should know the mechanics behind all of these things. Then, later, if you wish to use a calculator to save yourself some time and effort (and to have less chance of making a simple error), that is fine. The calculator, however, should NOT be an arcane and mysterious device which is relied upon like a crutch. Letting it do your busy work is one thing; letting it do your thinking is quite another. The dangers of letting machines do all of our thinking for us is aptly illustrated in any of the Terminator or Matrix movies!

5 With all that being said, let us now move on to numbers. When you first learned to count, you probably counted on your fingers and used the numbers one through ten. In set notation, this group of numbers would be written {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Notice that for set notation we just listed all the numbers, separated by commas, inside curly braces. If there are a lot of numbers in the set, sometimes we use the notation .. , called periods of ellipsis, to take the place of some of the numbers. For example, the set of numbers one through one hundred could be written {1, 2, .. , 99, 100}. When periods of ellipsis are used, the author assumes the reader can see a pattern in the numbers in the set and can figure out what is missing. If the periods of ellipsis occur after the last listed number, this means that the numbers go on forever.

6 An example of such a set of numbers is the first listed in this book that is important enough to have an official name (three names actually!): natural numbers = counting numbers = positive integers = {1, 2, 3, ..}. These numbers are the ones for which you probably have the best feel . For example, you can imagine seven pennies or eighty-two cows, although you probably get a little bit hazier for the bigger numbers. You will have heard of a million or billion, and possibly even a trillion (like the national deficit in dollars), but do you know the names of larger numbers? Try the following example: Example 1. Which of the following are natural numbers: (a) a quintillion, (b) a zillion, (c) a bazillion, (d) an octillion, (e) a google? INTRODUCTION 7. Solution 1. Did you say (a), (d), and (e)? (a) a quintillion = 1,000,000,000,000,000,000.

7 (b) a zillion is a real word (in the dictionary), but it represents an indeterminately large number ( a whole heck of a lot). (c) a bazillion is not in the dictionary; it is slang for a zillion (d) an octillion = 1,000,000,000,000,000,000,000,000,000. (e) a google = one followed by one hundred zeros. That's right! Before it was ever the name of an internet search engine, it was the name of a large natural number. The next number we would like to review is zero, another number of which you should have a good understanding. For example, if you have four quar- ters in your pocket and your brother takes all four to buy a bottle of iced tea, you are left with no quarters, or zero quarters. Once we add zero into the number set, the set gets a new name: whole numbers = nonnegative integers = {0, 1, 2, 3.}

8 }. IMPORTANT NOTATION: Prior to this course, you should have seen how to apply the four basic mathematical operations (addition, subtraction, multiplication or division) to whole numbers. This book will insert the word old-fashioned before any such operation whenever both the numbers AND. the result are whole numbers. For example, you are doing old-fashioned subtraction when you are subtracting two whole numbers and the bigger number is first. Note, even though we will illustrate how to do old-fashioned addition, subtraction, multiplication, and division without a calculator, the use or lack of use of a calculator plays no role in our using the adjective old-fashioned. The terminology is just to indicate the most basic use of the operation. Once we mix in negative numbers, fractions, etc., there will be new rules which will have to be applied.

9 Note that all four of these opera- tions require a number (or expression) both before and after the operation sign. Therefore, later when we see an expression like 2 , we will know that the sign here does NOT mean subtract, as there is no number before it. 8 CHAPTER 1. REAL NUMBERS AND THEIR OPERATIONS. Addition For addition, we will use the standard symbol of + . The two numbers being added are called the addends, while the answer is called the sum. A. property of addition which will be useful to know is that when you add two numbers, the order in which you add them doesn't matter. Thus: 3+5 = 5+3. This is the commutative property of addition which we will study in more detail in section ; for now, you just want to know that you can switch the order. To do old-fashioned addition without a calculator, you should have seen the tower method.

10 When using the tower method, make sure that the place-values of the two numbers are aligned, the one's digit of the top number is over the one's digit of the bottom number, the ten's digit of the top number is over the ten's digit of the bottom number, etc. Example 2. Add 357 + 4, 282. Solution 2. Scratch work: 1. 3 5 7. + 4 2 8 2. 4 6 3 9. So: 357 + 4, 282 = 4, 639. Notice that when we added the five to the eight and got thirteen, we carried the one to the next column. It is not unusual for you to have to carry when using the vertical tower method. Some other comments on the last example: First, it will be a good practice to have a separate line for your solution versus your actual work space, especially later when you may need to adjust the sign of your solution. Second, you should clearly indicate your solution for your instructor.