# 1. The Fly In The Ointment

Size: px
Start display at page:

Transcription

1 Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent the quotient as a common fraction or as a decimal fraction. More specifically, independently of how we elect to write the answer, ƒ 2 means the number that when multiplied by 2 yields as the product. However if we elect to write the quotient as a common fraction we use the notation 2 ; while in the language of decimal fractions, the quotient is represented by 0.5. One possible disadvantage of using common fractions rather than decimal fractions is that there are many different common fractions that name the same rational number. For example, while ƒ 2 œ, it is also equal to, ,,, etc. On the other hand, when we use decimal fractions there is only one way to represent the quotient ƒ 2; namely, as 0.5. On the other hand, using common fractions allows us to express the quotient of any two whole numbers rather quickly. For example, in the language of common fractions, 9ƒ 20 œ However, representing this quotient in the form of a decimal fractions presents a new challenge to us. Herbert and Ken Gross 999 Actually, there is a fine point that we are ignoring. Namely while ƒ 2 œ 0.5, it is also equal to 0.50, 0.500, etc. However to avoid this technicality we will agree to end the decimal representation after the last non-zero digit. For example, we will write 0.25 rather than , etc. This is analogous to what we do when we use common fractions. Namely we pick the fraction which has been reduced to lowest terms. For example, even though œ 4, we do not usually write that ƒ 2 œ 4. 2 More generally, if m and n are whole numbers and n Á 0 then mƒ n œ m n.

2 Arithmetic Revisited: Lesson 5 Part 5: Page 2 To see why let's see what happens when we try to represent the quotient 5ƒ 6 as a decimal fraction. Clearly it is a simple task to write the quotient as a common fraction, namely: 5ƒ 6 œ 5 6. Proceeding as we did in the previous part of this lesson we can use the division algorithm, to obtain: and no matter how many 0's we use to augment the dividend, we are faced with the fact that each time the algorithm tells us that 6 goes into 20 three times with a remainder of 2. In other words, the decimal fraction that represents 5ƒ 6 consists of a decimal point followed by an 8 and an endless number of 's! This is where the fly gets into the ointment. In this particular example, we saw that a problem arises when we try to express the common fraction 5 6 as an equivalent decimal fraction. Unfortunately, this situation turns out to be the general rule rather than the exception. More specifically: Given two whole umbers chosen at random, it is highly likely that the decimal fraction that represents their quotient will be an endless decimal. Why this is the case is the topic of the next section. 2. When Will the Quotient Be a Terminating Decimal? For the common fraction to represent a decimal that terminates, its denominator must be a power of 0. That is, a terminating decimal must be a whole number of tenths, hundredths, thousandths etc. However, the only prime factors of powers of 0 are 2 and 5. Hence if a divisor *i.e., the denominator) has even one prime factor other than 2's and 5's, the decimal representation will never come to an end.

3 Arithmetic Revisited: Lesson 5 Part 5: Page Illustrative Example: 8 Express as an equivalent common fraction whose denominator is a power of 0. We may decompose 8 into a product of prime numbers by writing it as If we multiply 2 by 5 we get 0. Hence if we multiply each of the three 2's by 5 we get 0 (or,000) which is, of course, a power of 0. More specifically: œ (2 5) (2 5) (2 5) œ œ 0 However, if we multiply the denominator of a fraction by 5 5 5, we must also multiply the numerator by 5 5 5; otherwise we change the value of the fraction ,000 Hence we may rewrite in the form or. 75 As a decimal,000 is 0.75 Commentary: If we wanted to do this problem strictly by using decimals, we know that 8 is the answer to ƒ 8 Þ If we now proceed as we did earlier, we see that: 0Þ Þ Notice that in doing the division we finally arrived at a point where the reminder was 0 when we subtracted.

4 Arithmetic Revisited: Lesson 5 Part 5: Page 4 Practice Problem # Express the common fraction Solution: 40 as an equivalent decimal fraction. Answer: We observe that the only prime factors of 40 are 2 and 5. More specifically: 40 œ To obtain a factor of 0 we must multiply 2 by 5. Since 40 has three factors of 2 but only one factor of 5, we need two more factors of 5 in order to obtain a power of 0. With this in mind we multiply both numerator and denominator of 40 by 5 5 to obtain: # # # & 0 0 0,000 Notes: œ œ 5 5 œ œ œ We could again have obtained the same result by long division. Namely: 0Þ Þ Once again notice that we finally arrived at a point where the remainder was 0 when we subtracted. We might also have observed that,000ƒ 40 œ 75. Hence,000 thousandths ƒ 40 œ 75 thousandths In the language of decimal fractions the above equality becomes,000ƒ 40 œ 0.075

5 Arithmetic Revisited: Lesson 5 Part 5: Page 5 Summary, So Far: Definition: A decimal is said to terminate if it has a last non-zero digit. -- Another way of saying this is to say that beyond a certain number of places, the decimal has nothing but 0's. What we have shown in this problem is that the fraction 8 can be represented by the terminating decimal 0.75 (as well as 0.750, , etc.). The important thing is that in order for a fraction to be represented by a terminating decimal, it must be equivalent to a common fraction whose denominator is a power of 0. However, the only prime numbers that are factors of 0 are 2 and 5. Hence, if when written in lowest terms, a common fraction has a prime factor other than 2 and/or 5, it will never be equivalent to a fraction whose denominator is a power of 0. One might think that since we can write as many 0's as we wish after the decimal point that we are eventually bound to come to a place where the decimal will terminate. Unfortunately, as we mentioned earlier, this will happen only if the denominator has no prime factors other than 2 and/or 5. To get an idea of what happens if the denominator does contain a prime factor other than 2 or 5, look at the following example. Illustrative Example Is there a common fraction whose denominator is a power of 0 that is equivalent to the common fraction? The key observation for answering this question is to recall that a number is divisible by if and only if the sum of it's digits is divisible by. Since any power of 0 has as its digits a followed only by 0's, the sum of its digits will always be and hence never divisible by.

6 Arithmetic Revisited: Lesson 5 Part 5: Page 6 Commentary: When we talk about common fractions we will always assume (unless specifically stated to the contrary) that they are expressed in lowest terms. For example, the denominator of is 6; which has a prime factor other than 2 or 5. However, if we reduce to lowest terms it becomes, which is represented by the terminating decimal 0.5. The result has an interesting interpretation if we think in terms of decimal fractions. Namely: 0Þ Þ The above process shows us that at each step in the division process, we are saying goes into 0 three times with a remainder of. Thus no matter how many places to the right we extended our quotient, there would always be a remainder of ; and, hence, never a remainder of 0. The fact that there are many more numbers whose prime factorization contains prime numbers other than 2's and or 5's means that when we express rational numbers as decimal fractions, the decimals will most likely be non-terminating. Practice Problem #2 Will the decimal fraction that is equivalent to the common fraction 9 5 terminate? Answer: Yes For example, if we look at the whole numbers that are greater than but less than 00, we see that only #ß%ß&ß"!ß"'ß#!ß#&ß \$#ß%!ß&! and 80 have no other prime factors other than just 2's and/or 5's.

7 Arithmetic Revisited: Lesson 5 Part 5: Page 7 Solution: This can be a bit tricky. Namely, the denominator has a factor other than 2 or 5. however the fraction is not in lowest terms. In fact, we can cancel the common factor from the numerator and denominator and see that: Note: œ œ œ œ 0.6 We could have obtained the same result by the division algorithm. Namely: 0Þ 6 5 9Þ However, we wanted to emphasize the role of the denominator in converting a common fraction into a decimal fraction. Practice Problem # Will the decimal fraction that is equivalent to the common fraction 2 5 terminate? Solution: Answer: No This time the fraction is in lowest terms. Therefore, since any power of 0 has only 2's and/or 5's as prime factors, 5, which contains as a prime factor can never be a divisor of any power of 0.

8 Arithmetic Revisited: Lesson 5 Part 5: Page 8 Note: Once again we could have gone directly to the division algorithm and obtained: 0Þ 5 2Þ It soon becomes clear that we can go on forever saying 5 goes into 50 three times with a remainder of 5. This leads in a natural way into our next section.. An Interesting Property of Decimal Fractions. When we tried to write as a decimal fraction we saw that the division yielded an endless sequence of 's. When we tried to write as a decimal fraction we saw that after the digit in the tenths place (namely, ) we again obtained an endless sequence of 's. The fact is that when any rational number is written in decimal form, the decimal will either terminate or else it will eventually repeat the same cycle of digits endlessly. 4 As an example 6 (but not a proof of our assertion), let's try to represent 7 as a decimal fraction. Using the short division format we see that 0Þ Þ % 5 6 Notice that when we started the division process our dividend was a 6 followed by nothing but 0's; and after the sixth decimal digit our dividend was again a 6 followed by nothing but 0's. Hence the cycle of digits will repeat endlessly. 2 5 ÞÞÞÞ 4 In fact if we think of a terminating decimals as eventually repeating 0's endlessly we may say that all rational numbers can be expressed as non-terminating, repeating decimals.

9 Arithmetic Revisited: Lesson 5 Part 5: Page 9 Was it a coincidence that in the two examples we picked in which the decimal fraction did not terminate the decimal eventually repeated the same cycle of digits endlessly? The answer is that it wasn't a coincidence. The proof is actually quite elementary. It is based on what is known as the Dedekind Chest-of-Drawers Principle. The example below gives a specific application of the principle (and why it has the name that it does!) Illustrative Example Imagine that in a bureau drawer there are 00 separate white socks and 00 separate black socks. The room is dark and you want to make sure that you pick a matched pair of socks (that is, either 2 white socks or 2 black socks). What is the least number of socks that you can take out of the drawer and still be sure that you have such a pair? If all you took were 2 socks you might have a matched pair but you may also have picked one white and one black sock from the drawer. It's possible that the first two socks you picked gave you a matched pair. However if this wasn't the case, it means that you have chosen one white and one black sock. Therefore, since the next sock you choose must be either while or black, you will then have either a black pair or a white pair. The point is that since there are only two colors, you are guaranteed to have a mixed pair if you pick any three socks. 5 As a second example, suppose there are 67 people in a room. Then even allowing for a Leap Year, at least two of the people in the room must have been born on the same day (but not necessarily in the same year). Namely, let's look at the case in which there are 66 people in the room. If we already can find 2 who do then we've proven our claim. So suppose no two of them celebrate their birthday on the same day. Since there are 66 people in the room, it means that between them they have used up every possible day. Therefore, in this case if another person enters the room, the birthday of this 67th person must be celebrated on the same day that one of the others is celebrating. 5 Notice that we aren't saying what color the matched pair is. It would be a quite different if we had said What is the least number of socks you can take from the drawer to make sure that you have a white pair?.

10 Arithmetic Revisited: Lesson 5 Part 5: Page 0 Practice Problem #4 Imagine that in a bureau drawer there are 00 separate white socks, 00 separate brown socks, 00 separate yellow socks and 00 separate black socks. The room is dark and you want to make sure that you pick a matched pair of socks. What is the least number of socks that you can take out of the drawer and still be sure that you have such a pair? Solution: Answer: five There are only 4 different colors. So if you took out only 4 socks, there is a chance that by a stroke of either good or bad luck, you managed to get 4 different colored socks. However,even in this worst case scenario, no matter how lucky or unlucky you deem yourself to be, if you now take out a fifth sock, it has to match one of the other four socks. These rather simple illustrations are a form of Drawers Principle. More specifically: Dedekind's Chest-of- Dedekind's Chest of Drawers Principle: If you have more items than you have drawers to put them in, at least one of the drawers must hold more than item (actually, all of the items could be in the same drawer; but at least two of them must be).

11 Arithmetic Revisited: Lesson 5 Part 5: Page 4. Dedekind's Principle and Non-Terminating Decimal Fractions Let's now apply this idea to converting common fractions into decimals. In particular, let's revisit the case of. When we divide a number by there are possible remainders; namely, 0,, or 2. 6 This concept applies to any denominator we may be using. For example, if there were 7 books in a carton, when all the books were packed, there could be at most 6 books left because if there were 7 we could have filled another carton. That is, when we divide a whole number by 7, there are 7 possible remainders; 0,, 2,, 4, 5, or 6. 7 In general when we divide a number by any non-zero whole number n, there are n, possible remainders. Hence when we write the common m fraction n as a decimal fraction, the decimal part must begin to repeat th no later than by the time we get past the n place in the decimal. The fact that the decimal fraction that represents a rational number eventually repeats the same cycle of digits gives us a nice way to write such a decimal. Namely, we simply place a bar over the repeating cycle of digits. For example, 0. stands for 0...(where the dots indicate the decimal never ends) 0.26 stands for stands for Non-terminating decimal fractions are an interesting intellectual topic but they are not necessary in the real world. For example, even though we can't express exactly as a terminating decimal, we can use terminating decimals to get very good approximations. For example, œ œ and œ Þ Hence 0. œ œ 6 An easy way to see this is to think of packing books into cartons, each of which holds books. When all the books are packed, there can be either none, or 2 books left because if there were at least books left, we could have filled another carton. 7 (Be careful here, there are not 6 remainders but 7. That is, there are 6 whole numbers between (and including) and 6 but there are 7 whole number between 0 and 6.

12 Arithmetic Revisited: Lesson 5 Part 5: Page 2 -- Thus, for example, if our measuring instrument cannot measure to closer than the nearest hundredth of an inch, since the difference between and 0. is 00 (that is, an error part per 00) we may in this case use 0. as a sufficiently accurate approximation for. -- And if we had a more sensitive measuring instrument, say one that could measure to the nearest millionth of an inch, we could use 0Þ as an approximation for because:,,000,000 œ,000, ,999,000,000,000,000 œ,000,000 and an error of part per,000,000 is less of an error than part per million. In other words, when a common fraction cannot be expressed exactly as a terminating decimal, we can chop off the decimal (the technical term is that we say are truncating the decimal) after a sufficient number of places and use this as an approximation for the exact answer. 8 A Note About Using the Bar There is a tendency for some people to write... to stand for and so on. The trouble with this notation is that what and so on means can vary from person to person. One nice example is the following sequence of numbers., 0,, 0,,, 0,, 0,,,... Based on what rule I am using to generate the above sequence of numbers, do you think you can you guess the next number? If you think you can, be warned that the next number is neither 0 nor. More specifically, the above list represents the number of days in a month starting with March in 8 In fact, we do this quite often in mathematics. For example, the number (which is the ratio between the circumference and diameter of a circle) was known to the Ancient Greek mathematician and 0 philosopher, Archimedes, to be more than 7 but less than 7. It's actual value in decimal form begins with However, in many situations we use 7 as the value of. As you can easily verify, the decimal form of begins as Notice that to the nearest hundredth this is the same value of 7 when it's rounded off to the nearest hundredth. Hence, even though Á " (, whenever we do not need more than accuracy to the nearest hundredth as may replace by 7.

13 Arithmetic Revisited: Lesson 5 Part 5: Page a non-leap Year (in other words, the next number should be 28 because it is the number of days in February in a non-leap Year). 9 The point is that once we put the bar above the repeating cycle of digits, we no longer have to guess what and so on means. That is, if we see the notation , we know at once that it means ; where in this case and so on means that the cycle 67 repeats endlessly.. Practice Problem #5 Write the first 7 digits of the decimal fraction Þ0Þ847Þ Solution: Answer: The bar over the sequence of digits 874 means that the cycle of digits 847 repeats endlessly after the decimal point. Hence the first 7 digits are Notes: Again, the most common mistake students make is they simply ignore the bar and replace 0Þ847 by is a very good approximation for 0Þ œ 0Þ Þ!Þ847 Þ In fact: In the event that the above error was deemed to be too great, we could approximate the value of 0Þ847 by writing more digits, for example, etc. In the real world we world eventually reach a point where we couldn't even measure the error because it would be too small.. In other words even if we didn't know the exact value of 0Þ847, we would eventually find a terminating decimal that was close enough. With respect to this problem, rounded off to the nearest millionth,!þ)%()%() becomes 0Þ847848à and what we could say in this case is that rounded off to the nearest millionth, 0Þ847 œ 0Þ Notice that if you had been told to write the number of days in each month of a non-leap year starting with March you would not have had any trouble guessing that the missing number was 28; but without your knowing what the pattern was, the phrase and so on could be easily misinterpreted by you. In particular, there are man ways to justify why you got a different answer but it still wouldn't have been the one that was required by the author of the sequence.

14 Arithmetic Revisited: Lesson 5 Part 5: Page 4 Practice Problem #6 Solution: Write the first 7 digits of the decimal fraction 0Þ847 Answer: This resembles the previous problem except that the bar is only over the 4 and the 7. This means that the repeating cycle begins after the 8, Hence this time the decimal is 0Þ ; and rounded off to the nearest ten millionth it would be Practice Problem #7 Is there a rational number that is denoted by 0Þ874? Solution Answer: No This is analogous to asking Is it possible to go 4 miles further after you've gone as far as is humanly possible? In other words the bar over the 7 tells us that the 7 is repeated without end. Hence it cannot be followed by a A Closing Thought There are many valid reasons to support why students prefer decimal fractions to common fractions. However there is one overriding advantage of common fractions that cannot be ignored. Namely much of our work with rational numbers is involved with the concept of rates; and finding a rate requires that we divide two numbers. Such a quotient is easily expressed in the language of common fractions. However, as we have just seen, problems can arise when we use decimal fractions to represent the quotient of two whole numbers. Among other things, it is much easier to recognize the that 7 represents the number that is defined by 6ƒ 7 than it is to recognize that is the answer to 6ƒ 7. Moreover, if wed did want to represent 6ƒ 7 as a terminating decimal, the best we could do is round our answer off to a certain number of decimal places For example, computing 6ƒ 7 on my calculator yields as the answer, which we can round off to as many as 7 decimal place accuracy; but the resulting number, although adequate for most real-life applications, is just a good approximation to the exact answer...

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### **Unedited Draft** Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 3: Multiplying Decimals

1. Multiplying Decimals **Unedited Draft** Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 3: Multiplying Decimals Multiplying two (or more) decimals is very similar to how

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### **Unedited Draft** Arithmetic Revisited Lesson 4: Part 3: Multiplying Mixed Numbers

. Introduction: **Unedited Draft** Arithmetic Revisited Lesson : Part 3: Multiplying Mixed Numbers As we mentioned in a note on the section on adding mixed numbers, because the plus sign is missing, it

### 3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

### Lesson on Repeating and Terminating Decimals. Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu

Lesson on Repeating and Terminating Decimals Dana T. Johnson 6/03 College of William and Mary dtjohn@wm.edu Background: This lesson would be embedded in a unit on the real number system. The set of real

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### JobTestPrep's Numeracy Review Decimals & Percentages

JobTestPrep's Numeracy Review Decimals & Percentages 1 Table of contents What is decimal? 3 Converting fractions to decimals 4 Converting decimals to fractions 6 Percentages 6 Adding and subtracting decimals

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Mathematical goals. Starting points. Materials required. Time needed

Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;

### north seattle community college

INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Converting from Fractions to Decimals

.6 Converting from Fractions to Decimals.6 OBJECTIVES. Convert a common fraction to a decimal 2. Convert a common fraction to a repeating decimal. Convert a mixed number to a decimal Because a common fraction

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Introduction to Fractions, Equivalent and Simplifying (1-2 days)

Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### MATH 13150: Freshman Seminar Unit 10

MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### FRACTIONS MODULE Part I

FRACTIONS MODULE Part I I. Basics of Fractions II. Rewriting Fractions in the Lowest Terms III. Change an Improper Fraction into a Mixed Number IV. Change a Mixed Number into an Improper Fraction BMR.Fractions

### 47 Numerator Denominator

JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### DIVISION OF DECIMALS. 1503 9. We then we multiply by the

Tallahassee Community College 0 DIVISION OF DECIMALS To divide 9, we write these fractions: reciprocal of the divisor 0 9. We then we multiply by the 0 67 67 = = 9 67 67 The decimal equivalent of is. 67.

### An Introduction to Number Theory Prime Numbers and Their Applications.

East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

### DECIMAL COMPETENCY PACKET

DECIMAL COMPETENCY PACKET Developed by: Nancy Tufo Revised: Sharyn Sweeney 2004 Student Support Center North Shore Community College 2 In this booklet arithmetic operations involving decimal numbers are

### Number Systems and Radix Conversion

Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will

### CHAPTER 4 DIMENSIONAL ANALYSIS

CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.

### Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots

### Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### Dr Brian Beaudrie pg. 1

Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, 3 0.14 means add 0.14 three times, regroup, and simplify,

### Grade 4 Unit 3: Multiplication and Division; Number Sentences and Algebra

Grade 4 Unit 3: Multiplication and Division; Number Sentences and Algebra Activity Lesson 3-1 What s My Rule? page 159) Everyday Mathematics Goal for Mathematical Practice GMP 2.2 Explain the meanings

### 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

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### QM0113 BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION)

SUBCOURSE QM0113 EDITION A BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION) BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION) Subcourse Number QM 0113 EDITION

### Rational Number Project

Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson : Overview Students estimate sums and differences using mental images of the 0 x 0 grid. Students develop strategies for adding

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### Grade 5 Math Content 1

Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

### Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES

Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES. Introduction (simple) This helpsheet is concerned with the ways that we express quantities that are not whole numbers,

### 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

### Prime Factorization 0.1. Overcoming Math Anxiety

0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF

### DATE PERIOD. Estimate the product of a decimal and a whole number by rounding the Estimation

A Multiplying Decimals by Whole Numbers (pages 135 138) When you multiply a decimal by a whole number, you can estimate to find where to put the decimal point in the product. You can also place the decimal

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### The Euclidean Algorithm

The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

### This explains why the mixed number equivalent to 7/3 is 2 + 1/3, also written 2

Chapter 28: Proper and Improper Fractions A fraction is called improper if the numerator is greater than the denominator For example, 7/ is improper because the numerator 7 is greater than the denominator

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Factoring Whole Numbers

2.2 Factoring Whole Numbers 2.2 OBJECTIVES 1. Find the factors of a whole number 2. Find the prime factorization for any number 3. Find the greatest common factor (GCF) of two numbers 4. Find the GCF for

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

Arithmetic 1 Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

UNDERSTANDING ALGEBRA JAMES BRENNAN Copyright 00, All Rights Reserved CONTENTS CHAPTER 1: THE NUMBERS OF ARITHMETIC 1 THE REAL NUMBER SYSTEM 1 ADDITION AND SUBTRACTION OF REAL NUMBERS 8 MULTIPLICATION

### 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

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### 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,

### 2.4 Real Zeros of Polynomial Functions

SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

### Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one.

What do fractions mean? Fractions Academic Skills Advice Look at the bottom of the fraction first this tells you how many pieces the shape (or number) has been cut into. Then look at the top of the fraction

### Division of whole numbers is defined in terms of multiplication using the idea of a missing factor.

32 CHAPTER 1. PLACE VALUE AND MODELS FOR ARITHMETIC 1.6 Division Division of whole numbers is defined in terms of multiplication using the idea of a missing factor. Definition 6.1. Division is defined

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### 1004.6 one thousand, four AND six tenths 3.042 three AND forty-two thousandths 0.0063 sixty-three ten-thousands Two hundred AND two hundreds 200.

Section 4 Decimal Notation Place Value Chart 00 0 0 00 000 0000 00000 0. 0.0 0.00 0.000 0.0000 hundred ten one tenth hundredth thousandth Ten thousandth Hundred thousandth Identify the place value for

### Chapter 13: Fibonacci Numbers and the Golden Ratio

Chapter 13: Fibonacci Numbers and the Golden Ratio 13.1 Fibonacci Numbers THE FIBONACCI SEQUENCE 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, The sequence of numbers shown above is called the Fibonacci

### 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

### The Crescent Primary School Calculation Policy

The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has

### LESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to:

LESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to: 1. Change fractions to decimals. 2. Change decimals to fractions. 3. Change percents to decimals.

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### BASIC MATHEMATICS. WORKBOOK Volume 2

BASIC MATHEMATICS WORKBOOK Volume 2 2006 Veronique Lankar A r ef resher o n t he i mp o rt a nt s ki l l s y o u l l ne e d b efo r e y o u ca n s t a rt Alg e b ra. This can be use d a s a s elf-teaching

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B

Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### 23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

### Sunny Hills Math Club Decimal Numbers Lesson 4

Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions into improper fractions, just to multiply and convert them back? Are you tired of reducing fractions

### Unit 11 Fractions and decimals

Unit 11 Fractions and decimals Five daily lessons Year 4 Spring term (Key objectives in bold) Unit Objectives Year 4 Use fraction notation. Recognise simple fractions that are Page several parts of a whole;

### 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,

### 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

### 0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents

Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A 3 4 0.75

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### Multiplying Fractions

. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

### Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations

Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)

### Irrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.

Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that