Chapter 1 Whole Numbers

Transcription

1 Chapter 1 Whole Numbers 1.1 Place Value and Rounding We begin our study of number systems with the whole numbers, which are the counting numbers starting with 0,1,2,3,. Our number system is referred to as a base 10 system, which refers to the fact that we use ten digits (symbols) to represent our numbers. In counting, after the tenth digit 9, we use place value to represent the next number 10, where the digit 1 is placed into the tens place. Similarly, after the number 99, the next number is 100, where the digit 1 is now placed into the hundreds place. These place-values continue to increase by the following naming convention: ones, tens, hundreds thousands, ten-thousands, hundred-thousands millions, ten-millions, hundred-millions billions, ten-billions, hundred-billions trillions, ten-trillions, hundred-trillions The first two examples illustrate how word names for numbers are used. Example 1 Write the word name for the number 23,168. Solution Following our naming convention, the digits are in the following place-value locations (listed in decreasing order): 2 ten-thousands 3 thousands 1 hundreds 6 tens 8 ones We group the names by threes, so our word name for 23,168 is twenty-three thousand, one hundred sixty eight. Note that the word and does not appear in the names of whole numbers (though it is used when we consider decimals in chapter 3). 1

2 Example 2 Write the word name for the number 1,624,859. Solution Following our naming convention, the digits are in the following place-value locations (listed in decreasing order): 1 millions 6 hundred-thousands 2 ten-thousands 4 thousands 8 hundreds 5 tens 9 ones Again grouping the names by threes, our word name for 1,624,859 is one million, six hundred twenty-four thousand, eight hundred fifty-nine. It is critical that you understand the significance of place-value in the whole number system. Consider the two numbers 1,368 and 8,361. Though both numbers use the same digits (1,3,6, and 8), yet the first number is much smaller than the second number. One way to see this is to use the word name for each number: 1,368: One thousand, three hundred sixty-eight 8,361: Eight thousand, three hundred sixty-one The word name indicates the second number is much larger (by almost seven thousand). We indicate the first number is smaller than the second number by writing an inequality symbol, in this case <, which is read is less than : 1,368 < 8,361 (1,368 is less than 8,361) We could also write the inequality using the inequality symbol >, which is read is greater than : 8,361 > 1,368 (8,361 is greater than 1,368) The other inequality symbols are variations of these: symbol meaning example < less than 12<45! less than or equal to 24! 24 > greater than 50>10 " greater than or equal to 40 " 20 = equal to 34=34 Note that 24 24, since the two numbers are equal (less than OR equal to), and that 40 20, since 40 > 20. 2

3 Example 3 Determine the appropriate inequality symbols which will make each statement true (more than one symbol may apply). a b c d. 1, e. 99, ,000 Solution a. Since 24 is less than 50, both < and symbols will make the statement true. b. Since 101 is larger than 100, both > and symbols will make the statement true. c. Since both numbers are equal, the symbols =,, and will make the statement true. Remember that and both include the or equal to portion of the inequality, which is why they produce a true statement. d. Since 1,000 is larger than 999, both > and symbols will make the statement true. e. Since 99,999 is smaller than 100,000 (remember to watch for place-value), both < and symbols will make the statement true. An alternative to word-names for numbers is expanded form, which utilizes place-value to write a number. The expanded form for our two numbers is: 1,368: ,361: Note how expanded form illustrates that 8,361 is nearly seven thousand greater than 1,368. 3

4 Example 4 Write the expanded form and word name for 24,597. Solution Again listing the digits in decreasing order, along with their associated value: 2 ten-thousands (20,000) 4 thousands (4,000) 5 hundreds (500) 9 tens (90) 7 ones (7) The word-name is twenty-four thousand, five hundred ninety-seven. The expanded form is 20, , Example 5 Give the value of the number 300, , Solution In expanded form, the number is 300, , Note that there are no ten-thousands (00,000). Thus the value is 304,287. A final point for this section involves the idea of rounding of numbers. Numbers are frequently referred to in rounded form, such as: 1. The national trade deficit was 23 billion dollars this month. 2. The stock market increased 200 billion dollars in value last week. 3. I paid 22 thousand dollars for my new truck. In these three statements, the number being referred to has been estimated, or rounded, to a particular place value. In statement 1, the place value used is billions. In statement 2, it is hundred-billions, and in statement 3 it is thousands. The basic idea of rounding is to choose the closest number accurate to the desired place value. For example, consider the number 148. If we desire to round this number to the closest hundred, we must choose which hundred (100 or 200) is closest to 148. Since 100 is closer, we say 148 rounded to the hundreds place is 100. If, however, we desire to round 148 to the closest ten, we must now choose which ten (140 or 150) is closest to 148. Since 150 is closer, we say 148 rounded to the tens place is 150. The general rule of thumb in rounding is: 1. Locate the place value to be rounded. 2. Look at the next (smaller place value) digit to the right. 3. If that digit is a 5 or larger, round up to the next place value digit. 4. If that digit is a 4 or less, leave the place value digit unchanged. 5. Replace all digits after the place value digit with 0. 4

5 Example 6 Round the number 29,748 to the indicated place value. a. tens b. hundreds c. thousands d. ten-thousands Solution a. The digit 4 is in the tens place. Looking to the next smaller place value (to the right), the digit is an 8. Since 8 is 5 or larger, we round up to 5 in the tens place. Finally, we replace the ones digit with 0. The rounded number is 29,750. b. The digit 7 is in the hundreds place. Looking to the tens place (to the right), the digit is a 4. Since 4 is 4 or smaller, we leave the digit 7 unchanged. Finally, we replace the tens and ones digits with 0. The rounded number is 29,700. c. The digit 9 is in the thousands place. Looking to the hundreds place (to the right), the digit is a 7. Since 7 is 5 or larger, we round up in the thousands place. The only way to round up 29 thousand is 30 thousand. Replacing the remaining digits with 0, the rounded number is 30,000. d. The digit 2 is in the ten-thousands place. Looking to the thousands place (to the right), the digit is a 9. Since 9 is 5 or larger, we round up to 3 in the ten-thousands place. Replacing the remaining digits with 0, the rounded number is 30,000. This last example illustrates an unusual point when you are presented with rounded numbers. In both parts c and d, the rounded value is 30,000. If you were presented the number 30,000 in a statement, you would naturally assume it had been rounded to the nearest ten-thousand, even though it could have been rounded to the nearest thousand, as in part c. The only way we can distinguish between these two ideas is with significant digits, which will be discussed later in Section 3.7. For now, we will assume the rounding performed (called the accuracy of the number) will be to the highest place value. That is, if presented with 30,000, we will assume the accuracy of the rounding is ten-thousands. Terminology place value inequality symbol rounding word name expanded form accuracy 5

6 Exercise Set 1.1 Determine the place value of the digit 2 in the following numbers , , , , , ,314, ,628, ,553, ,048, ,000,000 Write the word name for the following numbers , , , , , , ,824, ,340, ,300, ,408,500 Determine the appropriate inequality symbols which will make each statement true (more than one symbol may apply) ,000 1, ,999,999 6,000, , , , ,011 Write the expanded form for the following numbers , , , , , , ,950, ,564, ,800, ,658,509 6

7 Write the value of each of the following expressions , , , , , , , , , , , , , , Round each of the following numbers to the indicated place value ; tens ; hundreds 57. 1,457; hundreds 58. 1,457; tens ,849; thousands ,849; hundreds ,996; tens ,996; hundreds ,996; thousands ,996; ten-thousands 65. 5,846,994; tens 66. 5,846,994; hundreds 67. 5,846,994; thousands 68. 5,846,994; ten-thousands 69. 5,846,994; hundred-thousands 70. 5,846,994; millions ,998; tens ,998; hundreds ,998; thousands ,998; ten-thousands Determine whether each statement is true or false > ,999 < 200, > < Each of the following numbers has been rounded. In each case, give the accuracy of the rounded number (recall that the accuracy is the largest place value used in the rounding) , , , , ,800, ,000, ,000, ,030,000 7

8 Answer the following questions. 95. List all whole numbers x such that x < List all whole numbers x such that x < List all whole numbers x such that x List all whole numbers x such that x < Your friend tells you she spent $180 on textbooks this semester. Assuming this number is rounded, what possible range of whole numbers could she have spent on textbooks? 100. Your friend tells you he has $1,200 in his bank account. Assuming this number is rounded, what possible range of whole numbers could he have in his bank account? 101. In 1999, it was reported that Bill Gates was worth 90 billion dollars. What level of accuracy was used in reporting this number? 102. In 1999, a large mutual fund had assets of 84 billion dollars. What level of accuracy was used in reporting this number? 103. Assuming x represents a digit, what possible values could it have if 4x51 < 4498? 104. Assuming x represents a digit, what possible values could it have if 9x ? 105. Assuming x represents a digit, what possible values could it have if 4x,359 44,359? 106. Assuming x represents a digit, what possible values could it have if 10x,997 < 102,997? 107. Arrange the following numbers from smallest to largest using the < symbol: 8123, 8132, 8312, 8321, 8213, Arrange the following numbers from smallest to largest using the < symbol: 9834, 9843, 9483, 9438, 9384, Arrange the following numbers from largest to smallest using the > symbol: 5674, 5647, 5476, 5467, 5746, Arrange the following numbers from largest to smallest using the > symbol: 6789, 6798, 6879, 6897, 6978, 6987 For the next 8 questions assume Round(x) represents the rounded value of x Assume x < y. Is it possible for Round(x) Round(y)? Explain Assume x y. Is it possible for Round(x) Round(y)? Explain Assume Round(x) Round(y). Is it possible for x > y? Explain Assume Round(x) < Round(y). Is it possible for x y? Explain Assume Round(x) > Round(y). Is it possible for x y? Explain Assume Round(x) Round(y). Is it possible for x < y? Explain Assume x < y. Must it be the case that Round(x) < Round(y)? Explain Assume x > y. Must it be the case that Round(x) Round(y)? Explain. 8

9 The next four questions refer to the following figure which represents the population of a town (Anytown, USA) during a 20-year time period: 119. Rounded to the nearest ten-thousand, what was the population of Anytown in 1980? 120. Rounded to the nearest ten-thousand, what was the population of Anytown in 1990? 121. Rounded to the nearest ten-thousand, what was the population of Anytown in 2000? 122. Is the population of Anytown increasing or decreasing? Explain your answer using inequality symbols. The next four questions refer to the following figure which represents the sales of Brand X for a company during a 4-year time period: 9

10 123. Rounded to the nearest million, what were the sales of Brand X in 1996? 124. Rounded to the nearest million, what were the sales of Brand X in 1997? 125. Rounded to the nearest million, what were the sales of Brand X in 1998? 126. Rounded to the nearest million, what were the sales of Brand X in 1999? 127. Are the sales of Brand X increasing or decreasing? Explain your answer using inequality symbols. 10