Introduction to Whole Numbers

Transcription

1 Section 1.1 PRE-ACTIVITY PREPARATION Introduction to Whole Numbers Annette has just landed a terrific job as a middle school teacher. Instead of renting an apartment, she decides to buy a small condominium. When she reads the mortgage contract, she notices that the amount of the loan is written in two ways: in words and in digits. Her total loan amount is written as sixty-three thousand, two hundred seventy-nine dollars and as $63,279. Formal business documents frequently have numbers written in both word form and standard form (digits) for the purpose of cross-referencing and verifying dollar amounts. On handwritten business and personal checks, for example, using both forms ensures the that the digits have not been altered. Mathematics uses a language of numbers and symbols. When you understand what numbers mean, whether represented in standard form or in words, you will use them more effectively. LEARNING OBJECTIVES Read, interpret, and represent a number in digits as well as in words. Translate numbers between words and standard notation. TERMINOLOGY NEW TERMS TO LEARN base ten system decimal system digit expanded form infinite period place placeholder place value standard form standard notation value whole number word form 13

2 14 Chapter 1 Whole Numbers BUILDING MATHEMATICAL LANGUAGE The whole numbers are the number zero and all the counting numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and so on Note that the series of whole numbers is infinite that it goes on without end. You can always count one number higher. Whole numbers are most commonly written in standard form (or standard notation) using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to present the number, as in 63,279. The digits used to write a number in standard form take their proper positions in the number according to a system of writing numbers called the decimal (or base ten) system, wherein every position, or place has a corresponding and specific place value. The following is a partial decimal system place value chart for whole numbers. The number 63,279 is positioned correctly in the chart. Period millions thousands ones Place name hundredmillions ten-millions millions hundredthousands tenthousands thousands hundreds tens ones Place value 100,000,000 10,000,000 1,000, ,000 10,000 1, Example number , Notice that, as you move from right to left, each place value represents 10 times the place value to its immediate right: one ten is equal to 10 ones, one hundred is equal to 10 tens, one thousand is equal to 10 hundreds, one ten-thousand is equal to 10 thousands, one hundred-thousand is equal to 10 ten-thousands, one million is equal to 10 hundred-thousands, and so on. This pattern is the foundation of the decimal (base ten) system. It underlies not only the way numbers are written, but also how they are computed, as you will later study. Also note that the chart is partial because you could continue the pattern and get ten times larger with each place as you move past the millions to billions, trillions, quadrillions, quintillions, and beyond!

3 Section 1.1 Introduction to Whole Numbers 15 Furthermore, the position of a digit in a number determines its corresponding value in the number. Look at the example number in the chart: 63, in the ten-thousands place has a value of 6 ten-thousands or 60,000 3 in the thousands place has a value of 3 thousands or in the hundreds place has a value of 2 hundreds or in the tens place has a value of 7 tens or 70 9 in the ones place has a value of 9 ones or 9 The expanded form of a number breaks it down and calls attention to its parts by presenting the number as the total of its digits values. For example, the expanded form of the number 63,279 is 6 ten-thousands + 3 thousands + 2 hundreds + 7 tens + 9 ones or simply 60, , A whole number might be spoken aloud, in its word form. It might also be spelled out in word form, as for the previous example, sixty-three thousand, two hundred seventy-nine. Each whole number from 21 through 99 (other than 30, 40, 50, 60, 70, 80, and 90) requires the use of a hyphen in its written word form (twenty-one through ninety-nine). Starting from the ones place at the far right, each three-digit place value group makes up a period ones, thousands, millions, billions, and so on (refer again to the chart on the previous page). Notice that the ones period is actually made up of three place values ones, tens, and hundreds. Also notice how this pattern continues in the one thousands, ten-thousands, hundred-thousands, and so forth as you move left through the periods. For ease of reading or writing a number in words or digits, commas are used to separate the periods. When you add, subtract, multiply, or divide whole numbers, it is customary to drop the commas from their standard forms while doing the actual computations, resulting in a number with correctly positioned digits but no commas. In order to present the answer in standard form, start counting from the right, moving three places for each period. Insert commas at the period breaks. For example, an answer would be properly presented as millions thousands ones 5, 0 4 3, in its standard form. In numbers with only four digits, the comma between the thousands and the hundreds periods is optional, but the correct period word thousand is still required when reading or writing them. For example, the number 5,618 ( five thousand, six hundred eighteen ) is sometimes written as 5618 without the comma after the thousands place digit ( five thousand six hundred eighteen ).

4 16 Chapter 1 Whole Numbers An understanding of the proper use of the number zero (0) is essential for writing numbers in both standard and word forms A zero (0) is used as a placeholder in the standard form of a number to represent a value not accounted for in its word form. For example, consider forty-eight thousand, fi fty three. Because the word hundreds is neither written nor spoken, the digit 0 is used as the hundreds placeholder in its standard form: 48,053. The reverse is also true. A zero placeholder in the standard form of a number signifies a value of 0, hence a place that should not be represented by a word. Using the same example, the 0 in the hundreds place in 48,053 has a value of 0 hundreds, or 0. Therefore, hundreds is not presented in its word form, forty-eight thousand, fi fty-three. What does this mean for the expanded form of the number? The expanded form of 48,053 is 40, , or simply 40, , indicating that there are no hundreds in the number, only ten-thousands (4 of them), thousands (8), tens (5), and ones (3). The word and should not to be used when reading a three-digit whole number, although it is often mistakenly spoken in casual conversation. For example, the correct reading of 804 is eight hundred four (not eight hundred and four ). As you will learn in the chapters on decimal numbers and fractions, the word and is reserved for use when reading or writing a number containing a decimal point, indicating the placement of the decimal point, as in ( eight hundred four and seven tenths ) or when reading a mixed number such as 2½ ( two and one half ). For whole numbers of four or more digits, the final period name ones is never used. Also, in the English language, the singular forms of the larger period names are used when a number is spoken or written in words. For example, the number 3,201,536,794 is read, three billion, two hundred one million, five hundred thirty-six thousand, seven hundred ninety-four. Reading and Writing Whole Numbers Keeping in mind the base ten system place value chart, you can easily translate between the standard form and word form of any whole number, as demonstrated by the following techniques and models.

5 Section 1.1 Introduction to Whole Numbers 17 TECHNIQUE Translating a Number from Standard Form to Word Form Technique Working from left to right, say or write out the word name for the number in each period (indicated by comma breaks) followed by the period word. Use commas in the written form. Special Case: Three placeholder zeros in a period (see below, Model D) MODELS Translate each of the following numbers to its word form. A 7,243,603 millions thousands ones seven two hundred forty-three six hundred three Answer: seven million, two hundred forty-three thousand, six hundred three B 28,064,852,019 billions millions thousands ones twenty-eight sixty-four eight hundred fifty-two nineteen Answer: twenty-eight billion, sixty-four million, eight hundred fifty-two thousand, nineteen C 550,009 Answer: five hundred fifty thousand, nine Special Case: Three Placeholder Zeros in a Period When all three places in a period are zeros, that period is not named in the word form. D 4,000,975 THINK The thousands period consists of three zero placeholders, so thousands is not in the word form. Answer: four million, nine hundred seventy-five

6 18 Chapter 1 Whole Numbers TECHNIQUE Translating a Number from Word Form to Standard Form Technique Working from left to right, substitute digits that correspond to the number stated for each period and use commas in place of period names. Special Case: A particular period is not in the word form (see below, Models C & D) MODELS Write each of the following numbers in standard form. A fifty-three thousand, four hundred seventy two thousands ones Answer: 53,472 B seventy-eight million, three hundred fifty-five thousand, four hundred sixty-two millions thousands ones Answer: 78,355,462 Special Case: A Particular Period is Not in the Word Form Use three placeholder zeros to represent a period missing from the word form. C two million, one hundred twenty-eight THINK The thousands period is not in the word form. Use three zeros as placeholders for the thousands period. Answer: 2,000,128 D four hundred thousand Use three zeros as placeholders for the ones period. Answer: 400,000

7 Section 1.1 Introduction to Whole Numbers 19 ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Incorrectly translating periods of three zero placeholders when translating from standard form to word form 8,000,123 in word form is eight thousand, one hundred twenty-three t Keep in mind the place value chart. Use the commas in the standard form as clues to how many periods there are in the given number. THINK There are three periods in 8,000,123 as indicated by the commas millions, thousands, and ones. When a period consists of all zero placeholders, do not name that period in the word form. Zeros hold all three places in the thousands period. Do not name thousands in the word form. 8,000,123 in word form is eight million, one hundred twenty-three Incorrectly identifying the periods requiring zero placeholders when translating from word form to standard form six million, nine hundred sixty-five in standard form is 6,965,000 0 Keep in mind the place value chart. Use zeros as placeholders when entire periods are not in the word form. THINK The thousands period is the period not written in six million, nine hundred sixtyfive. Use three zeros as placeholders for the three places in the thousands period. 6,000,965 Incorrectly using the word and in the word form of a whole number 25,304 in word form is twenty-five thousand, three hundred and four The word and is not used when speaking or writing a whole number in word form. (It is reserved for use when reading a number containing a decimal point to indicate the placement of the decimal point.) 25,304 in word form is twenty-five thousand, three hundred four

8 20 Chapter 1 Whole Numbers PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with reading and writing whole numbers the identification of place values the value of a digit with respect to its particular position in a number the meaning of zero as a placeholder the use of period words in reading and writing whole numbers the use of commas in writing numbers

9 Section 1.1 ACTIVITY Introduction to Whole Numbers PERFORMANCE CRITERIA Translating a number between words and standard notation use of correct place values and period names correct placement of commas correct spelling and use of hyphens CRITICAL THINKING QUESTIONS 1. How do you determine a digit s value in a number? Give an example. 2. When do you use the placeholder digit zero (0) in the standard form of a number? Give an example. 3. When you translate a number into words, what do you do with periods that contain all zeros? Give an example. 21

10 22 Chapter 1 Whole Numbers 4. What is the relationship between periods (place value groups) and commas when writing numbers in either words or digits? 5. How is the place value chart used to translate numbers between words and digits? 6. How can you assure that a translation of a whole number into a new representation is accurate? 7. When should you use numbers written in words rather than in digits?

11 Section 1.1 Introduction to Whole Numbers 23 TIPS FOR SUCCESS Keep in mind the place value chart to translate numbers written in words or digits. When translating from words, be sure to use 0 placeholders in the standard form to represent place values not accounted for in words. When writing a number in standard notation, use commas to separate the three-digit periods. Count the place values in groups of three, beginning with the ones place and moving left. To assure that your translation is correct, cover the original representation; then translate back to see if you get the same number, in digits or words. DEMONSTRATE YOUR UNDERSTANDING 1. Identify the place name as requested for the number 4,350,927. a) The 2 is in the place. b) The 4 is in the place. c) The 3 is in the place. d) The 0 is in the place. 2. Write the following numbers in standard notation. a) three thousand, six hundred eighty-one b) six hundred fifty-four thousand, twenty-three c) five million, twenty-eight thousand, four hundred four d) seven million, six hundred fifty 3. Write in word form. a) 2,745 b) 378,213 c) 1,003,030 d) 9,000,422,000

12 24 Chapter 1 Whole Numbers IDENTIFY AND CORRECT THE ERRORS Identify and correct the errors in the following translations. The first one has been done for you. Worked Solution What is Wrong Here? 1) Translate thirty-two thousand, four hundred fifty-six to standard form 320,456 2) Translate three million, three hundred to standard form. 3,300,000 3) Translate 5703 to word form. fifty-seven hundred and three Identify the Errors Should be 32 (two digits) in the thousands rather than 320. Correct Process Answer: 32,456 4) Translate 12,000,500 to word form. twelve thousand five hundred ADDITIONAL EXERCISES 1. Write in standard notation: a) eighteen thousand, five hundred forty-two b) seven million, one hundred two thousand, six hundred three c) thiry-nine million, seven hundred sixty d) one billion, two hundred million, three hundred forty-five thousand, one hundred 2. Write in words: a) 498,555 b) 3,002,080 c) 11,000,346 d) 5,018,234,010