Addition and Subtraction of Whole Numbers: Whole Numbers

Transcription

1 OpenStax-CNX module: m Addition and Subtraction of Whole Numbers: Whole Numbers Denny Burzynski Wade Ellis This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses many of aspects of whole numbers, including the Hindu-Arabic numeration system, the base ten positional number system, and the graphing of whole numbers. By the end of this module students should be able to: know the dierence between numbers and numerals, know why our number system is called the Hindu-Arabic numeration system, understand the base ten positional number system, and identify and graph whole numbers. 1 Section Overview Numbers and Numerals The Hindu-Arabic Numeration System The Base Ten Positional Number System Whole Numbers Graphing Whole Numbers 2 Numbers and Numerals We begin our study of introductory mathematics by examining its most basic building block, the number. Number A number is a concept. It exists only in the mind. The earliest concept of a number was a thought that allowed people to mentally picture the size of some collection of objects. To write down the number being conceptualized, a numeral is used. Numeral A numeral is a symbol that represents a number. In common usage today we do not distinguish between a number and a numeral. In our study of introductory mathematics, we will follow this common usage. Version 1.5: Feb 17, :02 pm

2 OpenStax-CNX module: m Sample Set A The following are numerals. In each case, the rst represents the number four, the second represents the number one hundred twenty-three, and the third, the number one thousand ve. These numbers are represented in dierent ways. Hindu-Arabic numerals 4, 123, 1005 Roman numerals IV, CXXIII, MV Egyptian numerals 2.2 Practice Set A Exercise 1 (Solution on p. 7.) Do the phrases "four," "one hundred twenty-three," and "one thousand ve" qualify as numerals? Yes or no? 3 The Hindu-Arabic Numeration System Hindu-Arabic Numeration System Our society uses the Hindu-Arabic numeration system. This system of numeration began shortly before the third century when the Hindus invented the numerals Leonardo Fibonacci About a thousand years later, in the thirteenth century, a mathematician named Leonardo Fibonacci of Pisa introduced the system into Europe. It was then popularized by the Arabs. Thus, the name, Hindu-Arabic numeration system. 4 The Base Ten Positional Number System Digits The Hindu-Arabic numerals are called digits. We can form any number in the number system by selecting one or more digits and placing them in certain positions. Each position has a particular value. The Hindu mathematician who devised the system about A.D. 500 stated that "from place to place each is ten times the preceding." Base Ten Positional Systems It is for this reason that our number system is called a positional number system with base ten. Commas When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three. Periods These groups of three are called periods and they greatly simplify reading numbers.

3 OpenStax-CNX module: m In the Hindu-Arabic numeration system, a period has a value assigned to each or its three positions, and the values are the same for each period. The position values are Thus, each period contains a position for the values of one, ten, and hundred. Notice that, in looking from right to left, the value of each position is ten times the preceding. Each period has a particular name. As we continue from right to left, there are more periods. The ve periods listed above are the most common, and in our study of introductory mathematics, they are sucient. The following diagram illustrates our positional number system to trillions. (There are, to be sure, other periods.) In our positional number system, the value of a digit is determined by its position in the number. 4.1 Sample Set B Example 1 Find the value of 6 in the number 7,261. Since 6 is in the tens position of the units period, its value is 6 tens. 6 tens = 60 Example 2 Find the value of 9 in the number 86,932,106,005. Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions. 9 hundred millions = 9 hundred million Example 3 Find the value of 2 in the number 102,001. Since 2 is in the ones position of the thousands period, its value is 2 one thousands. 2 one thousands = 2 thousand 4.2 Practice Set B Exercise 2 (Solution on p. 7.) Find the value of 5 in the number 65,000. Exercise 3 (Solution on p. 7.) Find the value of 4 in the number 439,997,007,010.

4 OpenStax-CNX module: m Exercise 4 (Solution on p. 7.) Find the value of 0 in the number Whole Numbers Whole Numbers Numbers that are formed using only the digits are called whole numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,... The three dots at the end mean "and so on in this same pattern." 6 Graphing Whole Numbers Number Line Whole numbers may be visualized by constructing a number line. To construct a number line, we simply draw a straight line and choose any point on the line and label it 0. Origin This point is called the origin. We then choose some convenient length, and moving to the right, mark o consecutive intervals (parts) along the line starting at 0. We label each new interval endpoint with the next whole number. Graphing We can visually display a whole number by drawing a closed circle at the point labeled with that whole number. Another phrase for visually displaying a whole number is graphing the whole number. The word graph means to "visually display." 6.1 Sample Set C Example 4 Graph the following whole numbers: 3, 5, 9. Example 5 Specify the whole numbers that are graphed on the following number line. The break in the number line indicates that we are aware of the whole numbers between 0 and 106, and 107 and 872, but we are not listing them due to space limitations. 0, 106, 873, 874 The numbers that have been graphed are

5 OpenStax-CNX module: m Practice Set C Exercise 5 (Solution on p. 7.) Graph the following whole numbers: 46, 47, 48, 325, 327. Exercise 6 (Solution on p. 7.) Specify the whole numbers that are graphed on the following number line. A line is composed of an endless number of points. Notice that we have labeled only some of them. As we proceed, we will discover new types of numbers and determine their location on the number line. 7 Exercises Exercise 7 (Solution on p. 7.) What is a number? Exercise 8 What is a numeral? Exercise 9 (Solution on p. 7.) Does the word "eleven" qualify as a numeral? Exercise 10 How many dierent digits are there? Exercise 11 (Solution on p. 7.) Our number system, the Hindu-Arabic number system, is a number system with base. Exercise 12 Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called. Exercise 13 (Solution on p. 7.) In our number system, each period has three values assigned to it. These values are the same for each period. From right to left, what are they? Exercise 14 Each period has its own particular name. From right to left, what are the names of the rst four? Exercise 15 (Solution on p. 7.) In the number 841, how many tens are there? Exercise 16 In the number 3,392, how many ones are there? Exercise 17 (Solution on p. 7.) In the number 10,046, how many thousands are there? Exercise 18 In the number 779,844,205, how many ten millions are there? Exercise 19 (Solution on p. 7.) In the number 65,021, how many hundred thousands are there? For following problems, give the value of the indicated digit in the given number.

6 OpenStax-CNX module: m Exercise 20 5 in 599 Exercise 21 (Solution on p. 7.) 1 in 310,406 Exercise 22 9 in 29,827 Exercise 23 (Solution on p. 7.) 6 in 52,561,001,100 Exercise 24 Write a two-digit number that has an eight in the tens position. Exercise 25 (Solution on p. 7.) Write a four-digit number that has a one in the thousands position and a zero in the ones position. Exercise 26 How many two-digit whole numbers are there? Exercise 27 (Solution on p. 7.) How many three-digit whole numbers are there? Exercise 28 How many four-digit whole numbers are there? Exercise 29 (Solution on p. 7.) Is there a smallest whole number? If so, what is it? Exercise 30 Is there a largest whole number? If so, what is it? Exercise 31 (Solution on p. 7.) Another term for "visually displaying" is. Exercise 32 The whole numbers can be visually displayed on a. Exercise 33 (Solution on p. 7.) Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34. Exercise 34 Construct a number line in the space provided below and graph (visually display) the following whole numbers: 84, 85, 901, 1006, Exercise 35 (Solution on p. 7.) Specify, if any, the whole numbers that are graphed on the following number line. Exercise 36 Specify, if any, the whole numbers that are graphed on the following number line.

7 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 2) Yes. Letters are symbols. Taken as a collection (a written word), they represent a number. Solution to Exercise (p. 3) ve thousand Solution to Exercise (p. 3) four hundred billion Solution to Exercise (p. 4) zero tens, or zero 4, 5, 6, 113, 978 concept Yes, since it is a symbol that represents a number. positional; 10 ones, tens, hundreds ten thousand 6 ten millions = 60 million 1,340 (answers may vary) 900 yes; zero graphing 61, 99, 100, 102