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Textbook by: Ron Stewart. E-mail the author

Whole Numbers

Module by: Denny Burzynski, Wade Ellis. E-mail the authorsEdited By: Math Editors

Summary: 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 difference 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.

Section Overview

• Numbers and Numerals
• The Hindu-Arabic Numeration System
• The Base Ten Positional Number System
• Whole Numbers
• Graphing Whole Numbers

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.

Sample Set A

The following are numerals. In each case, the first represents the number four, the second repre­sents the number one hundred twenty-three, and the third, the number one thousand five. These numbers are represented in different ways.

• Hindu-Arabic numerals
4, 123, 1005
• Roman numerals
IV, CXXIII, MV
• Egyptian numerals

Practice Set A

Exercise 1

Do the phrases "four," "one hundred twenty-three," and "one thousand five" qualify as numerals? Yes or no?

Solution

Yes. Letters are symbols. Taken as a collection (a written word), they represent a number.

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

0 1 2 3 4 5 6 7 8 9

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.

The Base Ten Positional Number System

Digits

The Hindu-Arabic numerals 0 1 2 3 4 5 6 7 8 9 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.

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 five periods listed above are the most common, and in our study of introductory mathematics, they are sufficient.

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.

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

Practice Set B

Exercise 2

Find the value of 5 in the number 65,000.

five thousand

Exercise 3

Find the value of 4 in the number 439,997,007,010.

Solution

four hundred billion

Exercise 4

Find the value of 0 in the number 108.

Solution

zero tens, or zero

Whole Numbers

Whole Numbers

Numbers that are formed using only the digits
0 1 2 3 4 5 6 7 8 9
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."

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 off 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."

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.

The numbers that have been graphed are

0, 106, 873, 874

Practice Set C

Exercise 5

Graph the following whole numbers: 46, 47, 48, 325, 327.

Exercise 6

Specify the whole numbers that are graphed on the following number line.

Solution

4, 5, 6, 113, 978

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.

Exercises

Exercise 7

What is a number?

concept

Exercise 8

What is a numeral?

Exercise 9

Does the word "eleven" qualify as a numeral?

Solution

Yes, since it is a symbol that represents a number.

Exercise 10

How many different digits are there?

Exercise 11

Our number system, the Hindu-Arabic number system, is a


number system with base

.

positional; 10

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

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?

Solution

ones, tens, hundreds

Exercise 14

Each period has its own particular name. From right to left, what are the names of the first four?

Exercise 15

In the number 841, how many tens are there?

4

Exercise 16

In the number 3,392, how many ones are there?

Exercise 17

In the number 10,046, how many thousands are there?

0

Exercise 18

In the number 779,844,205, how many ten mil­lions are there?

Exercise 19

In the number 65,021, how many hundred thousands are there?

Solution

0

For following problems, give the value of the indicated digit in the given number.

5 in 599

1 in 310,406

ten thousand

9 in 29,827

Exercise 23

6 in 52,561,001,100

Solution

6 ten millions = 60 million

Exercise 24

Write a two-digit number that has an eight in the tens position.

Exercise 25

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

How many three-digit whole numbers are there?

900

Exercise 28

How many four-digit whole numbers are there?

Exercise 29

Is there a smallest whole number? If so, what is it?

yes; zero

Exercise 30

Is there a largest whole number? If so, what is it?

Exercise 31

Another term for "visually displaying" is


.

graphing

Exercise 32

The whole numbers can be visually displayed on a


.

Exercise 33

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, 1007.

Exercise 35

Specify, if any, the whole numbers that are graphed on the following number line.

61, 99, 100, 102

Exercise 36

Specify, if any, the whole numbers that are graphed on the following number line.

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