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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
    Three diagrams in succession, each with a label below. Three short vertical lines, labeled strokes. One swirled line next to two horseshoe-shaped lines, next to three short vertical lines, labeled coiled rope, heel bones, and strokes. One flower-shaped drawing next to five vertical lines, labeled, lotus flower and strokes.

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
Three segments, labeled from left to right, hundreds, tens, and ones. Below the segments is a larger label, period.

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.
A series of groups of three segments, separated by commas. The segments are labeled, from left to right, trillions, billions, millions, thousands, and units.

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.)
A series of groups of three segments, separated by commas. The groups of segments are labeled, from left to right, trillions, billions, millions, thousands, and units. Each segment in the group of three has a label. From left to right, in each group, the segments are labeled hundreds, tens, and ones.

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.

Solution

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.
A line with arrows on the left and right. Along the line are evenly spaced dashes, numbered from 0 to 10 from the left to the right of the line.

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.

A number line from 0 to 11. There are dots on top of the dashes labeled, 3, 5, and 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.

A number line from 0 to 874, with not all whole numbers between 0 and 874 displayed. There are two jagged breaks in the line, one between 0 and 106, and one between 107 and 872. There are dots on the dashes for 0, 106, 873, and 874.

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.
A line with arrows on the left and right. The line has two jagged breaks.

Solution

A number line from 0 to 327, with not all whole numbers between 0 and 327 displayed. There are two jagged breaks in the line, one between 0 and 46, and one between 48 and 325. There are dots on the dashes for 46, 47, 48, 325, and 327.

Exercise 6

Specify the whole numbers that are graphed on the following number line.
A number line between 0 and 979, with not all whole numbers between 0 and 979 displayed. There are two jagged breaks in the line, one between 6 and 112, and one between 113 and 978. There are dots on the dashes for 4, 5, 6, 113, and 978.

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?

Solution

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
               
.

Solution

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?

Solution

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?

Solution

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.

Exercise 20

5 in 599

Exercise 21

1 in 310,406

Solution

ten thousand

Exercise 22

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.

Solution

1,340 (answers may vary)

Exercise 26

How many two-digit whole numbers are there?

Exercise 27

How many three-digit whole numbers are there?

Solution

900

Exercise 28

How many four-digit whole numbers are there?

Exercise 29

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

Solution

yes; zero

Exercise 30

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

Exercise 31

Another term for "visually displaying" is

               
.

Solution

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.

A number line from 0 to 34, with not all numbers between 0 and 34 displayed. There is a jagged break in the line between 4 and 29.

Solution

A number line from 0 to 34, with not all whole numbers between 0 and 34 displayed. There is a jagged break in the line, between 4 and 29. There are dots on the dashes for 1, 31, and 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.

A number line from 0 to 102, with not all whole numbers between 0 and 102 displayed. There are two jagged breaks in the line, one between 0 and 61, and one between 64 and 99. There are dots on the dashes for 61, 99, 100, and 102.

Solution

61, 99, 100, 102

Exercise 36

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

A number line from 0 to 87, with not all whole numbers between 0 and 87 displayed. There are three jagged breaks in the line, one between 1 and 8, one between 11 and 73, and one between 74 and 85.

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