1.1: Whole Numbers

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Learning Objectives

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
be able to identify and graph whole number

Numbers and Numerals

We begin our study of introductory mathematics by examining its most basic building block, the number.

Definition: 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.

Definition: 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

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

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

The Hindu-Arabic Numeration System

Definition: 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

Definition: Lenoardo 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

Definition: 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."

Definition: Base Ten Positional Systems

It is for this reason that our number system is called a positional number system with base ten.

Definition: Commas

When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three.

Definition: 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

Find the value of 6 in the number 7,261.

Solution

Since 6 is in the tens position of the units period, its value is 6 tens.

6 tens $=60$

Sample Set B

Find the value of 9 in the number 86,932,106,005.

Solution

Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions.

9 hundred millions = 9 hundred million

Sample Set B

Find the value of 2 in the number 102,001.

Solution

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

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

Answer: five thousand

Practice Set B

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

Answer: four hundred billion

Practice Set B

Find the value of 0 in the number 108.

Answer: zero tens, or zero

Whole Numbers

Definition: 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

Definition: 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.

Definition: 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.

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

Definition: 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

Graph the following whole numbers: 3, 5, 9.

$A number line from 0 to 11. There are dots on top of the dashes labeled, 3, 5, and 9.$

Sample Set C

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.

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

Solution

The numbers that have been graphed are

0, 106, 873, 874

Practice Set C

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

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

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

Practice Set C

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

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

Answer: 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 $\PageIndex{1}$

What is a number?

Answer: concept

Exercise $\PageIndex{2}$

What is a numeral?

Does the word "eleven" qualify as a numeral?

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

Exercise $\PageIndex{3}$

How many different digits are there?

Our number system, the Hindu-Arabic number system, is a number system with base ?

Answer: positional; 10

Exercise $\PageIndex{4}$

Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called.

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?

Answer: ones, tens, hundreds

Exercise $\PageIndex{5}$

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

In the number 841, how many tens are there?

Answer: 4

Exercise $\PageIndex{6}$

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

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

Answer: 0

Exercise $\PageIndex{7}$

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

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

Answer: 0

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

Exercise $\PageIndex{8}$

5 in 599

1 in 310,406

Answer: ten thousand

Exercise $\PageIndex{9}$

9 in 29,827

6 in 52,561,001,100

Answer: 6 ten millions = 60 million

Exercise $\PageIndex{10}$

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

Write a four-digit number that has a one in the thousands position and a zero in the ones position.

Answer: 1,340 (answers may vary)

Exercise $\PageIndex{11}$

How many two-digit whole numbers are there?

How many three-digit whole numbers are there?

Answer: 900

Exercise $\PageIndex{12}$

How many four-digit whole numbers are there?

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

Answer: yes; zero

Exercise $\PageIndex{13}$

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

Another term for "visually displaying" is ?

Answer: graphing

Exercise $\PageIndex{14}$

The whole numbers can be visually displayed on a .

Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34.

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

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

Exercise $\PageIndex{15}$

Construct a number line in the space provided below and graph (visually display) the following whole numbers: 84, 85, 901, 1006, 1007.

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

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

Answer: 61, 99, 100, 102

Exercise $\PageIndex{16}$

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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