1.1: Whole Numbers
- 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 .
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.
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.
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
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
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
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
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."
It is for this reason that our number system is called a positional number system with base ten .
When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three.
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.
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 \(=60\)
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
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
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
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.
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.
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."
Graph the following whole numbers: 3, 5, 9.
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.
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.
- Answer
-
Practice Set C
Specify the whole numbers that are graphed on the following number line.
- 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.
- Answer
-
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.
- Answer
-
61, 99, 100, 102
Exercise \(\PageIndex{16}\)
Specify, if any, the whole numbers that are graphed on the following number line.