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.

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.

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.

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."

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

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