Lesson 2

THE MEANING OF DECIMALS

A decimal -- 0.3865 -- is a proper fraction, which is a number less than 1. It is a part of number 1. To understand what that means, we must introduce the ordinal numbers: first, second, third, fourth, and so on. Because the ordinal numbers express division into equal parts.

The counting numbers have two forms called cardinal and ordinal.

A cardinal number answer the question How much? or How many?

An ordinal number answer the question Which one?  We will now see that the ordinal numbers express division into equal parts.  An ordinal number names which part.

example, into three equal parts, then we say that 5 is the third part of 15. We say that because 15 is the third multiple of 5:  5, 10, 15.  We use that same ordinal number to name the part.  If we divide a quantity into four equal parts, then we call each part a fourth; if into five equal parts, a fifth; if into one hundred equal parts, a hundredth; but if into two equal parts, we say half.

And so with the exception of half, an ordinal number names which part a smaller quantity is of its multiples.

We will go into this more in Lesson 14.

Since our numbering system is based on the powers of 10, it is called a decimal system. Decem in Latin means ten.  In the previous Lesson we learned about whole numbers.  Here we will learn about numbers that are less than 1. They are numbers we create when we divide 1 itself into equal parts.  They are numbers we will need for measuring rather than counting.  And since this is a decimal system, those parts of 1 will have the ordinal names of the powers of 10:  tenths, hundredths, thousandths, and so on.

First, we will divide One into ten equal parts, and so each part is called a Tenth. This is illustrated below.

 

If we divide each Tenth into ten equal parts, then One will be in one hundred equal parts, and so we have divided One into Hundredths.  If we divide each Hundredth into ten equal parts, then each tiny piece will be a Thousandth part of One.  And so on.  Those are called the decimal units.

The sequence of names of the decimal units follows the same sequence as the powers of 10.  They are those powers in their ordinal form.

412.387056

The decimal point signifies "and," in the sense of  "plus."

That is the whole number 412 plus 3 tenths, plus 8 hundredths, plus 7 thousandths; and so on.

Any number written with a decimal point is loosely called a decimal. A number less than 1 -- a proper fraction -- that is written with a decimal point we call a decimal fraction. The decimal digits signify a decimal fraction. A number written with a numerator and denominator we call a common fraction (Lesson 19).

The positions to the right of the decimal point are called the decimal places; the 1st, the 2nd, the 3rd, and so on.  Each decimal place refers to a decimal unit.  The decimal digits occupy those decimal places.

The digit at the 1st decimal place shows the number of Tenths.  In the number above, there are 3 tenths.

The 2nd decimal place shows the number of Hundredths.  In that number, there are 8 hundredths.  And so on.

The first digit to the left of the decimal point -- the 2 -- shows the number of Ones.

Each decimal digit has its place value:

Ones, tenths, hundredths, thousandths, ten-thousandths, . . .

As with the powers of 10, each place is ten times the place to its right.

One is ten tenths.

One-tenth is ten hundredths.

One-hundredth is ten thousandths.

One-thousandth is ten ten-thousandths.

And so on.

Example 1.  In this number

534.267

a)  there are how many ones?

Answer.   4.  The ones place is the first digit to the left of the decimal point.

b)  How many hundredths?

Answer.   6.  Hundredths (ordinal) is a decimal unit.  It falls to the right of the decimal point.

c)  How many hundreds?

Answer.   5.  Hundred (cardinal) is a whole unit.  It falls to the left of the decimal point.

Example 2.  Expanded form.   Just as the symbol for every whole number stands for a sum (Lesson 1), so does the symbol for every decimal.  Here is the expanded form of

534.267.

534.267 = 5 Hundreds + 3 Tens + 4 Ones + 2 Tenths + 6 Hundredths +

7 Thousandths.

Or, using common fractions:

In the first decimal place 2, notice that the denominator 10
has one 0. In the second decimal place 6, the denominator has two 0's. In the third place, three 0's; and so on. In the sixth decimal place, the denominator would have six 0's: 1,000,000. The sixth decimal place is millionths.

At this point, please "turn" the page and do some Problems.

or

Continue on to Section 2.

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