Lesson 3 THE MEANING OF DECIMALSA 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. In this Lesson, we will answer the following:
The countingnumbers have two forms: cardinal and ordinal. 



A cardinal number answers the question How many? 



An ordinal number answers the question Which one? We will now see that the ordinal numbers express division into equal parts. An ordinal number names which part  the third part, the fourth, the fifth, and so on. Division into equal parts If we divide 15, for example, into three equal parts  into three 5's  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 indicates into how many parts a quantity has been divided. A third, a fourth, a fifth, and so on, are the names of parts. Those names belong to language itself, not just to mathematics. Those ordinals are prior to the names of numbers we call fractions, which, we are about to see, are numbers we need for measuring, and are the parts of number 1. We will go into this more in Lesson 15. The decimal system 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 Lessons we learned about whole numbers, which are the repeated additions of 1: First, we will divide One into ten equal parts, and so each part is called a Tenth.
If we divide each Tenth into ten equal parts, then One will be in one hundred equal parts  count them 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 Tenths, Hundredths, Thousandths 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 of 1, plus 8 hundredths of 1, 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  written as a decimal, we call a decimal fraction. A number written with a numerator and denominator we call a common fraction (Lesson 20). The positions to the right of the decimal point are called the decimal 412.387056 places; the 1st, the 2nd, the 3rd, and so on. Each decimal digit occupies a decimal place. 



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. As with whole numbers, each digit has its place value: 2.3875. . . Ones, tenths, hundredths, thousandths, tenthousandths, . . . As with the whole numbers, each place is ten times the place to its right. One is ten tenths. Onetenth is ten hundredths. Onehundredth is ten thousandths. Onethousandth is ten tenthousandths. 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 numeral for every whole number stands for a sum (Lesson 2), so does the numeral 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. At this point, please "turn" the page and do some Problems. or Continue on to Section 2.
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