CHAPTER XII.

DECIMALS.

Their Nature, Manner of Statement, &c.

365. Decimals, or Decimal Fractions, are numbers representing the numerators of fractions, whose denominators are not written but understood, and which are decuples,* centuples,† &c., of 1, that is, 1 with one or more cyphers annexed, as 10, 100, &c.

366. The difference between a vulgar fraction and a decimal fraction consists in this, that in a vulgar fraction the denominator is always written, and may consist of any, or of all the nine figures, or of any combination of these figures; whereas in a decimal fraction, the denominator is not written, but shown by a point placed before the numerator, called the decimal point, which point shows the number of parts into which a unit is divided, such parts being always tenths, hundredths, thousandths, &c.

867. Notwithstanding the ease and readiness with which calculations are carried on by vulgar fractions, in many cases these calculations may be performed with greater ease and readiness by decimals. Decimals are also more simple in their mode of statement than vulgar fractions. To represent a vulgar fraction requires two lines of figures, one line representing the numerator, and the other the denominator; but in a decimal fraction, both numerator and denominator are represented in the same line. For instance, a half may be represented by any fraction whose denominator is twice the numerator, as

Now any of these fractions written as a decimal would stand thus, 5, that is five-tenths; the decimal point placed before the 5 shows that the denominator is to be considered as 10. This point always shows the amount of the denominator by occupying the same position in the numerator of the decimal, which it would have done in the denominator, if written as a vulgar fraction. Thus

*Decuples, numbers which are ten times as much as others, as 20 is a decuple of 2; from Deca, ten.

+ Centuples, numbers which are a hundred times as much as others, as 200 is a centuple of 2; from Centum, a hundred.

The number of figures pointed off for a decimal fraction is always the same as the number of cyphers in the denominator considered as tenths, hundredths, &c. Thus,

5%==5·5.

I

That is 5 units and 5 tenths of a unit,

55 5

55%==55·5, §§ 5·55, 555, &c.

555 158=

368. Decimal fractions are subject to the same laws, and gov erned by the same principles, as vulgar fractions. No two branches of arithmetic are more intimately connected than these, though they are generally represented as separate and distinct branches; and from the manner in which they are generally treated, the student would be led to infer that there was no connexion between them. Yet every rule, every principle applicable to vulgar fractions, is also, in substance, applicable to decimals, which will be shown as we proceed.

369. The student will have seen, that in order to prepare vulgar fractions for addition and subtraction, it is necessary to reduce them to a common denominator. One advantage attending decimals is, that they are always in a common denominator, or at least their common denominator is seen at once, for the denominator of a decimal always consists of such num. bers as 10, 100, 1000, &c. A decuple of any number is changed into a centuple of the same number, by merely annex · ing a cypher, or contrariwise, by merely cutting off a cypher. Therefore 5 and ∙12, equal and are reduced to a common denominator by annexing a cypher to .5. It then becomes 50 equal. From this view of the subject, the student will perceive that cyphers on the right of a decimal do not alter its value; for 550,500 == 50 = 500 = ; consequently that they are reduced to a common denominator by annexing cyphers to the right hand, until they have all the same number of places, thus 4, 62, 180, and 16.34 when reduced to a common denominator, become ⚫400, 620, 180, and 16-340. He will perceive, also, that for all common purposes it is unnecessary to write cyphers to the right of a decimal, as it is equally well expressed without them, for 6, 60, and 600 are all of the same value.

On the contrary, cyphers on the left of a decimal diminish its ualue ten times for every cypher so placed.

•5 = √ = 1, ·05 = 15 =0,005

Thus

In this particular there is a striking difference between a whole number and the decimal parts of a number; for as a whole num ber is not altered in value by annexing cyphers to the left hand, but is increased tenfold by every cypher annexed to

the right hand; on the contrary, a decimal is not altered in value by annexing cyphers to the right hand, but diminished tenfold by every cypher annexed to the left hand; yet there is an intimate connexion between the notation of whole numbers and the notation of decimals.

370. To illustrate this connexion, let the student keep in mind that principle in figures put so conspicuously forward on all occasions throughout this work, and so particularly described in paragraphs 9 to 18, viz., that figures increase in a tenfold ratio as they advance from right to left, and consequently that they decrease in the same ratio as they recede from left to right. Applying this principle to the six threes in the margin, numbered 1, 2, 3, from the decimal point towards the left, and numbered also in like manner from the decimal point towards the right; if we take the three threes on the left of the decimal point first; the 3 marked 1 is three units, the 3 marked 2 is three tens, while the 3 marked 3 is three hundreds, and ten times as much as the 3 marked 2, and one hundred times as much as the 3 marked 1.

321 123 333.333

371. Now let us recede backwards, and mark how whole numbers and decimals harmonize in their decreasing ratio. The 3 marked 3, as just observed, is three hundred; the 3 marked 2 is one-tenth of the 3 marked 3; the 3 marked 1 is one-tenth of the 3 marked 2. If we proceed in the same manner beyond the decimal point towards the right, then the 3 marked 1 on the right of the decimal point is one tenth of the 3 marked 1 on the left of it; the 3 marked 2 is one-tenth of the 3 marked 1; and the 3 marked is one-tenth of the 3 marked 2. Thus it will be seen that any figure employed for expressing either a whole number or a decimal is always of what it would have been had it stood one space further towards the left hand. The six threes, represented as a mixed_number, would stand thus:

30

3

333333300 +30 +3 +3 +130 + 1000 = 300 +30 +3+300000 + 1100 + 7000 Compare the following fractions, with their corresponding decimals:

1000

100

то

186493186-493, 186493 =1864-93, 186493 — 18449-3, 186493 186493-0 186493 18.6493, 186493

I

10000

100000

=1.86493

372. It has been advanced, atparagraph 368, that every rule in vulgar fractions is also in substance a rule in decimals; that their principles are in most cases the same, but differing in their notation. In vulgar fractions, the denominator describes the number of parts into which unity is divided, and the

numerator the number of these parts taken: so also in decimals; for the decimal 17 shows that unity is divided into 100 parts, and that 17 of them are taken. The value of a vulgar fraction depends on the proportion existing between the numerator and the denominator, so also does the value of the decimal, for 5 is ten times as much as '05 .and twice as much as 25. In a vulgar fraction the denominator is considered as a divisor, and the numerator as a dividend; and in cases where the fraction is an improper fraction, dividing the numerator by the denominator, gives its value as 55 In a decimal the division is represented as completed, and the quotient wrote down as 5.5. From the above, the student will perceive that where the denominator of an improper fraction consists of 1, and any number of cyphers, such fraction may be reduced to a decimal, by merely dividing the numerator by the denominator; thus

1893 18.93 1674 167·4, &c.

The student must be careful to distinguish between the decimal point and the point which is the sign of multiplication. The point which is the sign of multiplication is written at the foot of the figures, thus 2.4=8; while the decimal point is written near the top, thus 5.75.

373. QUESTIONS FOR EXAMINATION,

IN CHAPTER XII.

1. What are Decimal Fractions? In what consists the difference between Decimal and Vulgar Fractions?

2. Have Decimal Fractions any denominator?

3. What determines the value of a decimal fraction?

4. In what do decimal fractions resemble vulgar fractions? 5. Is the value of a decimal altered by annexing cyphers to the right hand? Is its value altered by annexing cyphers to the left hand?

6. Is there any resemblance between the notation of decimals and that of whole numbers ?

7. What effect has the removal of the decimal point towards the right, or towards the left, on the value of the decimal?

8. Are the principles and rules of vulgar fractions similar to those of decimals?

CHAPTER XIII.

REDUCTION OF DECIMALS.

CASE 1.

374. It has just been shown above, that if a fraction be an improper fraction, and its denominator be a decuple of 1, of 10, of 100, &c., it may be reduced to a decimal, by dividing the numerator by the denominator, which is merely to point off as many figures from the numerator for a decimal as there are cyphers in the denominator; as

166166, 166 166, 131 13.21, &c.

=

375. But as fractions of the description of the above are seldom met with, and as we have proper as well as improper fractions, and as we cannot divide a lesser number by a larger, and as in proper fractions the denominator is always larger than the numerator, we must have some other rule for the reduction of fractions; that rule is this:

:

376. RULE.-Divide the numerator by the denominator in cases where it can be done; where it cannot, annex cyphers to the numerator to complete the division, and point off in the quotient for decimals as many figures as there have been cyphers annexed to the numerator; and if there be not as many figures in the quotient as there have been cyphers annexed to the numerator, annex cyphers to the left of the quotient to supply the requisite number. The following examples will exemplify the rule :

377. The reason for the rule may be thus explained: From the knowledge which the student will have acquired of the relationship of multiple, sub-multiple, and common multiple, of measure, and of common measure, as explained in paragraphs 120 to 185, he will readily see, that if one number is to be