Tut. Although vulgar fractions can be applied in all cases where they are wanted, and that correctly, yet from the nature of numeration, it must appear plain to you that if fractions were made to decrease in the same manner that whole numbers increase, the labour of performing on fractional numbers would be greatly facilitat ed, and rendered equally easy with whole numbers. Such is the nature of decimal fractions.

11. As it is a law of whole numbers that they increase from right to left in a tenfold proportion, so it is a law of decimal fractions, that they decrease to the right in a tenfold proportion. You have been taught (2. Con. I.) that a unit is the smallest whole number that we have, and that every time we remove it one place to the left, we increase it ten times.

12. Now in order that fractions may be made to correspond with this law, a unit is made to consist of ten parts, and each of these tenth parts to consist of tenths parts or hundredth parts of a unit, and each of these hundredth parts to consist of tenths parts, or thousandths of a unit, and so on, no part ever becoming so small but that it may be divided into ten parts.

The unit is equivalent to

10 tenths;
10 hundredths;

One of these tenths is equivalent to
One of these hundredths is equivalent to 10 thousandths;

And so on to infinity, one part of the preceding being equal to ten of the next succeeding, decreasing in the same manner to the right in which whole numbers increase to the left; or it merely is a continuation of decrease from left to right in the same manner as though whole numbers and decimals were all alike, and decreased to the right, down to the most minute parts.

Pup. I now think I understand what a decimal fraction is, and it appears to me that they will entirely supersede the use of vulgar fractions: for I cannot see why they may not answer every purpose equally as well, and make the work much more easy.

Tut. They will answer all the purposes for which you will want to use fractions. But, notwithstanding, it was

necessary that you should understand vulgar fractions, or the principles of them, in order to understand decimals; for there is not so much difference between them as you probably imagine. Vulgar fractions always have a de. nominator expressed; but decimal fractions, having their denominators increase in a uniform ratio, do not require that they should be expressed.

Suppose you wish to write three and half in decimal form, you would do it thus, 3,5, which is read, three and five tenths. This may be changed into a mixed number by expressing the denominator, thus 3

If it were required to write 27 and 3 tenths, 40 and 22 hundredths, 12 and 11 thousandths, they should stand as follows,

27,3

40,22
12,011

11

13. If the fractions were expressed with their denominators they would stand as follows, 27, 40, 120, so that you will see plainly that there is no need of expressing the denominators to decimals, as their denominators are always 1, with as many ciphers annexed as there are decimal figures, consequently their denominators are always known. You can always reduce a decimal to a vulgar fraction, by placing, for its denominator, a unit with as many ciphers annexed as there are figures in the decimal expression.

That you may be able to enunciate, or reduce to language, numbers with decimal expressions, I will give the following table.

The number 26,4 is read 26 and 4 tenths;

10,02 is read 10 and 2 hundredths;

0,24 is read 24 hundredths;

0,024 is read 24 thousandths;

0,246 is read 246 thousandths;

0,0000042 is read 42 ten millionths;

0,5 is read 5 tenths;

0,50 is read 50 hundredths;

0,500 is read 500 thousandths.

From this table you will see that it makes no differencein the value of decimals, whether ciphers are placed at their right or not; for 0,5 bears the same proportion to its denominator that 50, 500 and 5000 do to theirs. (2. Con. III.)

Pup. From this explanation of decimal fractions, I cannot see as operations with them are performed any way different from operations with whole numbers.

14. Tut. All the difference, in general, will be in pointing, for the decimals must be separated from the whole numbers by a comma, or you cannot tell how many decimals there are. And you must know what the value of the figures is separately, or you cannot tell what it is collectively. Ciphers placed at the left of decimals decrease their value in tenfold ratio. Thus decimals are governed by the same law of increase and decrease that whole numbers are in regard to the operations that are to be performed by them.

To add numbers accompanied by decimals;

Place the numbers so that the commas may stand perpendicularly one under the other. Add as in whole numbers, and point of so many places for decimals, as shall be equal to the greatest number of decimals in any of the numbers added.

Add 126,428, 69,0327 and 12,004 together.

1 2 6, 4 2 8

6 9, 0 3 2 7

1 2, 0 0 4

2 0 7, 4 6 4 7

Add 426,342, 6,05, 92,0001 and 14 together.

Ans. 538,3921.

The rule for the subtraction of whole numbers will apply to the subtraction of decimals, as well as the rule for the addition of whole numbers to the addition of decimals.

Hence, to subtract decimals ;

Write the larger number, and directly under it write the smaller, so that the comma of the latter may be exactly under the comma of the former. Then proceed as in whole numbers, pointing off so many places for decimals as shall be equal to the greatest number of decimals in either of the given numbers.

Subtract 0,246231 from 0,468..

0, 4 6 8

0,2 4 6 2 3 1

0, 2 2 1 7 6 9

Here you will perceive that the number to be subtracted consists of more figures than the other number; but as ciphers, placed at the right of decimals, do not alter their value, so the subtrahend in this case is not larger in value than the minuend, although it contains more figures. The minuend here is 468 thousandths, and the subtrahend is 246231 millionths. But 468 thousandths can be reduced to millionths without altering its value, thus,

15. In the multiplication of decimals, the only varia tion from the manner of multiplication of whole numbers is in pointing the product. The comma in the expression of a whole number with decimals, shows the place where the whole numbers and decimals begin. By removing the comma towards the right, figures, which composed the fractional part, are made to represent whole numbers, and in this manner the given number is increased. On. the contrary, if the comma is removed towards the left, E. 2.

the figures, which before represented whole numbers, are made to represent fractions; and thus the value of the given number is decreased. In the expression, 124,682, if you remove the comma to the right two places, the expression will read 12468,2; the units will have become hundreds, the tenths tens, the hundredths units, the thousandths tenths. But if the point is removed towards the left two places, 1,24682, the expression is decreased in value; the tens will have become tenths, the units hundredths, the tenths thousandths, &c. from which you must be sensible of the importance of keeping the comma in its proper place. We will first consider the multiplication of decimals by whole numbers.

16. Suppose it were required to multiply 0,5 by 2, the product would be 10, which is 10 tenths. 0,5 is 5 tenths, equal to one half, which multiplied by 2 gives a product of 1; consequently the product of 0,5 by 2 is 1, (5. Con. III.) Again, if it were requested to multiply 42,6 by 3, you would begin as in whole numbers, and multiply 6 by 3, the product of which is 18, and 6 being tenths, 18 must be tenths, and as 10 tenths are equal to 1 in the place of units, you must carry 1 to the unit figure, and set

down the excess of tens.

The product of 42,6 by 3 is 127,8; now if you strike out the comma, you reduce this number to tenths, and as the product is in tenths, by dividing it by 10, you reduce it to units; and dividing by 10 is merely cutting off the right hand figure; and whatever the decimal expression is, whether tenths, hundredths, or thousandths, &c. the product will be in the same value, and must be divided by 10, 100, 1000, &c.

From what I have said, you will see plainly that—

To multiply any number, accompanied by decimals by a whole number, multiply the numbers together as though they were all whole numbers, and cut off so many figures from the right of the product as there are decimals in the factor.

Multiply 246,02 by 8.

Multiply 32,1509 by 15.

Multiply 0,840 by 840.

Ans. 1968,16.

Ans. 482,2635.

Ans. 705,6.